Our lives are often dominated by math.

 

Math can be funny.

Puzzles

Torture your friends by giving them these puzzles about arithmetic.

Apples If you have 5 apples and eat all but 3, how many are left? Kids are tempted to say “2,” but the correct answer is 3.

Birds If you have 10 birds in a tree and shoot 1, how many remain in the tree? Kids are tempted to say “9,” but the correct answer is 0.

Corners If you have a 4-sided table and chop off 1 of the corners, how many corners are left on the table? Kids are tempted to say “3,” but the correct answer is 5.

Lily pads In a lake, a patch of lily pads doubles in size every day. It takes 48 days for the patch to cover the lake. How long would it take for the patch to cover half the lake? Kids are tempted to say “24 days,” but the correct answer is 47 days.

Baseball A bat and a ball cost a total of $1.10. The bat costs $1 more than the ball. How much does the ball cost? Kids are tempted to say “10¢,” but the correct answer is 5¢.

Seven How do you make seven an even number? Remove the “s”.

Eggs Carl Sandberg, in his poem Arithmetic, asks this question:

Missing dollar Now that you’ve mastered the easy puzzles, try this harder one:

Ask your friends that question and see how many crazy answers you get!

Here’s the correct answer:

Adding what the girls spent ($27) to what the bellboy got ($2) doesn’t give a meaningful number. But that nonsense total, $29, is close enough to $30 to be intriguing.

Here’s an alternative analysis:

Ten sticks Arrange 10 sticks (or pencils or pens) like this:

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That’s 4 then 3 then 2 then 1.

Here’s the puzzle: move just 1 stick, so you have the reverse order: 1 then 2 then 3 then 4.

Here’s the solution: move the second stick, to fill the gap before the last stick:

ççççççççççççç

Warning: when you set up that trick for your friends, make sure the gap before the last stick is narrow enough so it gets completely filled when you move a stick there.

Ten coins Try this task:

“5 rows of 4 coins” would normally require a total of 20 coins, but if you arrange properly you can solve the puzzle. Hint: the rows must be straight but don’t have to be horizontal or vertical. Ask your friends that puzzle to drive them nuts.

Here’s the solution:

Running A teacher told her 3^rd-grade students to solve this problem:

The teacher thought the answer is 6, but some parents disagree, because “4 times more than” is vague:

Statistics

Courses in statistics can be difficult. That’s why they’re called “sadistics.”

Lies Statisticians give misleading answers.

For example, suppose you’ve paid one person a salary of $1000, another person a salary of $100, another person a salary of $10, and two other people a salary of $1 each. What’s the “typical” salary you paid? If you ask that question to three different statisticians, they’ll give you three different answers!

Which statistician is right? According to the Association for Defending Statisticians (started by my friends), the three statisticians are all right! The most common salary ($1) is called the mode; the middle salary ($10) is called the median; the average salary ($222.40) is called the mean.

But which is the “typical” salary, really? Is it the mode ($1), the median ($10), or the mean ($222.40)? That’s up to you!

If you leave the decision up to the statistician, the statistician’s answer will depend on who hired him.

Which statistician is telling the whole truth? None of them!

A century ago, Benjamin Disraeli, England’s prime minister, summarized the whole situation in one sentence. He said:

Intransitive voting Suppose you have 3 numbers, called A, B, and C. If A is bigger than B, and B is bigger than C, then A is bigger than C. That’s because “bigger” is transitive.

But voting is not transitive. Here’s why.

Suppose A, B, and C are candidates in an election. Suppose most voters prefer A over B and prefer B over C. You can’t conclude most voters prefer A over C.

For example, suppose you have 3 voters.

Here’s the result.

Logic

A course in “logic” is a blend of math and philosophy. It can be lots of fun — and also help you become a lawyer.

Beating your wife There’s the old logic puzzle about how to answer this question:

Regardless of whether you answer that question by saying “yes” or “no,” you’re implying that you did indeed beat your wife in the past.

Interesting number Some numbers are interesting. For example, some people think 128 is interesting because it’s “2 times 2 times 2 times 2 times 2 times 2 times 2.” Here’s a proof that all numbers are interesting:

Surprise test When I took a logic course at Dartmouth College, the professor began by warning me and my classmates:

Then he told the class to analyze that sentence and try to deduce when the surprise test would be.

He pointed out that the test can’t be on the semester’s last day — because if the test didn’t happen before then, the students would be expecting the test when they walk into class on that last day, and it wouldn’t be a surprise anymore. So cross “the semester’s last day” off the list of possibilities.

Then he continued his argument:

Continuing in that fashion, he said, more and more days would be crossed off, until eventually all days would be crossed off the list of possibilities, meaning there couldn’t be a surprise test.

Then he continued:

Mathematicians versus engineers

The typical mathematician finds abstract concepts beautiful, and doesn’t care whether they have any “practical” applications. The typical engineer is exactly the opposite: the engineer cares just about practical applications.

Engineers complain that mathematicians are ivory-tower daydreamers who are divorced from reality. Mathematicians complain that engineers are too worldly and also too stupid to appreciate the higher beauties of the mathematical arts.

To illustrate those differences, mathematicians tell 3 tales.…

Boil water Suppose you’re in a room that has a sink, stove, table, and chair. A kettle is on the table. Problem: boil some water.

An engineer would carry the kettle from the table to the sink, fill the kettle with water, put the kettle onto the stove, and wait for the water to boil. So would a mathematician.

But suppose you change the problem, so the kettle’s on the chair instead of the table. The engineer would carry the kettle from the chair to the sink, fill the kettle with water, put the kettle onto the stove, and wait for the water to boil. But the mathematician would not! Instead, the mathematician would carry the kettle from the chair to the table, yell “now the problem’s been reduced to the previous problem,” and walk away.

Analyze tennis Suppose 1024 people are in a tennis tournament. The players are paired, to form 512 tennis matches; then the winners of those matches are paired against each other, to form 256 play-off matches; then the winners of the play-off matches are paired against each other, to form 128 further play-off matches; etc.; until finally just 2 players remain — the finalists — who play against each other to determine the 1 person who wins the entire tournament. Problem: compute how many matches are played in the entire tournament.

The layman would add 512+256+128+64+32+16+8+4+2+1, to arrive at the correct answer, 1023.

The engineer, too lazy to add all those numbers, would realize that the numbers 512, 256, etc., form a series whose sum can be obtained by a simple, magic formula! Just take the first number (512), double it, and then subtract 1, giving a final result of 1023!

But the true mathematician spurns the formula and searches instead for the problem’s underlying meaning. Suddenly it dawns on him! Since the problem said there are “1024 people” but just 1 final winner, the number of people who must be eliminated is “1024 minus 1,” which is 1023, so there must be 1023 matches!

The mathematician’s calculation (1024-1) is faster than the engineer’s. But best of all, the mathematician’s reasoning applies to any tournament, even if the number of players isn’t a magical number such as 1024. No matter how many people play, just subtract 1 to get the number of matches!

Prime numbers Mathematicians are precise, physicists somewhat less so, chemists even less so. Engineers are even less precise and sometimes less intellectual. To illustrate that view, mathematicians tell the tale of prime numbers.

First, let me explain some math jargon. The counting numbers are 1, 2, 3, etc. A counting number is called composite if you can get it from multiplying a pair of other counting numbers. For example:

A counting number that’s not composite is called prime. For example, 7 is prime because you can’t make 7 from multiplying a pair of other counting numbers. Whether 1 is “prime” depends on how you define “prime,” but for the purpose of this discussion let’s consider 1 to be prime.

Here’s how scientists would try to prove this theorem:

Actually, that theorem is false! All odd numbers are not prime! For example, 9 is an odd number that’s not prime. But although 9 isn’t prime, the physicists, chemists, and engineers would still say the theorem is true.

The physicist would say, slowly and carefully:

The chemist would rush for results and say just this:

The engineer would be the crudest and stupidest of them all. He’d say the following as fast as possible (to meet the next deadline for building his rocket, which will accidentally blow up):

Goldbach’s conjecture

Now most textbooks define “prime number” to be “a non-composite number greater than 1.” In 1742, Christian Goldbach made a guess (called Goldbach’s conjecture), which in modern jargon is:

For example:

Is it really true that every even integer greater than 2 is the sum of 2 primes? Mathematicians still don’t know! It’s the oldest famous unsolved math problem! If you want to become famous, prove Goldbach’s conjecture or find an exception. So far, computers have checked every even integer less than 4´10¹⁸; none of those integers are exceptions.

Why proofs?

Unlike physicists, chemists, engineers, and other scientists, mathematicians don’t trust a bunch of experiments. Mathematicians demand proofs. Here are 2 examples of why.

Euler’s lucky number Try computing x²-x+41, when x is different integers. Is x²-x+41 always prime?

For example:

Every x you try (0, 1, 2, 3, 4, 5, etc., up to 40) gives a result that’s prime. Pretty impressive, huh? So maybe x²-x+41 is always prime?

That’s enough evidence to convince a lazy scientist or engineer, but not a mathematician, because when x is 41, x²-x+41 is not prime: it’s divisible by 41.

That example was discovered by Euler, who called 41 a
lucky number.

Here’s a different way to express the problem:

Regions in a circle Try this experiment. Draw a circle. Put 4 points (dots) on the circle (not in the circle). Draw segments (short lines) connecting each of those 4 points to the other 3 points, like this:

 

 

 

 

Those point-to-point segments are called chords. Now the circle is divided into 8 regions (areas). So 4 points produced 8 regions.

What happens if you have more points, or fewer points? Try it! You get these results:

So it seems that each time you add a point, the number of regions doubles. It seems that if p is number of points, the number of regions is 2^p-1. It seems 6 points should generate 32 regions. But they don’t! If the 6 points are equally spaced, they generate just 30 regions; if the 6 points are not equally spaced, they generate 31 regions; but they never generate 32.

If the points are very irregularly spaced (so no 3 chords meet each other at the same spot inside the circle), here’s the result:

The number of regions is not always 2^p-1; instead, the number of regions can be proved to be always p(p-1)(p²-5p+18)/24+1, which by coincidence just happens to equal 2^p-1 when p is 1, 2, 3, 4, or 5.

What’s the formula when the points are equally spaced, so 6 points create just 30 regions? I haven’t yet met a mathematician smart enough to create that formula. As far as I know, that problem is still unsolved.

Logger

Every few years, authors of math textbooks come out with new editions, to reflect the latest fads. Here’s an example, as reported (and elaborated on) by Reader’s Digest (in February 1996), Recreational & Educational Computing (issue #91),
John Funk (and his daughter), ABC News Radio WTKS 1290 (in Savannah), and others:

Winston Churchill

Winston Churchill (who was England’s prime minister) said:

Terrorist mathematicians

A colleague passed me this e-mail, forwarded anonymously:

Letters

These letters appeared on the Internet.

 

Euclid poems

2 famous poems have been written about Euclid.

In 1922, Edna St. Vincent Millay wrote this poem praising him and titled “Euclid alone has looked on Beauty bare”:

But back in 1914, Vachel Lindsay wrote this simpler poem, titled just “Euclid”:

Like many poets, Vachel Lindsay committed suicide.

 

On the beach

Here’s my very abridged version of Arthur Koestler’s story about an event on the beach:

Arthur Koestler committed suicide.

For the full version of that Pythagoras story, labeled “Pythagoras and the Psychoanalyst,” read Clifton Fadiman’s anthology, “Fantasia Mathematica.”

Pseudo-geometry theorems

Modern math: the number of horn blasts in a traffic jam equals the sum of the squares at the wheels.

Proof that a line is a lazy dog:

Poster

Michael O’Donoghue was one of the founding writers of National Lampoon magazine, which in 1974 (or thereabouts) published a poster he invented. The poster shows a photo of a beautiful, bikinied girl jumping for joy at the beach, with this caption:

After National Lampoon magazine, Michael O’Donoghue started a newer form of comedy: in 1975, he became Saturday Night Live’s first head writer.

Math frustration

Math can be frustrating. Here’s my feeling:

 

 

Because of our culture, certain integers make people emotional.

This old man

Kids sing a counting song whose first verse is:

There are 10 verses, all similar, except for changing “1” to other numbers and changing the word that rhymes:

As a result, kids associate 1 with thumb, 2 with shoe, 7 with heaven, etc. So 7 is considered a heavenly place to be!

English digits

What do these digits have in common:

Answer: they all sound like English words:

Do you speak a non-English language? In your language, which digits sound like words? Please tell me!

Here are other associations from our culture….

0

If somebody calls you a “zero,” that person thinks you’re worthless, a loser. Trump thinks most Democrats are “losers,” so Democrats should retaliate by proudly wearing badges that say “0.”

When a rocket is going to blast off, people say:

So 0 is the blast-off number!

Some people hate negative numbers: they’ll stop at nothing to avoid them.

1

If somebody calls you “number 1,” it means you’re the winner, the best, the boss. Since “one” is pronounced the same as “won,” you can have fun saying this:

But “1” is also the symbol for being single, unmarried, unattached, and lonely. A famous song begins by singing:

That song, titled “One,” was written by Harry Nilsson in 1968 and sung by Three Dog Night in 1969, which you can hear here:

Try this experiment.

Here are examples of that procedure:

That unsolved problem (“Do all positive integers lead to 1?”) is called the 3x+1 problem (because odd numbers are tripled then you must add 1). It’s also called the Collatz conjecture., because it was first mentioned by Lothar Collatz in 1937.

Computers have shown that every positive integer less than 5´2⁶⁰ leads to 1, but what about integers that are even bigger? Unknown!

Some numbers are really horrible. For example, if you start with 27, you need 111 steps to finally get to 1; and you might feel discouraged along the way, since at one point you get up to 9232 before coming back down toward 1.

 

2

If somebody is called the “number 2,” that person is the main assistant to the “number 1.” For example, that person might be the vice-president or secretary or administrative assistant or “COO” (Chief Operating Officer).

But a group of 2 is considered heartwarming & productive. People say “It takes 2 to tango.” It takes 2 people (a male + a female) to create a child. A famous song joyfully recommends “tea for 2” so “we can raise a family.” So 2 is better than 1.

2 is part of you, since your body has 2 eyes, 2 ears, 2 arms, 2 legs, 2 nostrils, and 2 lungs. If you’re female, you get a bonus: 2 breasts. People say “There are 2 sides to every question.” By holding up a mirror, you can make anything become 2.

Computer circuits use the binary number system (based on the number 2) instead of the decimal system (based on the number 10) because most things in life have just 2 states: for example, an electric circuit is either on or off; magnetic poles are either north or south; a card is either whole or has a hole punched through it. If you give a computer a math problem written in the decimal system, the computer translates it into the binary system, does the computation in binary, then converts the answer back to the decimal system so you can understand it.

2 tablespoons make an ounce. 2 cups make a pint. 2 pints make a quart.

2 U.S. Presidents were named “Adams,” 2 were named “Harrison”, 2 were named “Johnson,” 2 were named “Roosevelt,” and 2 were named “Bush.” Some people think Trump is 2 much to handle.

2 is pronounced the same as “to” and “too” but stupidly spelled “two.” A text saying “I like U2” is praising either you or an Irish rock band.

3

3 is the number of the Christian Trinity: Father, Son, and Holy Ghost. There are also 3 feet in a yard, 3 teaspoons in a tablespoon, and 3 colors on a traffic light (green, yellow, and red). Shamrocks, poison ivy, and many other plants have 3 leaves.

If somebody asks you whether a certain statement is true, there are 3 possible answers: “true,” “false,” or “indeterminate” (which means “not enough information to decide yet”).

In many cultures, even numbers (such as 2) are female, but odd numbers (such as 3) are male. That’s because 2 represents a woman + her child, or her 2 breasts, or the other symmetries in her body, whereas a man’s body is similar but adds a prick.

Michelin makes tires but also judges which restaurants to travel to. Michelin awards the best restaurants 3 stars: «««. So in the Michelin guide, a 3-star restaurant means “the best” (because it’s better than 2-star or 1-star or unmentionable). Chefs are proud to be called “3-star.”

There are 3 guys in the standard joke (such as “3 guys walk into a bar. The first guy said… The second guy said… But the third said…”). Example of a standard joke:

Many jokes have 3 examples, such as:

A variant of that joke is:

Joyce Kilmer wrote a poem whose main verses are:

John Atherton wrote a parody, whose main verses are:

Warren Buffett said:

Similarly, many T-shirts say:

Here’s a similar thought (from the Shoe comic on 11/12/2020):

4

Since Independence Day (when the Declaration of Independence from England was signed) is July “the 4^th,” Americans consider 4 to be patriotic, revolutionary, fiery, and full of firecrackers. The Irish consider 4 to be lucky, because finding a 4-leaf clover is a rare joy.

But the Chinese consider 4 to be unlucky, because in Chinese it’s pronounced “sì,” which sounds similar to “sĭ,” which is the Chinese word for death. So the Chinese try to avoid house numbers & phone numbers containing the number 4. In many Chinese apartment buildings & hotels, the floor above 3 is called “3A” or “5” instead of 4.

Japanese is similar: in Chinese-influenced Japanese (called Sino-Japanese), 4 and death are both pronounced “shi,” so 4 is unlucky, so some buildings don’t have a 4^th floor: the floor above 3 is “3A” or “5.”

Fear of 4 is called “tetraphobia.”

A deck of cards has 4 suits (hearts, diamonds, clubs, spades).

4 cups make a quart, and 4 quarts make a gallon, so those measurements are both 4tunate. Life would be even more 4tunate & simpler if we banned the word “pint,” which gets in the way.

A human has 5 fingers on each hand, but the typical
cartoon character (such as Mickey Mouse) has just 4 fingers on each hand (a thumb plus 3 more) and those fingers are all wide. That’s because 5 thin fingers take too long to draw and, Walt Disney said, look too much like a bunch of bananas. In the Simpsons, every character has 4 fingers per hand except God, who appears rarely and is powerful enough to have 5.

In 1941, President Franklin Roosevelt said everybody in the world should get 4 freedoms:

He said the U.S. should protect those freedoms and oppose tyrants who squelch them. Norman Rockwell made 4 paintings to illustrate those 4 freedoms.

Maps The number 4 has tortured mathematicians because of the 4-color problem, which is this problem in topology (similar to geometry):

 

Here are the rules:

This map requires 4 colors, because each of its 4 countries rubs against the other 3:

                                                 1

 

 

                                          2            3

 

                                                 4

Does any map require 5 colors? In 1852, Francis Guthrie noticed that 4 colors seemed to always be enough, but he couldn’t prove it. In 1890, Percy Heawood proved that no map required more than 5 colors, but did any map require 5? In 1976, mathematicians at the University of Illinois finally proved no map requires more than 4 colors; 4 colors are always enough, so the
4-color problem became the 4-color theorem.

How many letters? 4 is the only number that’s as big as its spelling: 4 is spelled “four,” which has 4 letters. By contrast, “five” doesn’t have 5 letters; “six” doesn’t have 6 letters. Just “four” is so nice.

Here’s a more detailed analysis….

In English, “four” is the only number that has correct length: it contains 4 letters.

All integers bigger than 4 contain insufficient letters:

All numbers less than 4 contain excessive letters:

All rational numbers lead to 4:

Four fours In the 1890’s, math nerds began having fun trying to solve this puzzle: compute each digit by combining four fours. So in the list below, make each equation become true by filling in the blanks, using just the symbols for addition (+), subtraction (-), multiplication (´ or * or ·), division (¸ or /), and parentheses.

Here’s a solution (using scientific order of operations):

To continue much beyond 9, you must permit factorials:

You must also permit either square roots (Ö4 = 2) or shifted numbers (44 and .4). Then you get:

To continue past 32, the square root symbol isn’t good enough to bail you out, so you must permit .4 or something wacky, such as “4!!” (which is called “4 double factorial” and means “multiply just the even numbers up to 4,” so it’s “2 times 4,” which is 8).

Paul Dirac won a Nobel Prize for physics in 1933, but he liked math puzzles too. In the 1930’s, he invented a devilish way to construct every integer by using four fours! He cheated: he used logarithms. Here’s his method….

To write n by using four fours, put n square-root signs in front of 4, then write “log_(Ö4)/4 log₄” before all that.

For example, to write 7 by using four fours, put 7 square-root signs in front of 4, so you get ÖÖÖÖÖÖÖ4, then write
“log_(Ö4)/4 log₄” before that, so you get:

That’s because:

But writing numbers without using logarithms is still a challenge. The first difficult number is 99. Another is 113. To win, you must cheat.

5

A business that’s perfect, top-notch, is called “5-star.” The symbol “«««««” is the top rating on many Websites, such as Yelp, Trip Advisor, and Google. Many businesses (such as Home Depot) let customers rate individual products, with the top rating being “«««««.”

In business, “5” is a popular digit to put at the end of a price. For example, instead of charging $4, a business will charge $3.95.

5 is the number for drinking tequila, since the biggest holiday celebrated by Mexican bars in the United States is “Cinco de Mayo,” which means “5^th of May,” which is the date “5/5”.

The alphabet has 5 vowels: a, e, i, o, and u.

To divide by 5, use this trick: divide by 10 (by moving the decimal point to the left), then double the result. For example, to divide 93 by 5, move the decimal point to the left (to get 9.3), then double it, to get the final answer, 18.6.

5 is the first number the French pronounce like an English word. In French:

6

6 is the sexiest number.

Mathematicians call a number “perfect” if the sum of its factors equals the number itself. For example, the factors of 28 (the numbers that go into 28) are 1, 2, 4, 7, and 14; if you add them up, 1+2+4+7+14, you get 28, so 28 is called “perfect.”

Chinese consider 6 to be good because it’s pronounced “liù,” which sounds similar to “liū,”which is the Chinese word for “smooth sailing, nicely slick, calm, peaceful, trouble-free trip.”

Though Chinese consider 6 good and mathematicians consider 6 “perfect,” religious folks consider 6 “devilish,” a failed attempt to be perfect, since 6 is 1 less than 7, which religious folks consider perfect.

In a game of dice, each die has 6 numbers. A Jewish star has 6 points. If a tough dog owner gets angry at you, he “sics” his dog on you.

6 is the only digit that becomes bigger if you turn it upside-down: it becomes 9.

6 is the jolly number! An old song about a British soldier’s pay begins:

Then he sings the math: 6=2+2+2:

After more singing about his joy, the song takes a darker turn: he repeats the verse, except “six” becomes “four,” and “2 pence to take home” becomes “no pence to take home.” Then he repeats the song again, except “six” becomes just “two” (with appropriate math). Then he repeats the song again, except all the numbers become “no pence,” because all his money disappeared, so his jolliness becomes cynical, as he sings:

How sad!

7

Many people consider 7 to be perfect, since:

Literature loves to be perfect, so it loves to have 7:

When playing dice, people yell “7, come, 11!” because they win if 7 or 11 comes on the first roll.

7 is not a multiple or divisor of any other counting number from 1 to 10, so it’s the most unique of those counting numbers.

When you ask Americans “What’s your favorite number?” the answer is more likely to be “7” than any other number. (“3” gets second place, so “37” is also popular.)

In chemistry, a pH of 7 is neutral, like water: it’s neither acid nor base. So in chemistry, 7 is the safest number.

Seven is the easiest number to make even: just erase its “s.”

7 and 0 are the only digits that force you to say two syllables, to emphasize their bizarre importance.

For more details about why 7 is popular, see Davy Derbyshire’s article, published online at The Daily Mail. It’s at —

but you can type just:

Also see Emma Taubenfeld’s article, published online at Reader’s Digest. It’s at:

8

The Chinese consider 8 to be a lucky number because it’s pronounced “bā,” which sounds similar to “fā,” which is the Chinese word for wealth. Even luckier is 88. Even luckier is 888.

There are 8 full planets in the solar system, since Pluto was downgraded to be called just a “dwarf planet.” (The other main dwarf planet is Eris.)

8 ounces make a cup. 8 bits make a byte.

The Jewish holiday of Hanukah lasts 8 nights.

According to the Beatles, a week has 8 days, not 7. Their song “8 Days a Week” says their love is bigger than 7:

In normal English, 8 is pronounced like “ate,” so 8 is the most popular number to stick at the end of a text-message word:

That also creates this math horror:

But I once met a Texas girl who pronounced 8 as “ott.”

9

Optimists say 9 is the age when can begin calling yourself a “tween.” (Pessimists say you must wait until you’re 10.)

Hey, kids! Having trouble memorizing your multiplication tables? To multiply 9 by any digit (from 1 to 9), use this handy trick:

Here’s that rule, expressed & defended algebraically:

Hey, guys! If you ask a German girl for a date, you’ll probably hear her say “9.” That’s because the German word for “no” is “nein,” which is pronounced the same as the English word “nine.”

A cat has 9 lives. The Supreme Court has 9 justices.

In Chinese, 9 (pronounced jiŭ) a means “a long time.” To say “forever,” say 99 (jiŭjiŭ). For example:

10

Our whole number system is based on 10, because we have 10 fingers. We also have 10 toes.

A woman’s body has 10 major holes: 2 ears, 2 nostrils, 1 mouth, 2 nipples, 1 urethra, 1 vagina, and 1 asshole.

If you’re wonderful, you’re called a “perfect 10.” If you’re blasting into outer space, the countdown can begin at 10.

10 is the only number that becomes a word when spelled backwards: 10 becomes “net.”

11

To multiply 11 by a digit, just write the digit twice. For example, 11 times 7 is 77.

To multiply 11 by a two-digit number, write the two digits but put their sum between them. For example, to multiply 11 by 53, write the 5 and the 3 but also write their sum (8) between them, so you get 583. Exception:

To compliment someone, say:

World War 1 ended in 1918 on the date of 11/11 at 11 o’clock. That’s when hostilities on the Western Front officially ended. That date became Armistice Day, whose name the U.S. later changed to Veterans Day, a holiday every year on 11/11.

Since “11/11” looks like four singles, China celebrates 11/11 as Singles Day, to celebrate singles who were smart enough to not get married yet and can therefore still shop around and buy presents for themselves. That holiday was invented in Nanjing University but popularized by Alibaba’s Websites (Tmall & Taobao) as an excuse to sell goods at a discount, earlier than America’s Black Friday sale.

12

12 months make a year. 12 inches make a foot. The typical jury has 12 jurors. 12 is how high you can get when you roll a pair of dice. 12 pennies made a shilling (until the British government changed that “12” to “5,” and Australian & New Zealand changed it to “10”). 12 anything make a dozen.

You see 12 numbers on a clock. Porn movies try to show 12 inches on a cock. (But the average guy’s erect cock is just 5 inches, according to surveys.)

Jesus had 12 apostles, who ate with him at the Last Supper (though Judas turned out to be a jerk, leaving just 11 apostles who were good). At Christmas, you can sing a song about “The 12 Days of Christmas.”

In Chinese, 12 can be pronounced “yāo èr.” When you mumble that, it resembles “yào ài,” which means “want love,” so 12 is the secret Chinese code for “want love.” Many Chinese weddings took place on December 12, 2012, because the Chinese write that date as 2012.12.12, which means “20 times want love, want love, want love!”

Math would be much simpler if we had 12 fingers. When I was a high-school kid, I did calculations that indicated 12 is the best number to use as a base for a number system. 12 is much better than 10! Specifically:

When I entered Dartmouth College as a freshman, I showed my research to the math department’s chairman (John Kemeny). After pausing just a few seconds, he scribbled on the blackboard a formal proof that my conclusion was correct: of all numbers, 12 has the highest factor-to-square-root ratio. I was blown away by his brilliance. Alas, I don’t remember what he scribbled.

13

13 is considered an unlucky number now because 13 people were sitting at the Last Supper (Jesus and the 12 apostles, 1 of whom decided to kill Jesus soon). But actually, 13 was considered an unlucky number before the apostles: in Norse mythology, 12 gods sat down to a feast that was interrupted by a gate-crasher and, in the ensuing scuffle, the most beloved god was killed. Historians view Christ’s “The Last Supper” as just copying the Norse legend. (Gee, I thought everything in the Bible was real and original. The apostles were plagiarists? How upsetting!)

If you’re afraid of the number 13, you have triskaidekaphobia (which comes from the Greek words for “three-and-ten fear”). To avoid scaring travelers, many hotels skip the 13^th floor: the floor after the 12^th is called the 14^th or “floor 12½.”

The United States began with 13 states. Afterwards, more states joined and got luckier, until Trump made us feel unlucky again.

13 is the first number whose name ends in “teen.” So to be a “teenager,” you traditionally must be at least 13. If you’re 10, 11, or 12, you’re called a “pre-teen” or “tween.” (Exception: some psychologists call a 13-year-old a “tween” instead of a “teenager,” since the typical 13-year-old hasn’t reached puberty yet.)

13 is the age when a Jewish boy is considered to be an “adult,” old enough to be responsible for his actions, so he undergoes a ceremony & celebration on his 13^th birthday, called “Bar Mitzvah,” which is Hebrew and means “Son of the Commandments.”

Since girls mature faster, girls can be “Bat Mitzvah” (daughter of the Commandments) a year earlier, when they’re 12; but that doesn’t matter much, since the Jewish religion doesn’t take women very seriously (hah!), so 13 is still considered generally the magic age to become a Jewish adult. A boy who dates such a girl can be called a “Bat man.”

When playing card games, there are 13 cards in each suit: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.

In a perfect calendar, each of the 4 seasons would include 13 weeks, making a total of 52 weeks in the year.

13 is called a baker’s dozen. Why? Theories are at:

14

The French consider 14 to be patriotic & revolutionary, because Bastille Day (when the French stormed the Bastille) is July “the 14^th”.

Chemistry students think 14 is the highest pH you can get (though you can get even higher if you try hard).

16

When chatting about weight, 16 ounces make a pound; but when chatting about volume, 16 ounces make a pint.

16 is the age when girls are considered to be sweetly sexy and somewhat adult. In many states, 16 is the age when a girl is allowed to get married. U.S. federal law says 16 is the age when a girl is mature enough to say “yes” to sex without having the male get jailed for “statutory rape” (having sex with a minor), so 16 is called the “age of consent.”

When a U.S. girl turns 16, people often throw a
“sweet-16 party.” “16 Candles” is a 1958 song about that birthday party, which you can hear here:

In Hispanic countries, girls mature faster (hah!), so they throw a “sweet-15 party” instead (called “quinceañera” in Spanish, “festa de debutantes” in Brazilian Portuguese).

“16 tons” is the name of a song (written by Merle Travis, sung by Tennessee Ernie Ford & Johnny Cash) about a coal miner who’s overworked & underpaid. The chorus is:

17

At age 17, a girl can blossom further. In 1962, the Beatles wrote a song about that, called “I Saw Her Standing There,” whose main thoughts are:

Aside from that song, 17 is a relatively boring number, so 17 is the number most commonly used in this joke:

18

The constitution’s 26^th Amendment says you can vote in federal & state elections if you’re at least 18 years old. (Some states & towns are kinder: they let kids vote even if they’re just 17 or 16.)

Federal law says you must be at least 18 years old to buy cigarettes (but younger kids can smoke cigarettes if they receive them as a gift).

Many states make the buying age even higher: 21. Alaska has a compromise: 19.

Some states also ban kids under 18 from smoking in public.

21

21 is how old you must be to buy alcohol. So if you want to kill yourself legally, by getting drunk, you must be 21.

22

“Catch-22” is Joseph Heller’s novel about the craziness of World War 2. The title refers to military rule 22, which seems simple but has a catch.

The rule says you can bomb just if you’re sane. But here’s the catch: to want to bomb, you must be a bit crazy. If you’re not feeling mentally well today and therefore reasonably ask to avoid bombing today, that request proves you’re reasonable, so you’re not insane, so you must bomb.

The term “catch-22” is now used to describe any contradictory rule. For example, Mary Murphy said:

When writing the novel, Heller wanted to call the rule “catch 18,” because 18 is the Jewish code number for “alive.” But his publisher rejected 18, because a novel by Leon Uris already had “18” in its title. 11 and 17 were rejected because movies already had those numbers in their titles (“Ocean’s 11” and “Stalag 17”). 14 was rejected because it wasn’t funny enough. So Heller eventually settled on 22.

Details about “Catch-22” are at:

23

23 is famous for being the number in the birthday surprise (which is also nicknamed the “birthday paradox”).

In that sentence:

That statement is true for the number 23 but not for the number 22. You need at least 23 people to make the probability be more than half.

If you have a lot more than 23 people in the room, the probability of matching birthdays is a lot more than half. For example, if you have 367 people in the room, the probability of matching birthdays is 100%, since there are just 366 possible days in the year (including February 29), so 367 people would have a duplicate somewhere.

So next time you’re in a room holding at least 23 people, try this experiment: have each person shout a birthday. The probability is greater than half that somewhere in the room, 2 people’s birthdays will match each other. If the room holds a lot more people, the probability of success goes way up. If the people are all single, maybe they’ll get excited, and the matching people will marry.

24

24 hours make a day.

24 is number of the confusing President.

Who was the 24^th?

Confusing, eh?

24 is an important number to me because May 24^th is
my birthday. When is yours?

25

December 25 is Christmas. Priests say yay for Jesus, and kids say yay for presents, so 25 is a yay day.

According to Men’s Health magazine, the typical woman wants to be 25 years old. More precisely:

Yeah, that’s sexist. So is America!

26

The alphabet has 26 letters, so 26 is the most literate number — if you speak modern English.

Some languages use fewer letters:

Some languages use more letters:

I often feel overloaded. A day doesn’t contain enough hours to accomplish everything I’m supposed to. Sadly, I tell myself:

Most people say “There are only 24 hours in a day,” but 26 sounds better and uses the fact that I’d get up 2 hours later each day if there were no clock to yell at me.

28

On a lunar calendar, each month has exactly 4 weeks, so 28 days. There are also 28 days in a normal menstrual cycle.

On a normal calendar (solar), February normally has 28 days. 29 days make a leap month. 30 make a banker’s month (since bankers usually say “you have 30 days to pay”). 31 make the longest month.

Like 6, 28 is perfect. So God made women be perfect!

29

February usually has 28 days but during leap years has 29, so 29 feels like a bonus, a leap up!

30

30 is the smallest number that’s the product of 3 primes: it’s “2 times 3 times 5.”

31

October 31 is Halloween. Scary!

32

32 is the icy number, because 32 degrees Fahrenheit is when water freezes.

In the 1950’s, a team of famous comedians (Carl Reiner, Mel Brooks, Neil Simon, and others) wrote skits for the TV show “Your Show of Shows.” They tried to decide which number is the funniest. They decided the funniest number to say is “32,” so they had a performer (Imogene Coca) often say “thiiirrrrty-twooo.” The studio audience laughed every time.

36

A yard is 36 inches. A full-size classical piano has 36 black keys.

Since 6 is the first perfect number, and 36 is 6 squared, 36 is the first squared perfection.

During the 1950’s, the ideal shape for a woman’s body — the shape considered the sexiest —consisted of a big chest (& breasts), slim waist, and wide hips (& ass). That’s called the hourglass figure. The extreme example was Marilyn Monroe, who at the height of her fame measured 36-22-36. Other examples are Katy Perry (36-25-36) and Hedy Lamarr (36-25-36). So 36 means “delightfully big boobs & buns.”

37

37 degrees Celsius is considered to be the “normal” temperature of a human. But that’s just an approximation. 36 degrees Celsius is a bit cold, 38 degrees is a fever, but anything more than 36 and less than 38 is okay.

If you convert “37 degrees Celsius” to Fahrenheit, you get exactly “98.6 degrees Fahrenheit,” so Americans are taught to strive for 98.6 degrees Fahrenheit, but 98.6 degrees Fahrenheit is no better than 98.5 or 98.7.

38

In the United States, a roulette wheel (used in gambling) has 38 pockets (where the ball could land). They’re numbered 1 through 36, plus 0, plus 00. So if you gamble on a number, your chance of winning is just 1 out of 38.

In Europe, a roulette wheel lacks 00, so your chance of winning is slightly better: 1 out of 37. Moral: if you want to gamble, go to Europe.

Chinese In Chinese slang, calling a person (male or female) “38” means “acting like an obnoxious woman.”

That’s partly because the Chinese are cynical about International Women’s Day, which is March 8, which is 3/8. It’s also because “three eight” is pronounced “sān bā,” which sounds similar to the Chinese word for “stupid” in some dialects.

The Chinese say “38” to refer to all kinds of obnoxious women: “gossipy” or “bitchy” or “a mean mama” or “too concerned about fashion” or “dumb like an American blonde” or “a man who acts too feminine.”

39

Traditionally, when you turn 40, you’re called “middle aged,” so 39 is the last year you can claim you’re “young.”

Jack Benny was an elderly comedian who, whenever people asked how old he was, would jokingly answer “39.” So 39 is the Jack Benny age.

For actresses, 39 is typically the last fuckable year (the last year you can be in a movie that shows you fucking), unless you’re lucky enough to look young when you’re even older, according to this cynical video, called “Last Fuckable Day”:

40

40 is the beginning of being “middle-aged.” Is being middle-aged good or bad?

An old expression is “life begins at 40.” That thought began in the 1800’s, was expressed more clearly in 1917, became a book title in 1932, and became a song in 1937. The basic idea is that, especially for women, the drudgery of raising kids can end when a woman turns 40; then she can relax! Details about that history are at:

40 is the smallest vaguely big number: the Bible and other ancient books often say “40” when they really mean just “many.” For example, the Bible says Noah experienced rain for “40 days and 40 nights,” but it means just “many days and many nights,” not exactly 40. More examples of “40” meaning just “many” are at:

A standard workweek has 40 hours. That’s considered “full time.” If you work more than 40 hours, your boss must pay you overtime.

The path around a Monopoly board has 40 spots to land on. (The most famous is Boardwalk; the most infamous is “Go To Jail.”)

40 is the most misspelled number. The correct modern spelling is “forty,” not “fourty” (which was an older spelling that started getting phased out about 200 years ago, in 1821). Dictionaries still have “four,” “fourteen,” and “four hundred” but not “fourty.”

A child says “40” when seeing “T T T T.” A proper British lady says “40” when you’ve invited to her house “for tea.”

The Wall Street Journal wrote an article saying that when the typical “expert” warns us a calamity might happen, he says there’s a “40% chance” it will happen. That way, if the calamity does happen, the expert brags he warned us; and if the calamity does not happen, the expert brags he said the chance was just 40%, which is less than half. So either way, the expert can brag, even if the expert is really an idiot.

41

Health departments require restaurant refrigerators to cool below 41 degrees Fahrenheit.

Why 41 instead of 40 or 42? Because “5 degrees Celsius” is a reasonably scientific rule (more reasonable than 0 degrees or 10 degrees), and “5 degrees Celsius” happens to be exactly 41 degrees Fahrenheit. So the “41 rule” is rough science that pretends to be precise.

42

The Japanese consider 42 unlucky, because in Japanese “four two” is pronounced “shi ni,” which sounds like the Japanese word for death.

New York City’s main numbered street is 42^nd Street. It’s considered the center of New York’s excitement. It runs through Times Square (along with Broadway and 7^th Avenue). “42^nd Street” is the name of an exciting song, a dance number, a musical, and a movie; in them, the main line is:

In “The Hitchhiker’s Guide to the Galaxy” (a novel by Douglas Adams), a supercomputer called Deep Thought spends 7.5 million years computing the “answer to the ultimate question of life, the universe, and everything.” Its final answer is:

Truth imitates art! Mathematicians actually tried to make computers solve a difficult math problem about 42, and computers spent many days working on it. Here’s the problem:

During the beginning of President Trump’s reelection campaign, he sent emails demanding a $42 donation from each of his supporters. Is that because he considers himself a supercomputer, the “answer to the ultimate question of life, the universe, and everything”? His competitor, Joe Biden, asked for just $3. Is that because Democrats are poorer?

45

During the 1950’s and 1960’s, the most popular kind of
phonograph record was 45rpm: it did 45 revolutions per minute. Folks would say:

Its diameter was 7-inch.

Trump was the 45^th president, so people called him “45.” Toward the end of his reelection campaign, he requested $45 from each donor.

48

For many years (starting in 1912), the United States included 48 states. But in 1959, Alaska & Hawaii joined us and complicated who we are. We still say that the United States contains 48 “contiguous states” (or “conterminous states” or “lower states”), plus those 2 far-off weirdos.

50

Now the United States has 50 states. So does my brain: its 50 states vary from wonderful to horrible.

In the Danish language, 50 is the first crazy number. Most other numbers are reasonable. Here’s how to say numbers in traditional Danish:

For 50, 60, 70, 80, and 90, modern Danes are too lazy to write or say the ending (“inds-tyve”) so they have just:

Danish kids memorize just those short forms, which are easy; most Danes don’t know the long forms they came from. Danes have trouble remembering how to spell “halvtreds,” “tres,” and “halvfjerds,” because in Danish the “d” in “ds” is silent: “ds” is pronounced as just “s”. To communicate with other Scandinavians (Norwegians and Swedes), Danish bankers write simpler words on checks, using base-10 instead of base-20:

More details are at:

52

A deck contains 52 cards (4 suits, each containing 13 cards). A year contains 52 full weeks (plus 1 or 2 days, depending on whether it’s a leap year). A full-size classical piano contains 52 white keys.

57

Heinz makes ketchup, pickles, and more. In 1896, Heinz began bragging that it made 57 varieties of pickles. Actually, it made more than 60 varieties of things, but Henry Heinz thought 57 sounded more upbeat, because 7 was a lucky number.

On a Heinz ketchup bottle, the ketchup will come out fastest if you tap the “57” on the label.

57 doesn’t seem to be divisible by anything, since its last digit is 7 (not 0 or 2 or 4 or 5 or 6 or 8). So 57 seems to be prime. But here’s a surprise: 57 is not prime: it’s “3 times 19.” Mathematicians jokingly call 57 the Grothendieck prime, because the mathematician Alexander Grothendieck is rumored to have mistakenly said 57 is prime, although others claim it was just Herman Weyl who said 57 is prime.

60

“Sixty” is the classic answer to this famous riddle —

or this shorter version:

Another correct answer is “sixth” but not “eight.”

60 seconds make a minute.

60 minutes make an hour. “60 minutes” is also the name of a CBS TV news show, so 60 minutes means investigation.

When you turn 60, you can brag you’re a sexagenarian, so can sound sexy, and keep bragging about that until you turn 70, when you become a shitty septic: a septuagenarian.

65

In American culture, 65 is the age when you’re supposed to retire.

When I was a kid taking tests, I needed to get a score of at least 65 to pass.

69

69 is the symbol for oral sex, since it’s the same shape as 2 people lying side-by-side, licking each other’s naughty parts.

70

70 is like 69 but means a sexual threesome, since it’s “69 plus 1 more.”

70 is the first big number the French language lost. In school, the French are taught no word for “70” or “80” or “90”.

But French speakers got “liberated” in Switzerland & Belgium (and the Belgian Congo and a few villages in France & Canada). They got disgusted by classical French’s “40 times 20, plus 10,” so they invented a new word for 90: “nonante.”

75

In Japan, you’re not called “elderly” until you turn 75.
If you’re between 65 and 74, you’re called “pre-old.”

76

76 is called the trombone number, because of the song
“76 Trombones” in the musical movie “The Music Man.”

86

86 means “out.” Examples:

Why does 86 mean “out”? Here are many reasons:

Details about the history of 86 are at:

88

A full-size classical piano has 88 keys. So to a pianist, 88 means full, complete, wonderful, not crap.

In Chinese, 88 is a lucky number, because 8 is a lucky number, as I explained in my discussion of “8”. So in Chinese, a classical piano is a lucky instrument! The Chinese are trying to make Father’s Day be August 8, because August 8 is 8/8, which is pronounced bābā, sounding similar to the Chinese word for father (bàbà).

89

On June 4, 1989, China experienced the Tiananmen Square massacre, where the Chinese government massacred protesters. Now any mention of that date is censored on China’s Internet. “1989” is censored, and so is just “89.” In China, if you search for “89,” your results are censored, especially each year near June 4, to prevent further protests.

90

Where I live (the Northeast United States), a day is considered “hot” if its temperature is at least 90 degrees Fahrenheit, and a “heat wave” is defined to be 3 consecutive days of at least 90 degrees.

Other parts of the world define a “hot” day differently: “much hotter than what’s normal there.” Details are at:

99

When schoolkids get stuck on a long bus ride, they like to kill time (and annoy the bus driver) by singing:

The song continues until it reaches:

Then the song repeats. More details about the song are at:

100

On most school tests, a score of 100 means you’re perfect.

101

In college, the first course in a topic is numbered 101. For example, the first course in psychology is called “Psychology 101.” So 101 is the most elementary course, for beginners, such as freshmen. 101 means “elementary.”

101 is called the dalmatian number, because of the book & movies “101 Dalmatians.”

110

Instead of saying “try really hard” or “try your best” or “give it your all,” people say “Give it your 110%” or simply “Give it 110.”

144

144 is 12 dozen. It’s also called a gross.

If somebody disgusts you, you can say:

If you’re 72 years old, you can say:

196

A number (or word) is a palindrome if it’s the same forward as backwards. For example, 121 is a palindrome. So is 646.

A number is palindromable if, when you add the number to its reverse, you get a palindrome eventually. Examples:

Is every counting number palindromable? Nobody knows! If you answer that question, you’ll become famous! Mathematicians have checked many numbers. All the numbers from 1 to 195 have been proved to be palindromable, but
nobody knows whether 196 is palindromable.

Some numbers require many steps to get to a palindrome. For example, the number 89 requires 24 steps to get to a palindrome. Mathematicians found huge numbers that require 289 steps to get to a palindrome.

A number that’s not palindromable is called a Lychrel number. The unsolved problem is: do Lychrel numbers exist? Details are at:

211

In the United States, dialing 211 gets you free help from United Way, the nonprofit that helps people in distress.

212

212 is the boiling number, because 212 degrees Fahrenheit is when water boils.

212 is the traditional area code for the main part of Manhattan, which is the most important place in the United States, according to Manhattanites.

360

360 seconds make an hour. A circle has 360 degrees.

365

On the usual calendar (solar), there are 365 days in the year, unless it’s a leap year, which has 366.

411

In the U.S., 411 is the phone number to call to get directory assistance, to speak to a human who’ll help you look up a phone number. So 411 means “give me information, please.” 411.com and 411.info are Websites to look up phone numbers.

Those services and Websites all charge money for the info.

420

420 is the secret slang number for marijuana. That tradition began in 1971, when students at California’s San Rafael High School met after school, at 4:20PM, to hunt for a field of marijuana. They didn’t find the field, but they had fun and starting saying “420” to mean “get together to smoke marijuana.” Details are at:

In honor of 420, April 20^th (4/20) is celebrated every year as Weed Day. It’s also Hitler’s birthday.

451

“Fahrenheit 451” is a novel about book-burning, because the author (Ray Bradbury) thought 451 degrees Fahrenheit is the temperature at which book paper burns. Actually, book paper burns at a temperature somewhere between 424 & 475, depending on what the paper is made of.

500

How many grams are in a pound? The usual answer is: a pound is defined to be 453.59237 grams. That’s the official international definition, used in the United States, England, and most other countries, since July 1, 1959. It’s called the avoirdupois pound.

But many people in Germany define a “pound” (written “Pfund”) more simply: exactly 500 grams. That’s called the metric pound. It’s used in Germany, Austria, Switzerland, and even in Denmark. It’s slightly heavier than an avoirdupois pound.

That confuses people. I remember a German woman who got on a scale and wondered why the scale reported she gained a lot of weight. The answer: the scale was using avoirdupois pounds, not the metric pounds she grew up with.

The same happens in China: the Chinese use the metric pound (which they called “jīn”), 500 grams. Half a jīn is 250 grams. In Chinese, if you want to call somebody stupid, you call him a “250,” which means “half a jīn,” which means he acts like he has just half a brain. If you’re too lazy to say “250,” say just “2” (which is pronounced “èr”). So “èr” means “acts stupid” (or “happily silly” or “trying to act funny but just being silly”).

511

In many parts of the United States, dialing 511 gives you info about traffic conditions.

520

In Chinese, 520 is pronounced “wŭ èr líng.” When you mumble that, it resembles the popular phrase “wŏ ài nĭ,” which means “I love you,” so 520 is the secret Chinese code for “I love you.” Every year on 5/20 (which is May 20), the Chinese celebrate an “I love you” day (similar to Valentine’s Day). That’s when you’re encouraged to express your love to the person you secretly or publicly admire: go out with that person, or propose marriage! It’s a popular day to give flowers, chocolates, and beyond, while mumbling “wŭ èr líng” or “wŏ ài nĭ.”

555

In Thailand, 555 is popular, because 5 is pronounced “ha,” so 555 is pronounced “ha-ha-ha.” In a Thai text message, “555” means “ha-ha-ha, funny, laughing out loud.”

800

On the math part of the Scholastic Aptitude Test (SAT), the highest score you can get is 800, so that’s considered the “perfect” score and means you’re brilliant.

When I was a kid, I got that score myself, and so did all my friends. You get that score even if a few of your answers are wrong, because the 800 means just “you’re at least 3 standard deviations better than the norm.”

811

Before you dig a hole in your yard, the U.S. government requires you to phone 811, to make sure you don’t accidentally hit a cable or pipe hidden underground.

Exception: if you’re a dog burying a bone, you don’t have to phone 811.

911

In the U.S., 911 is the phone number to call in case of emergency (to get police or the fire department or an ambulance). 9/11 is also the date of the terrorist attack on the U.S. So 911 means danger, a desperate cry for help.

For milder help, dial 211 instead. That works in many parts of the United States and gets you a free phone counselor to help with homelessness, recovery, or any other kind of personal crisis. The counselor will refer you to a local organization to help you. So 211 means caring. Details are at www.211.org.

988

Wanna kill yourself? I have good news! In the U.S., starting in July 2022, the phone number for the national suicide hotline will be 988. Just dial 988 to avoid suicide. (Before then, you must dial 800-273-TALK.)

996

In China, young people complain about “996”: how they’re expected to work from 9AM to 9PM, 6 days a week. They’d rather be free to relax!

1001

1001, like 40, is a vaguely big number, especially in Arabic, such as “1001 nights,” which means just “many nights,” not exactly 1001. Modern English books are sometimes titled “1001 Uses For…,” meaning just “many uses for….”

1089

1089 is the magic number, because math magicians use it to create this trick.…

Write down any three-digit number “whose first digit differs from the last digit by more than 1.” For example:

Take your three-digit number, and write it backwards. For example, if you picked 852, you have on your paper:

You have two numbers on your paper. One is smaller than the other. Subtract the small one from the big one:

Take your answer, and write it backward:

Add the last two numbers you wrote:

Notice the final answer is 1089.

1089 is the final answer, no matter what three-digit number you started with (if the first and last digits differ by more than 1).

Here’s another example:

Here’s another example:

Yes, you always get 1089!

Proof To prove you always get 1089, use algebra: make letters represent the digits, like this.…

To subtract the bottom (C B A) from the top (A B C), the top must be bigger. So in the hundreds column, A must be bigger than C. Since A is bigger than C, you can’t subtract A from C in the ones column, so you must borrow from the B in the tens column, to produce this:

Now you can subtract A from C+10:

In the tens column, you can’t subtract B from B-1, so you must borrow from the A in the hundreds column, to produce this:

Complete the calculation:

Don’t burn your arm I call 1089 the “don’t burn your arm” number, because of this trick suggested by Irving Adler in The Magic House of Numbers:

Variants That procedure (reverse then subtract, reverse then add) gives 1089 if you begin with an appropriate 3-digit number. If you begin with a 2-digit number instead, you get 99.

If you begin with a 4-digit number instead, you get 10989 or 10890 or 9999, depending on which of the 4 digits are the biggest. If you begin with a 5-digit number, you get 109989 or 109890 or 99099. Notice that the answers for
4-digit and 5-digit numbers — 10989, 10890, 9999, 109989, 109890, and 99099 — are all formed from 99 and 1089.

1314

In Chinese, 1314 (pronounced yī sān yī sì) sounds like yī shēng yí shì, which means “1 life 1 whole life,” which means “my whole life.”

1492

Christopher Columbus discovered “America” in 1492 on October 12, which Americans celebrates as “Columbus Day.”

On that date, Chris landed just on an island in the Bahamas. He thought he’d reached Japan. He didn’t reach American mainland until later trips, when he reached the coasts of Central America & Venezuela. He never got to North America.

He wasn’t the first to visit Americas. 491 years earlier, in the year 1001, Leif Erikson beat him.

But the Native Americans before Chris & Leif beat them all.

1760

A mile contains 1760 yards.

1776

America’s “Declaration of Independence” from “evil England” was signed on July 4, 1776.

1922

1922 is the only number having a joyous song. The song’s main line, sung joyously, is:

The song is about life in 1922 and how wonderfully modern our lives became then, compared to previous years.

The song is in a musical comedy called “Thoroughly Modern Millie,” which takes place in 1922 and became a movie starring Julie Andrews. Here’s a sample verse, where she sings about the joys of living today (in 1922):

You can hear the song (and see Julie Andrews enjoying 1922’s culture & fashion) at:

Another seductive year is 1928, which appeared in a song called “Let’s Misbehave,” written by Cole Porter in 1927. The song includes this thought:

2000

In the United States, 2000 pounds is called a “ton” (or a “short ton”). In England, a ton is 2240 pounds instead (and called a “long ton”). In most other countries, a “tonne” (also called a “metric ton”) is 1000 kilograms instead, which is about 2205 pounds.

Those are weights. But the word “ton” is also used to describe the volume of a ship or freight car — and the explosive power of a bomb (such as a “megaton bomb”).

In Chinese restaurants, tons are edible and called “wontons.”

5280

A mile contains 5280 feet.

6174

To understand what’s unusual about the number 6174, you must learn how to find a number’s heart. Here’s how:

Try this experiment. Write a 4-digit number that’s not a multiple of 1111. Find your number’s heart. (If the heart seems to be less than 4 digits, make it 4 digits by putting zeros in front.) Then take that heart and find its heart. Then find the heart of that. Then find the heart of that.

For example, starting with 1925, you get:

No matter what number you start with, you’ll reach 6174 within 7 steps. So 6174 is where all hearts lead! It’s the heartiest 4-digit number!

6174 is called Kaprekar’s constant, because it was discovered by D.R. Kaprekar (an Indian mathematician) in 1946.

If you start with a 3-digit number (instead of a 4-digit number), you eventually get to 495 (instead of 6174).

By the way, each heart is a multiple of 9. (That’s because the heart is created by subtracting 2 numbers that have the same digits, so those 2 numbers have the same remainders when divided by 9.)

3,628,800

If you multiply together all the integers from 1 to 10 (1 times 2 times 3 times 4 times 5 times 6 times 7 times 8 times 9 times 10), you get 3,628,800. That’s called 10 factorial. Mathematicians write it with an exclamation point, like this:

By coincidence, it’s how many seconds are in 6 weeks.

Billion

What’s a billion? In the United States, a billion has always been a thousand millions. It’s 1 followed by 9 zeros. It’s 10⁹. Here’s the chart:

 

So the smallest counting number (positive integer) that includes sex is 1,000,000,000,000,000,000,000. Guys, beware: if a girl promises to include sex, she might just write that number and tell you to get lost.

In Great Britain, a billion used to be defined differently: a million millions. Here was the British chart:

So the British had to wait much longer to get sex! But in 1974 the British government officially decided to switch to the American definitions, to reduce international confusion, so even the British now define a billion to be 10⁹. So do the Irish, Australians, and New Zealanders.

Unfortunately, most European countries (and Russia) still use the old British definitions and define a “billion” to be 10¹².

Canada is a bilingual mess: Canadians who speak English define a billion to be 10⁹ (like the Americans), but Canadians who speak French define a billion to be 10¹² (like the French).

Details are at:

The Chinese use a totally different naming system, based on a name for 10,000 instead of 1,000:

Though 100,000,000 is officially called “yì,” some folks say “wàn wàn” instead (which means 10,000´10,000), because yì (which means 100,000,000) sounds too much like yī (which means 1).

 

4 irrational numbers have become famous. Here they are.

Square root of 2

(which is about 1.4)

What number, multiplied by itself, is 2? The answer is called the “square root of 2.”

The answer is about 1.4, but not exactly (since 1.4 times itself is just 1.96, which is a hair less than 2). The square root of 2 is also about 99/70, but not exactly (since 99/70 times itself is 9801/4900, which is a teeny-weeny hair more than 9800/4900, which is 2).

The simplest way to draw the square root of 2, exactly, is to draw a square whose sides each have length 1. The length of the square’s diagonal will be exactly the square root of 2. That can be proved by the Pythagorean theorem.

But if you try to measure that diagonal, by using a ruler, you’ll see that the diagonal’s length is not any simple decimal or fraction.

The square root of 2 is not exactly any rational fraction (integer divided by an integer), so the square root of 2 is called irrational. Yes, the square root of 2 is irrational, just like the personalities of most mathematicians.

To prove the square root of 2 is irrational, mathematicians show the opposite leads to a contradiction, an absurdity (a technique called reductio ad absurdum, which means “reduction to the absurd”):

The square root of 2 can be computed in many ways. Here are 3 cute methods.

Multiply-forever method The square root of 2 is:

That means: multiply (1+1/1) by (1-1/3), then by (1+1/5), then by (1-1/7), etc., forever, or until you get tired. The longer you continue before you get tired, the closer you’ll be to the exact square root of 2.

Average-forever method Guess what the square root of 2 is. (Any guess bigger than 0 will work. For example, guess 1.5.) Call your guess “G.” Then get a better guess by averaging G with 2/G, so the better guess is:

Do that method repeatedly, so you get better & better guesses, closer & closer to the exact square root of 2.

Continued-fraction method Guess what “the square root of 2, minus 1” is; but make your first guess be .5. Call your guess “G.” Then get this better guess:

Do that method repeatedly, so you get better & better guesses, closer & closer to the exact “square root of 2, minus 1.”

Mathematicians write that method as a “continued fraction”:

Bragging Using those 3 methods (and others that are similar), mathematicians have computed the square root of 2 to many decimal places.

Here’s how they bragged:

I’m sorry, but my book isn’t big enough to show you Ron Watkins’ 10 trillion decimal places. But here are the first 66 digits of the square root of 2:

Pi

(which is about 3.14)

In the Greek alphabet, the letter “p” is written as π. Americans lacking Greek typewriters write π as “pi” and pronounce it the same as the apple “pie” you eat, but Greeks pronounce π the same as the letter “p” and the American word “pee.” To be correct, American mathematicians ought to pronounce “π” as “pee”; but they’re scared of acting pissy and getting pissed on, so they say “pie.”

Mathematician define pi, written as π, to be a circle’s circumference divided by its diameter. (That “Greek p” letter, π, was chosen because a circle’s circumference is also called its “periphery,” and p stands for “periphery.”) Pi is also a semicircle’s curved length divided by its radius, so it’s also the curved length of a semicircle whose radius is 1. It’s also the area of a circle whose radius is 1.

Pi is about 3.14, but not exactly. Pi is about 22/7, but not exactly. Like the square root of 2, pi is irrational, so it can’t be written exactly as a decimal or fraction.

Here again are the first 3 digits of pi:

Hold them up to a mirror and see what they spell.

Nerds celebrate pi day every year, on March 14 (which is 3/14), by baking pies.

Here are the first 6 digits of pi:

Those digits are famous and often used as an approximation. That approximation is not exact, of course, but cynical engineers say it’s “close enough for government work.”

Nerds like to scream “3.14159,” so the football cheer at the Massachusetts Institute of Technology (M.I.T.) includes “3.14159,” along with calculus & trigonometry:

Variants of that football cheer are used at other nerd universities also (RPI, Worcester Polytech, and Rice).

Pizza What’s the volume of a pizza whose radius is Z and whose altitude (height, thickness) is A? According to math, the volume of that “almost-flat cylinder” is “pi times the radius squared times the altitude,” so the volume is:

T-shirt Math nerds like to wear a T-shirt saying —

because it means “i eight sum pi”.

Computing pi Trigonometry says π/4 is the arctangent of 1, but calculus says the arctangent of x is

so:

Here are other popular ways to compute π:

Using those formulas (plus better formulas that converge faster), mathematicians computed many digits of pi.

 

Memorizing pi Here are the first 32 digits of pi:

To memorize the first 7 of them, just memorize this sentence (by C. Heckman), and count how many letters are in each word:

To memorize the first 8 digits, memorize this sentence instead (reported by Martin Gardner):

To memorize the first 9 digits, memorize this sentence instead (reported by Presh Talwalkar) —

or this (by M. Amling):

To memorize the first 31 digits, memorize this poem instead (by Michael Shapiro):

In that poem, “endeavour” must be spelled with the British ending (“vour”), not the American ending (“vor”).

To memorize the first 32 digits, memorize this instead (by James Jeans & S. Bottomley):

That’s as far as math guys can memorize without cheating, since the 33^rd digit of pi is zero.

A group called “ASAP Science” wrote this song about the first 100 digits:

A group called “College Humor” created this longer video, where the singers gradually go insane and die:

A contest is held often, to see who can memorize and recite correctly the most digits of pi (after the decimal point). Here are the winners:

Here are details about the 3 recent winners:

Bible The Bible says pi is 3. Specifically, the Old Testament (first book of Kings, chapter 7, verse 23) says King Solomon made a circular metal object that had a diameter of 10 cubits and a circumference of 30.

But maybe the writer meant “30” was just an approximation? 31 would have been more accurate.

Indiana In 1897, Indiana’s House of Representatives passed a bill declaring that pi is exactly 3.2, not less, so textbooks must say pi is exactly 3.2, and schoolkids must learn that pi is exactly 3.2. The bill passed unanimously: all 67 congressmen voted yes.

Fortunately, Indiana’s senate rejected that bill, so the bill never became a law.

Here’s how that bill came about:

Powers Is this number an integer:

Probably not. But nobody knows for sure, because the number is too big for today’s computers to compute accurately enough.

Tau Some mathematicians think tau is more useful than pi. Tau is the symbol τ, which is the Greek letter for t.

Some mathematicians define tau to be the circle’s circumference divided by its radius (instead of diameter), so tau would be twice as big as pi. Other mathematicians think tau should be the length of a 45-degree arc of a circle whose radius is 1, so tau would be a quarter of pi.

Such definitions of tau would make some math formulas shorter but make other math formulas longer.

 

Golden ratio

(which is about 1.6)

Here’s a puzzle for you:

To solve that puzzle, the typical student might begin by guessing simple numbers, such as 0, 1, 2, 3, -1, -2, or -3. Each of guesses fails. Then the student might try fractions (such as ½) or mixed numbers (such as 1½). They fail also. No matter what normal number the student tries, the student will fail.

If you’re a teacher, you can give that puzzle to your students, then ignore them for the rest of the hour, while you pick your nose. I call that the “pick your nose” puzzle.

Eventually, the students will realize that if the puzzle has a solution, it must be wacky.

If the students studied algebra enough, they’ll eventually realize the problem can be rewritten this way:

If their algebra class included the “quadratic formula,” they’ll realize the problem can be solved by rewriting that equation as —

then applying the quadratic formula, which gives this final result:

Since I feel negatively about negative numbers, I’ll ignore the negative choice, so my answer is:

Since that answer include a square root, it has the same annoying property as the square root of 2 and the square root of 5:
the answer is irrational (can’t be written as a simple decimal or fraction).

The answer is about 1.6, but not exactly. (If you square 1.6, you almost get 2.6, but not exactly: you get a hair less, 2.56.) A closer approximation to the answer is 1.618 (whose square is just a teensy-weensy hair less than 2.618). An even closer approximation is 1.6180339887.

The answer is called phi (which resembles pi and is a Greek letter whose symbol is j). It’s also called the golden ratio. Religious folks call it the divine proportion. I’ll explain why shortly.

Here’s another puzzle:

Written in algebra, that becomes:

To solve that equation, multiply both sides by x, so you get
“x² = x+1.” But that’s the same equation as the previous puzzle, so it has the same answer: phi.

Here’s the puzzle that forced mathematicians to get interested in phi:

Does today’s technology use phi a lot? No. Phi is about 1.618, but people use these ratios instead:

So according to that chart, the most artistically pleasant people are lawyers and crap-readers!

Like the square root of 2, phi can be written as a continued fraction; and phi’s is even prettier:

Here’s the proof:

So you can compute phi by using this method:

Phi arises in trigonometry.

Phi arises in exponent equations.

 

Phi arises in the Fibonacci series.

One mathematician believed that in a perfectly proportioned woman, her total height divided by the height of her navel would be phi. (In other words, her total height would be about 1.6 times the height of her navel.) Ladies, go measure yourselves….

Euler’s number, e

(which is about 2.7)

In Switzerland long ago, in the year 1690, Jacob Bernoulli tried to solve a puzzle about a bank giving compound interest. His puzzle has tortured math students ever since, since his answer is discussed in most courses about calculus and most college courses about statistics (which are sadistic). His answer is called “e,” which should stand for “eek!” but actually stands for “Euler” (a Swiss mathematician who analyzed Bernoulli’s answer a lot).

Here’s the puzzle.

Suppose you deposit money in a ridiculously generous bank that gives you an interest rate of 100% per year! To keep things simple (and because you don’t trust the bank), suppose you deposit just $1. How much money will you have after 1 year?

The answer is you’ll have the $1 you deposited, plus $1 (which is 100%) in interest, so you’ll have a total of $2.

Suppose the bank is even more generous: it compounds your interest quarterly, so you get a quarter of 100% (which is 25%), 4 times per year, immediately redeposited. How much money will you have at the end of the year? Here’s the solution:

Hey, that’s better than just $2!

Suppose the bank is even more generous: it compounds your interest monthly, so you get a twelfth of 100%, 12 times per year. At the end of the year, you’ll have slightly more than $2.61, which is even better!

Suppose the bank is even more generous: it compounds your interest daily, so you get a 365^th of 100%, 365 times per year (if it’s not a leap year). At the end of the year, you’ll have slightly more than $2.71, which is even better, but just slightly, just 10 cents more.

Here’s the final puzzle: suppose the bank is super-duper generous: it compounds your interest continuously (every tiny fraction of every second of every minute of every hour of every day). At the end of the year, how much will you have? The answer is just very slightly more than $2.71. In fact, it will be this many dollars:

That number is called e. If the bank is mean, it will round that down to the nearest penny and give you just $2.71, the same as if it compounded interest just daily instead of continuously.

That number, e, which is about 2.718, shows up in many branches of math. From that banking example, you can see that e is what you get when you compute —

and n gets very big. Mathematicians say it’s “the limit of that expression, when n approaches infinity.” They write:

Mathematicians who are nervous about infinity write the equation this way instead (using “x” to stand for “1/n”) —

or this way, which looks cooler:

A faster way to compute e is to use this trick:

Unfortunately, e is irrational: its digits never end and never completely repeat. But e’s first 16 digits are easy to memorize, if you view them in pairs:

The 5 digits after that are 23536, which I memorize by saying to myself, “2 plus 3 is 5, but times 3 is 6.” That gives 21 digits. I was the only kid in my high school who was weird enough to memorize e. I was e-specially eerie.

The e has this weird property: if you grab a piece of graph paper and graph the function “y = e^x,” you get a curve whose “slope up” at every point is the same as y. For example, at the curve’s point where y reaches 5, the curve’s “slope up” (rise divided by run) is also 5. That’s why e is fundamental to calculus, which is the study of slopes (which calculus calls “derivatives”).

Statisticians like to draw a “bell curve” showing most people & situations are middle-of-the-road. That bell curve is just a heavily modified version of “y=e^x,” to make the curve symmetric.

 

Let’s look closely at math’s wackiness.

Pythagorean theorem

The most amazing math discovery made by Greeks is the Pythagorean theorem. It says that in a right triangle (a triangle including a 90° angle), a²+b²=c², where c is the length of the hypotenuse (the longest side) and a&b are the lengths of the legs (the other two sides). It says that in this diagram —

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c’s square is exactly as big (has the same area) as a’s square and b’s square combined.

The Chinese discovered the same truth, perhaps earlier.

Why is the Pythagorean theorem true? How do you prove it?

You can prove it in many ways. The 2^nd edition of a book called The Pythagorean Proposition contains many proofs (256 of them!), collected in 1940 by Elisha Scott Loomis when he was 87 years old. Here are the 5 most amazing proofs.…

3-gap proof Draw a square, where each side has length a+b. In each corner of that square, put a copy of the triangle you want to analyze, like this:

 

 

 

 

 

 

 

 

 

 

Now the square contains those 4 copied triangles, plus 1 huge gap in the middle. That gap is a square where each side has length c, so its area is c².

Now move the bottom 2 triangles up, so you get this:

 

 

 

 

 

 

 

 

 

 

The whole picture is still “a square where each side has length a+b,” and you still have 4 triangles in it; but instead of a big gap whose area is c², you have two small gaps, of sizes a² and b². So c² is the same size as a²+b².

1-gap proof Draw the same picture that the 3-gap proof began with. You see the whole picture’s area is (a+b)². You can also see that the picture is cut into 4 triangles (each having an area of ab/2) plus the gap in the middle (whose area is c²). Since the whole picture’s area must equal the sum of its parts, you get:

In this proof, instead of “moving the bottom 2 triangles,” we use algebra. According to algebra’s rules, that equation’s left side becomes a² + 2ab + b², and the right side becomes 2ab + c², so the equation becomes:

Subtracting 2ab from both sides of that equation, you’re left with:

1-little-gap proof Draw a square, where each side has length c. In each corner of that square, put a copy of the triangle you want to analyze, like this:

 

 

 

 

 

 

 

The whole picture’s area is c². The picture is cut into 4 triangles (each having an area of ab/2) plus the little gap in the middle, whose area is (b-a)². Since the whole picture’s area must equal the sum of its parts, you get:

According to algebra’s rules, that equation’s right side becomes 2ab + (b² - 2ba + a²). Then the 2ab and the -2ba cancel each other, leaving you with a² + b², so the equation becomes:

1-segment proof Draw the triangle you’re interested in, like this:

 

 

 

Unlike the earlier proofs, which make you draw many extra segments (short lines), this proof makes you draw just one extra segment! Make it perpendicular to the hypotenuse and go to the right angle:

 

 

 

The original big triangle (whose sides have lengths a, b, and c) has the same-size angles as the tiny triangle (whose sides have lengths x and a), so it’s “similar to” the tiny triangle, and so the big triangle’s ratio of “shortest side to hypotenuse” (a/c) is the same as the tiny triangle’s ratio of “shortest side to hypotenuse” (x/a). Write that equation:

Multiplying both sides of that equation by ac, you discover what a² is:

Using similar reasoning, you discover what b² is:

Adding those two equations together, you get:

Since x+y is c, that equation becomes:

1-segment general proof Draw the triangle you’re interested in, like this:

 

 

 

As in the previous proof, draw one extra segment, perpendicular to the hypotenuse and going to the right angle:

 

 

 

Now you have 3 triangles: the left one, the rightmost one, and the big one.

Since the left triangle’s area plus the rightmost triangle’s area equals the big triangle’s area, and since the 3 triangles are similar to each other (“stretched” versions of each other, as you can prove by looking at their angles), any area constructed from “parts of the left triangle” plus the area constructed from “corresponding parts of the rightmost triangle” equals the area constructed from “corresponding parts of the big triangle.” For example, the area constructed by drawing a square on the left triangle’s hypotenuse (a²) plus the area constructed by drawing a square on the rightmost triangle’s hypotenuse (b²) equals the area constructed by drawing a square on the big triangle’s hypotenuse (c²).

Which proof is the best? The 3-gap proof is the most visually appealing, but it bothers mathematicians who are too lazy to draw (construct) so many segments. (It also requires you to prove the gap is indeed a square, whose angles are right angles, but that’s easy.)

The 1-gap proof uses fewer lines by relying on algebra instead. It’s fine if you like algebra, awkward if you don’t. The 1-little-gap proof uses algebra slightly differently.

The 1-segment proof appeals to mathematicians because it requires constructing just 1 segment, but you can’t understand it until you’ve learned the laws of similar triangles. This proof was invented by Davis Legendre in 1858.

The 1-segment general proof is the most powerful because its thinking generalizes to any area created from the 3 triangles, not just square areas. In any right triangle:

That proof was invented by a 19-year-old kid (Stanley Jashemski in Youngstown, Ohio) in 1934.

Ugliness

To understand the concept of math ugliness, remember these math definitions:

Now you can tackle the 3 rules of ugliness:

More precisely:

In different branches of math, those same 3 rules keep cropping up, using different definitions of what’s “ugly” and “nice.”

The rules apply to people, too:

How math should be taught

I have complaints about how math is taught. Here’s a list of my main complaints. If you’re a mathematician, math teacher, or top math student, read the list and phone me at 603-666-6644 if you want to chat about details or hear about my other complaints, most of which result from research I did in the 1960’s and 1970’s. (On the other hand, if you don’t know about math and don’t care, skip these comments.)

Percentages Middle-school students should learn how to compute percentages (such as “What is 40% of 200?”); but advanced percentage questions (such as “80 is 40% of what?” and “80 is what percent of 200?”) should be delayed until after algebra, because the easiest way to solve an advanced percentage question is to turn the question into an algebraic equation by using these tricks:

Graphing a line To graph a line (such as “y = 5 + 2x”), students should be told to use this formula:

So to graph y = 5 + 2x, put a dot that’s a distance of 5 above the origin; then draw a line that goes through that dot and has a slope of 2.

The formula “y = h + sx” is called the “hot sex” formula (since it includes h + sx). It’s easier to remember than the traditional formula, which has the wrong letters and wrong order and looks like this:

Imaginary numbers Imaginary numbers (such as “i”) should be explained before the quadratic formula, so the quadratic formula can be stated simply (without having to say “if the determinant is non-negative”).

Factoring Students should be told that every quadratic expression (such as x² + 6x + 8) can be factored by this formula:

For example:

As you can see from that example, the a (which in the example is 3) is the average of the two final numbers (4 and 2). That’s why it’s called a.

The d (which is 1) is how much each final number differs from a (4 and 2 each differ from 3 by 1). That’s why it’s called d. You can call d the difference or divergence or displacement.

Here’s another reason why it’s called d: it’s the determinant, since it determines what kind of final answer you’ll get (rational, irrational, imaginary, or single-root). You can also call d the discriminant, since it lets you discriminate among different kinds of answers.

Quadratic equations To solve any quadratic equation (such as “x² + 6x + 8 = 0”), you can use that short factoring formula. For example:

Another way to solve a quadratic equation is to use “Russ’s quadratic formula,” which is:

That’s much shorter and easier to remember than the traditional quadratic formula, though forcing an equation into the form “x² = 2bx+c” can sometimes be challenging. Here’s an application:

Prismoid formula Students should be told that the volume of any reasonable solid (such as a prism, cylinder, pyramid, cone, or sphere) can be computed from this prismoid formula:

That formula can be written more briefly, like this:

For example, the volume of a pyramid (whose height is H and whose base area is L times W) is:

The volume of a cone (whose height is H and whose base area is πr²) is:

The volume of a sphere (whose radius is r) is:

In the prismoid formula, V = H (T + B + 4M)/6, the “4” is the same “4” that appears in Simpson’s rule (which is used in calculus to find the area under a curve). The formula gives exactly the right answer for any 3-D shape whose sides are “smooth” (so you can express the cross-sectional areas as a quadratic or cubic function of the distance above the base). To prove the prismoid formula works for all such shapes, you must study calculus.

Balanced curriculum Math consists of many topics. Schools should reevaluate which topics are most important.

All students, before graduating from high school, should taste what statistics and calculus are about, since they’re used in many fields. For example, economists often talk about “marginal profit,” which is a concept from calculus. Students should also be exposed to other branches of math, such as matrices, logic, topology, and infinite numbers.

The explanation of Euclidean geometry should be abridged, to make room for other topics that are more important, such as coordinate geometry, which leads to calculus.

Like Shakespeare, Euclid’s work is a classic that should be shown to students so they can savor it and enjoy geometric examples of what “proofs” are; but after half a year of that, let high-school students move on to other topics that are more modern and more useful, to see examples of how proofs are used in other branches of math.

Too much time is spent analyzing triangles.

 

I’ve invented several new ideas. I figure I should get a Nobel prize for them, except the ideas are half-baked: they need further research to make them fleshed out, complete, and fully useful. So I beg you: improve on these ideas, so you can get a Nobel prize. If you mention me in a footnote, I’d appreciate that. We can split the Nobel prize: you get the Bell prize, and I get No prize.

There’s just one little hitch in our plan to split a Nobel prize:

Derived happiness

What makes people happy? Several centuries ago, the “meaning of happiness” was considered a philosophical problem. Nowadays, it’s considered a psychiatric problem: happiness is whatever makes your happiness hormones increase. In the future, it will become a math problem; here’s why....

To begin our fancy-schmancy math analysis, let’s do the same thing physicists do when analyzing motion: oversimplify! Later, we’ll discuss all the complications of the “real world,” such as friction.

Physicists begin by assuming objects move in a vacuum, then later add the effects of friction. We’ll begin by assuming happiness consists of having lots of money, then later add the effects of interpersonal friction (good & bad relationships with other people) and God friction (good & bad relationships with the desire to have a meaningful life). I’ll start with money, rather than frictions, because money is easier to measure.

Zeroth-derivative happiness Let’s start with the simplest situation:

That’s all we know about Joe & Tim so far. They’re both American males, so we don’t know any cultural differences between Joe & Tim yet. On the basis of what we know so far, Joe is probably happier than Tim, since Joe is wealthier.

This explanation is going to get mathy, and I’m even going to say jargon from calculus! But to avoid scaring the anti-math part of your brain, I promise to explain all math jargon simply.

Using math jargon, we say that Joe is higher up on the “wealth function” than Tim. That stupidly simple explanation is called the zeroth-derivative function.

First-derivative happiness Now let’s complicate the situation slightly, by peeking at the past:

Now the happiness seems different. Tim is happy because his money doubled. Joe is unhappy because Joe’s money halved. Even though Joe still has more money than Tim, Joe feels unhappy because Joe’s “life is going downhill,” so his future looks grim, whereas Tim is thrilled because Tim’s “life is going uphill” so his future looks bright.

Compared to yesterday, Tim gained $50, whereas Joe lost $200. In calculus jargon, we say:

So Tim’s slope is better than Joe’s slope. Slope is also called the derivative. More precisely, it’s called the first derivative. So to figure out a person’s happiness, you should look at the person’s slope (first derivative).

Second-derivative happiness Now let’s complicate the situation further, by peeking further into the past:

Who’s happier: Ann or Sue?

Ann has more money than Sue (since Ann has $305 while Sue has just $55). Ann’s recent slope is also better than Sue’s recent slope (since Ann’s recent slope was $5 per day, while Sue’s recent slope was minus $5 per day).

But in spite of all that good news for Ann, she probably feels depressed, because her recent raise (the $5 raise from $300 to $305) is worse than her previous raise (the $100 raise from $200 to $300). Her raise decreased by $95 (since the $100 raise dropped to $5). She feels her life isn’t improving as much as it used to. She fears her life will, in the future, improve less and less and finally go downhill. She’s depressed that she has less pride now (going from $300 to $305) than she had before (going from $200 to $300). She feels she’s no longer a star on the rise. She’s a has-been with probably a depressing future. She wants to commit suicide, because the great part of her life is over.

Sue, by contrast, is feeling relieved. Although her money dropped recently (a $5 drop, since $60 became $55), the drop wasn’t as dramatically bad as the period before (a $40 drop, from $100 to $60). She’s happy she didn’t drop $40 again. She’s happy her drop this time was just slight, almost insignificant, so her losses are “stemming” (becoming less significant). She feels her life is “turning the corner” and might soon rise. Her slope improved: it was minus $40 per day previously but became minus $5 per day for the recent day.

Comparing old slopes against new slopes is called
computing the second derivative. Since Ann’s slope got worse (decreased), her second derivative is negative, and Ann feels depressed; since Sue’s slope got better (not as bad as before), her second derivative is positive, and Sue feels relieved.

So according to that theory, happiness is the second derivative of the wealth function.

If you graph the history of Ann’s money and Sue’s money, you see that Ann’s graph looks like the left half of a cap (which has no visor); Sue’s graph looks like the left half of a cup (which has no handle). A cap graph means the second derivative is negative; a cup graph means the second derivative is positive. So according to that happiness theory, happiness is a cup.

To improve that theory further, we should make modifications....

Logarithms The first improvement is to use logarithms. Here are the details.

Compare these two people:

We don’t know enough of the past to compute a second derivative. According to the previous theory, Bud should be happier than Sam, since Bud has more money ($115) and a bigger slope ($15 per day). But in reality, Sam is more thrilled than Bud, since Sam’s money doubled (from $10 to $20), whereas Bud’s money went up by just a small percentage of what Bud had before (15%). Sam can brag to himself & friends that his money doubled, whereas Bud hasn’t much to brag about. Bud is happy (since Bud’s money went up, not down), but Sam is thrilled.

So to measure happiness, we should measure the percentage by which money increased. To do that, we can choose two methods, each giving the same result:

The two methods give the same result because, according to calculus, “the derivative of log f(t)” equals “f'(t) divided by f(t).”

Use the percentage method (or the equivalent logarithm method) to compute first-derivative happiness and second-derivative happiness.

Blended derivatives If your second derivative and first derivative are both negative, you might feel depressed. But if you start whining about them, your friends might remind you that you shouldn’t feel so bad, because you still have enough money to live on. For example, if you had 4 billion dollars but then had just 3 billion and then just 1 billion, your second and first derivatives are both negative; but your friends might remind you that you still have a billion dollars left and you’re still better off than most other people, so cheer up!

How important to your happiness are the first and second derivatives in relation to the amount of money you actually have? Your happiness is actually a blend of all that data. Your happiness might even be affected by the third derivative (which measures how much your second derivative is better than it was before). Maybe the happiness of people (and other animals) having impaired memory isn’t influenced much by derivatives, second derivatives, and third derivatives. Experiments should be done to determine how much the various derivatives contribute to the happiness of various kinds of people.

Beyond money Besides money in your pocket, these other things can give you happiness: investments, things you own, food, shelter, health (and being pain-free), beauty, intelligence, good relationships (with people, pets, and the environment), love, sex, feeling useful (in your career or by volunteering or by helping friends & family), feeling powerful, feeling moral, and — alas! — taking mood-enhancing drugs (alcohol, nicotine, marijuana, heroin, and beyond). Your happiness is affected by how much you have of all those things, how much more you have than your neighbors, and how much fame you have for what you do. Your happiness is a blend of all those factors. Experiments should be done to determine how important those factors are in the blend.

Focus Maybe most factors in your life are okay, but one factor is bugging you at the moment. Maybe it’s a test you must take tomorrow (and you haven’t studied for yet), or a friend who’s dying, or a lover you’re in the middle of breaking up with, or you’re being arrested and transported in a paddy wagon to the police station, or you’re having a medical emergency and need help fast.

Or maybe one factor is thrilling you at the moment. For example, maybe you’ve just won an award, or won a lottery, or had an orgasm.

During those especially bad or good moments, your attention focuses on one thing and nearly ignores everything else; but those other things still have some effect on your happiness then, though maybe just slightly. To compute your overall happiness in that situation, we must invent a formula that’s a weighted average of your feelings about everything: that formula must emphasize (give more weight to) the extreme feelings (feelings that are extremely positive or extremely negative) and de-emphasize the feelings that are closer to neutral (and therefore nearly ignored).

Please finish this explanation and get a Nobel prize.

Simplest infinitesimals

In elementary school, you learned how to count: 1, 2, 3, etc. Later, you learned about other kinds of numbers: zero, negative numbers, and fractions. If you took 2 years of high-school algebra, you also learned about “imaginary” numbers, such as “i”, which is the square root of minus one.

During the last 3,000 years, whenever new kinds of numbers were invented, critics laughed at the inventors:

The critics got silenced when inventors drew pictures:

When Germany’s Gottfried Leibniz and England’s Isaac Newton invented calculus in the 1600’s, they thought about an “infinitesimal number,” which is a number so tiny that it’s less than every fraction of integers (less than ½, less than 1/10, less than 1/100, less than a millionth, less than a trillionth, etc.) but is still more than zero. But since an “infinitesimal number” was hard to picture, it was hard to discuss confidently, so mathematicians later did calculus a different way, involving “limits” and awkward phrases such as “for every epsilon there exists a delta such that….” Those long-winded phrases make students want to cry, or give up and just sleep through the calculus lectures, or snore.

Mathematicians wish there were an easy, confident, pictorial, accurate way to mention infinitesimals, but that goal has eluded them. In 1966 at Yale University, Professor Abraham Robinson became famous for inventing what he called
non-standard analysis, which is his own way to do calculus by using infinitesimals, but it’s hard to understand. In the year 2000 at the University of Wisconsin, Professor H. Jerome Keisler invented a simpler way to explain Robinson’s work, but mathematicians complain that Keisler’s explanation seems sloppy.

Here are my own 2 ways to explain infinitesimals: the
zillions method and the minimal method. Each has its own advantages and disadvantages. Neither is completely satisfactory. I hope someday you or your friends can improve on what I’ve done and get a Nobel prize.

Zillions method This way to start doing calculus is understandable even to kids in elementary school. Just use the word “zillion.” As most elementary kids already know, “a zillion” means “a lot of,” “ridiculously many,” as in “I have a zillion chores to do.”

The word “zillion” has been popular for many years. According to the Merriam-Webster Dictionary
(at merriam-webster.com/dictionary/zillion), the word “zillion” has been used for many decades, even back in 1934, and some folks have been saying “jillion” instead, beginning in 1942.

To do calculus, consider a zillion to be more than a million, more than a billion, more than a trillion, more than every other “illion” you ever heard of. Make the symbol for a zillion be ∞. You can call that number “infinity” if you like, but people get scared about the word “infinity,” whereas kids use the word “zillion” all the time.

Like a trillion, a zillion is a number that obeys all the normal rules of arithmetic and algebra. It pleases mathematicians because, like normal numbers, it all obeys the commutative and associative laws and all the other laws of an “ordered field.” It just happens to be even bigger than a trillion.

The only “law” a zillion doesn’t obey is the “Archimedes principle,” since you can’t reach a zillion by counting 1, 2, 3, etc. in a finite amount of time, though you can reach it in a zillion amount of time. In other words, a zillion can’t be generated by starting at 0 and then adding 1 repeatedly in a finite amount of time; it can’t be generated by multiplying two finite numbers together. But that disappointment about zillion doesn’t affect any computations used in high-school algebra or calculus, so don’t worry about it.

A zillion is not the biggest number, since “a zillion plus one” is even bigger (and written “∞+1”), and “two zillion” is bigger yet (and written “2∞”), and “a zillion times a zillion” is bigger than those (and written “∞∞” or “∞²”), and “a zillion to the zillionth power” is bigger than all those (and written “∞^∞”).

An example of an infinite number that’s slightly smaller than a zillion is “a zillion minus one” (written “∞-1”). An even smaller infinite number is “the square root of a zillion.”

Just like a “million” has a reciprocal called “a millionth,” a zillion has a reciprocal called a zillionth, which is the fraction 1/∞. That fraction is an example of an infinitesimal, since it’s tinier than any normal fraction but still bigger than 0. Mathematicians like to call that fraction “epsilon” (which is the Greek letter for “e” and written “є”), but that Greek jargon confuses young kids and makes them complain “It’s Greek to me!” so obey the warning of AIDS advisors: don’t do Greek.

A zillionth isn’t the only infinitesimal number. A slightly bigger infinitesimal number is “two zillionths” (which is twice as big as a zillionth and written “2/∞”).

In elementary school, kids learn how to round numbers. Examples:

 

In calculus, mathematicians round using a method I call calculus round (cRound).

In old-fashioned calculus, the word “limit” is defined in a long-winded way, starting with “for every epsilon there exists a delta such that.” But in my zillion calculus, we can define “limit” to mean just cRound. More precisely, define “the limit, as x approaches p, of f(x)” to mean the result of this 3-step procedure:

So here’s the definition:

That definition requires no “delta”! That definition works if p is ∞ or -∞ or a normal number (such as 7).

In my zillion calculus, we can define “the derivative of f(x)” to mean just the cRound of “f(x+є)-f(x), all that divided by є,” like this:

That definition involves no “delta,” no “limit,” and no “p,” so it lets you compute the derivative much faster than old-fashioned methods.

Minimal method Gee, infinity can be scary: so many kinds of infinite numbers! To do elementary calculus simply, fuck infinity: let’s have no infinite numbers at all! Let’s have just the minimal necessary to do elementary calculus: a special number, called epsilon (written “є”).

Epsilon is tiny. It’s tinier than any fraction you encountered in elementary school: it’s tinier than 1/10, tinier than 1/100, tinier than 1/1000, etc. It’s so tiny that when you multiply it by itself, it disappears, poof! Here’s the equation: є²=0. Physicists brag about “black holes,” where things seem to disappear, but we mathematicians have epsilon, whose square really does disappear!

So how do you make a number system that includes epsilon and lets you do calculus, all in a reasonable way? It’s easy! It’s even easier than the crap they teach in high school’s algebra 2 class about “imaginary numbers.” In algebra 2, they teach you to draw a horizontal ruler (an x axis) labeled 0, 1, 2, etc., and draw a vertical ruler (a y axis) labeled 0, 1i, 2i, 3i, etc. Do the same thing for my minimal method, but write “є” instead of “i"”, so the vertical ruler is labeled 0, 1є, 2є, 3є, etc. In algebra 2, they teach you to invent numbers of the form x+yi, such as 3+7i; in my minimal method, invent numbers of the form x+yє, such as 3+7є. In algebra 2, they teach you to add, subtract, and multiply numbers in the obvious way, but remembering that i²=-1; in my minimal method, you can add, subtract, and multiply numbers in the obvious way, but remember that є²=0.

Inventing “i” simplified algebra, by making the quadratic formula more understandable. Inventing є simplifies calculus, by making derivatives more understandable.

For you math nerds, here’s a formal explanation….

To use є, construct the extended real numbers, which consist of numbers of the form a + bє (where “a” and “b” are ordinary “real” numbers). Add and multiply extended real numbers as you’d expect (bearing in mind that є² is 0), like this:

For example:

You can define order:

Those definitions of addition, subtraction, multiplication, and order obey the traditional “rules of algebra” except for one rule: in traditional algebra, every non-zero number has a reciprocal (a number you can multiply it by to get 1), but unfortunately є has no reciprocal.

If x is an extended real number, it has the form a + bє, where a and b are each real. The a is called the real part of x. For example, the real part of 3 + 7є is 3.

A number is called infinitesimal if its real part is 0. For example, є and 2є are infinitesimal; so is 0.

Infinitesimals are useful because they let you define the “derivative” of f(x) easily, by computing f(x+є):

Then calculations & proofs about derivatives and limits become easy, especially when you define sin є to be є and define cos є to be 1.

I was proud I invented that minimal method. But recently, I discovered the same method was invented in 1873 by William Clifford and named the dual-number system. Damn!

Measure theory

People think math is impersonal, but it can get very personal. A math theorem changed my whole life and personality, in ways I didn’t expect. Here’s my story.…

When I was in elementary school, junior high, and high school, I was good at math, won many awards, got a perfect score on the math SAT, and so got admitted to an Ivy League college (Dartmouth), graduate school (Harvard), and intensely personal further graduate school (Wesleyan University in Connecticut). I invented new ways to do many branches of math and got praised for my discoveries. I thought I was hot stuff. I thought I’d become a top mathematician.

Most of my discoveries got kinda ignored, but finally, in 1973, I made a discovery I figured would make me world-famous: I discovered a way to reduce all of geometry to just 5 axioms, instead of all those stupid axioms Euclid’s gang invented. I figured: hooray, now I’ll definitely be famous! Wow! This is it!

My 5 axioms let you compute the length, area, and volume of anything, even if the thing is an object that’s bent or weird or split into many parts.

Moreover, I proved my 5 axioms achieved math’s holy grail, they were perfect, satisfying the 3 goals of math axiomatics: I proved my 5 axioms were consistent (they didn’t create any contradictions), complete (no extra axioms would ever be necessary), and independent (all 5 axioms were necessary, none were redundant).

Since I was still just a graduate student, I showed my research to my advisor. My research belonged to a math specialty called “measure theory,” but my advisor was not an expert in that specialty, so he told me to mail my results to the expert in that field. I mailed. And waited.

Finally, I got a reply from that expert, who was a professor at Brown University. His letter went something like this:

I was crushed. I went to the library and looked up what Kolmogorov did. He didn’t actually say there were 5 axioms, but his thinking was very similar to mine; he just expressed his result differently. So he deserves at least 80% of the credit, and I deserve at most 20%.

What bugged the hell out of me was: here was a great result, which should have affected the teaching of geometry in a big way, but nobody knew and nobody cared. I noticed I could write the most profound things about math, and nobody cared, not even the math professors; but I could write the stupidest things about computers (such as where to find the power button), and everybody wanted a copy of them.

That experience convinced me that math was a dead-end occupation, so I switched my allegiance from math to computers and became famous for writing about computers, not math.

That was in the 1970’s, when math professors were losing their jobs (because people cared less about math and more about feminism, the environment, and antiwar protests), and so math professors had to learn about computers instead (which were just starting to become popular and personal).

But ever since, I’ve missed my old love: math! And now the tide has turned: everybody already knows about computers, but not enough people know enough about math, so I wish I could become a math guy again.

I’ll try.

Thanks for listening. Here are the 5 axioms.

But wait! The axioms are a bit hard for normal humans to understand, so (assuming you’re normal and not a mashed-up mathematician) I’ll start with a simplified version of the axioms first, then give you the whole gory story.

To simplify, I’ll start by talking about just area. (Later I’ll show how rewriting the axioms, just slightly, lets them also cover length & volume.) So to start, here are the 5 axioms about area.

But wait! I must confess: the axioms assume you own a sheet of graph paper, and you know how to plot points on it if I tell you the coordinates.

So here are the 5 axioms, as they apply to area:

Those axioms get you the area of any shape, even if the shape doesn’t fit on graph paper (because the shape is infinitely big or is a 3-dimensional curve or a cube or even a science-fiction type that involves more than 3 dimensions), and even if the shape is just a bunch of scattered dots (such as all points whose coordinates are integers, or all points whose coordinates are rational numbers, or all points whose coordinates are irrational numbers).

In those 5 axioms, if you change the word “area” to “volume,” you get the 5 axioms about volume. If you change the word “area” to “length,” you get the 5 axioms about length.

Instead of saying “length,” mathematicians often say “the 1-dimensional measure” or “m₁”. Instead of saying, “area,” mathematicians often say “the 2-dimensional measure” or “m₂”. Instead of saying “volume,” mathematicians often say “the 3-dimensional measure” or “m₃”.

Those 5 axioms can be applied to k-dimensional measure, m_k, no matter whether k is 1, 2, 3, or even (in science fiction) bigger than k. So here are those 5 axioms, written in the most general form:

When written that way, those 5 axioms are brief but powerful.

Are they powerful enough to really replace all of Euclid’s axioms? No, because they have 3 kinds of limitations:

Chat

If you want to chat about any of that stuff, call my cell phone (603-666-6644) anytime (24 hours). I’ll be glad to give more details, explain more clearly, or listen to your objections.

Here are the best definitions, axioms, and theorems for formalizing the elementary part of high-school algebra. It assumes you know the meaning of these logic words:

To save you time reading this yukky stuff, I won’t bother writing “proofs” and “examples” here, but phone me at 603-666-6644 anytime you want free help!

Equality

In “a=b”, the “=” is pronounced “is” or “equals” or “is equal to”. It’s primitive (which means it’s undefined).

It leads to these definitions:

Here are the axioms (fundamental properties):

Those definitions and axioms lead to these theorems (consequences that can be proved):

New:

One

“1” is pronounced “one”. It’s primitive.

Addition

In “a+b”, the “+” is pronounced “plus” or “added to” or “more than” or “increased by”. It’s primitive.

Definitions:

Axiom:

Negative

In “-a”, the “-” is pronounced “minus” or “negative”. It’s primitive.

Definitions:

New:

Axiom:

Simple theorems:

Theorems involving the associative law:

New:

Theorems about solving equations:

Theorems about double negatives:

Theorems involving three negatives:

Theorems about negating both sides:

Theorems about solving simultaneous equations:

Positivity

In the phrase “a is positive”, the “is positive” is primitive.

Definitions:

New:

Axioms:

Theorems about “positive”:

Theorems about “not”:

Theorems about “>”:

Theorems about “<”:

 

Theorems about “£:”

Theorem about “³”:

Theorems about “is negative”:

Theorems about “is real”:

Multiplication

In “a·b”, the symbol “·” (which is a dot, a raised period) is pronounced “times” or “multiplied by”. It’s primitive.

You can omit that symbol if there’s no confusion. Examples:

Definitions:

Axioms:

You get these theorems (about multiples of simultaneous equations), which you can prove without using the multiplication axioms:

Exponents

The symbol x^a is pronounced “x up a” or “x to the a” or “x, power a”
or “x, exponent a” or “x raised to the a”. It’s primitive.

Definitions:

Axioms:

New:

Theorems about exponent notation:

Theorems about multiplying:

New:

Theorems about exponent computation:

Theorems about 0:

Theorems about multiplying negatives:

Theorems about multiplying negativity:

Theorem about reality:

The FOIL theorem:

Advanced theorems about squaring:

Advanced theorems about cubing:

 

Theorems about “/”:

Theorems about solving equations:

Theorems relating exponents to “/”:

Theorem about advanced factoring:

Theorems about “/0”:

Theorems relating /a to 0:

Theorems about slashing different numbers:

Theorems about changing a fraction’s denominator:

Definition:

Using that definition, here’s how to rewrite that last batch of theorems:

Theorems about positivity:

Theorems using the multiply-exponents axiom:

Theorems about square roots:

Theorems about solving quadratic equations:

Theorems about i:

Logarithms The symbol “log_x a” (pronounced “the logarithm, base x, of a” or “log, base x, of a”) leads to these definitions:

Here are the axioms:

What’s different:

Those definitions and axioms lead to these theorems about logarithms:

Those definitions and axioms also lead to these theorems about exponents:

Theorems about the logarithm of 2 variables:

Theorems about changing the log base: