William Dunham, in his excellent book The Mathematical Universe: An Alphabetical Journey through the Great Proofs, Problems, and Personalities, gives a beautiful and simple example of a situation in which intuition and common sense are incorrect and logic is correct.
Consider a swimming pool that has a perimeter of 240 feet, say a swimming pool that is 100 feet long and 20 feet wide. Intuition tells us that the shape does not affect the area as long as the length of the perimeter remains unchanged (i.e., you can change the shape from a nonsquare rectangle to a square or even a hexagon without changing the area). The logic of mathematics demonstrates that this is incorrect.
Our first swimming pool is 100 feet by 20 feet. Its perimeter is 240 feet (100 + 20 + 100 + 20), and its area is 2,000 square feet (100 × 20).
Now let’s change the swimming pool to a square shape with the same perimeter of 240 feet. The square is 60 feet on each side. The swimming pool now has an area of 3,600 feet (60 × 60). So the length of the perimeter is the same, but the area is different.
It is even possible, as Dunham points out, for a longer perimeter to contain less area than a shorter one. For example, a swimming pool 2 yards wide by 12 yards long has a perimeter of 28 yards, but an area of only 24 square yards. A square pool measuring 5 yards on each side has a shorter perimeter of 20 yards, but a larger area of 25 square yards.
Pi (π), the ratio of the circumference of a circle to the circle’s diameter, is an unending number that begins 3.14159 … . For practical use, the number must be rounded off at some point.
A century ago, as students are often told, a legislator from Indiana attempted to have pi declared to equal precisely 3.2. Ever since, this has served as the quintessential example of an absurd attempt to impose a human wish on mathematical and scientific truth, the equivalent of a law declaring the earth to be flat.
While I would never admit it to a mathematician, this legislative foray—while no doubt dopey to the nth degree and potentially making a lot of creaky bridges and leaning buildings—never struck me as sufficiently stupid to carry the moral weight the story is meant to carry. The story is told as if the legislator were denying a truth where the practitioner of applied mathematics was asserting it. (In pure mathematics pi need not be rounded off.) This would be the case had the legislator declared pi to be a rational number, rather than the transcendental number that it is. Then he would have deserved the degree of ridicule the story attempts to convey; he would have committed a mathematical felony. (Do not worry about what a transcendental number is; all that matters here is that it is not a rational number.)
But the legislator did not deny truth where the mathematician asserts it; they both deny truth, the legislator only quantitatively more so—a misdemeanor. Pi is not 3 nor 3.14159; it is a number beginning with 3, 3.14, 3.141, 3.1415, 3.14159, and so on.
Many Americans have had the experience of traveling in a far-off American city and meeting someone who is a friend of a friend of theirs. As astonishing as this invariably seems, it is to be expected. Itheil de Sola Pool of MIT found that the probability is over 60 percent that any two randomly selected Americans can be linked by two or fewer intermediaries.
This finding is based on the assumption that the average American personally knows a thousand people well enough to recognize them and call them by name: friends from the sixth grade, all the doctors you have gone to more than once, and so on. (Celebrities known only through the media do not count.) Thus, if I randomly choose an American adult (the president of the United States or an alcoholic on a wharf in Southern California or a taxidermist in Idaho) and you are an American, the probability is better than 6 in 10 that you know someone who knows someone who knows the person I randomly selected. These figures apply to the continental United States as a whole and impose no further constraints.
If we are more rigorous in our definition of “know,” so that the average person knows only a hundred people, the number of intermediaries increases by two or three. If, on the other hand, a further constraint is added—say, both you and your friend are college graduates—the likelihood that you know someone who knows someone who knows your college graduate friend is greatly increased.
In practice, this last is the case when, for example, you realize that you know someone known to the person next to you on the plane. You and your travel acquaintance are both wealthy enough to fly, possibly residents of the same city, and so forth. What is surprising is not that you both know the same person, but that you discovered this fact during a short plane ride.
If all this seems too astonishing to believe, think about how quickly the network tree of acquaintances grows. You know a thousand people by name and face, and the thousand people know an average of a thousand each. That is a million. Those million know an average of a thousand each. That is a (US) billion. Even with duplications removed, that is a lot of people (four times the American population) to whom you are this closely “connected.” It does not seem so surprising that these people include 60 percent of the American population.
To see this all more clearly, picture a clockface that, instead of the 12 numbers around the face, has 150 million numbers. (It is a really big clock face, and the numbers are really small.) Each number is “connected” to (“knows”) 100 other numbers, with no 2 numbers sharing any other numbers. Because each number is connected to 100 other numbers, any 2 numbers will be connected to 10,000 numbers (100 × 100). Each of these 10,000 numbers is connected to 100 other numbers, so just three degrees of separation makes a million (100 × 100 × 100) connections. Add another 100 connections and we are up to 100 million (100 × 100 × 100 × 100). With just one more 100 numbers we have 10 billion connections. (There are only about 6 billion people in the world.)
However, there is a very big however. In our clockface example, we assumed that all of (the numbers equivalent to) Alan’s friends were different from (the numbers equivalent to) Bob’s friends, and all of (the numbers equivalent to) Bob’s friends were different from (the numbers equivalent to) Charlie’s friends, and so forth. In real life, many of Alan’s friends are also friends of friends of Alan’s (i.e., Bob is Charlie’s friend); thus, there are many, many duplications, which add nothing to the connectedness.
Nonetheless, as Cornell University mathematicians Duncan Watts and Steven Strogatz have shown, a very few long-range connections (say, the one friend you have in India, the one patient an Australian doctor had who now lives in England, etc.) reduce the number of connections needed to connect any two people more dramatically than would seem possible, and this generates the reality Itheil de Sola Pool found.
The small-world situation is still far from settled. There are statisticians who doubt the randomness of the selected samples (as well as one or another of the assumptions made), with one going so far as to suggest that the entire matter is an urban legend. Future research will likely settle the issue.
Gravity, at least in our universe, follows what is known as the inverse square law. Say you and a rock are floating in space x miles away from each other. When you double the distance, you reduce the gravitational force to 1/x2. This is because we live in a universe of three spatial dimensions.
If we lived in a universe of four spatial dimensions, things would be quite different. As the British cosmologist and astrophysicist Martin Rees points out, the reduced gravitational force would be 1/x3, and the planets would be slowed down sufficiently to cause them to plummet into the sun. Good-bye to us. (Whether there could be a universe of four spatial dimensions is not known.)
Rees explains why the inverse law holds. Think of your distance from the rock as the radius of a sphere, with the rock at the center (or a sphere with you at the center). As you increase the radius of the sphere, you spread the lines of gravitational force over a larger area, thereby diluting and weakening them as described above.
A number system that always permits addition, multiplication, and subtraction without introducing a new kind of number is called a “ring.” The positive and negative integers constitute a ring. A system that permits these and division is called a “field.” The integers do not constitute a field because one cannot always divide an integer by an integer and get an integer (i.e., 29 divided by 3 is not an integer). The rational numbers—numbers expressible as a fraction of two integers, such as 1/2 or 743/627 do constitute a field because dividing a rational number by a rational number always gives a rational number.
Julian Havil is a retired mathematics master of England’s Winchester College. In his Nonplussed! Mathematical Proof of Implausible Ideas, Havil gives two interesting examples of nonintuitive truths:
1.Consider two equally talented tennis players, each of whom wins 90 percent of the games in which he serves.
In tennis, scoring goes 15, 30, 40, win, with the winner having to be two points ahead. This is what is counterintuitive: a server who has a 40–30 lead is less likely to win the game than a server in a game that is just beginning (i.e., 0–0), and this is true for purely mathematical reasons.
This is why: on any point, the nonserver has only a 10 percent chance of winning. The probability that he will win three points in a row (to win the game he is losing 40–30) is greater than his winning any game.
2.If Andy is taller than Bob and Bob is taller than Charlie, then Andy must be taller than Charlie. This is pure logic.
You might think that this is true for any term substituted for “taller than.” But consider the kids’ game of Rock, Paper, Scissors. Scissors cuts (i.e., beats) paper, paper covers (i.e., beats) rock, and rock breaks (i.e., beats) scissors (where Charlie is not taller than Andy).
Where “taller than” is linear, Rock, Paper, Scissors is circular.
Counting one number per second,
| it will take | to count to |
| 1 second | 1 |
| 17 minutes | 1,000 (thousand) |
| 12 days | 1,000,000 (million) |
| 32 years | 1,000,000,000 (billion) |
| 32,000 years | 1,000,000,000,000 (trillion) |
| 32 million years | 1,000,000,000,000,000 (quadrillion) |
| 32 billion years | 1,000,000,000,000,000,000 (quintillion) |
Thirty-two billion (US terminology) years is about two or more times the current age of the universe, depending on the cosmological theory one uses to estimate the age of the universe.
a = b
So, ab = a2 = b2
So, ab + (a2 - 2ab) = a2 + (a2 - 2ab)
So, a2 - ab = 2a2 - 2ab
So, 1(a2 - ab) = 2(a2 - ab)
So, 1 = 2
What is going on here? Substitute numbers for a and b and you will find that the next-to-last step is 1 × 0 = 2 × 0 or 0 = 0, not 1 = 2. In other words, to get to 1 = 2, both sides of 1(a2 - ab) = 2(a2 − ab) were divided by 0. That is not allowed.
In everyday life, we usually assume that there are few of something that is “rare.” People who can run a mile in under three minutes and fifty seconds are not merely rare in frequency, but in absolute numbers. This assumption is valid because we live in a finite world.
However, it is worth remembering that in mathematics “rare” usually means “exceedingly low frequency.” Rare numbers may be infinite in number. For example, numbers evenly divisible by 1010923 are very rare in frequency. But there are an infinite number of them. There are, of course, numbers that are both relatively and absolutely rare. For example, 2 is the only even prime, and 6 is the only perfect, perfect number. (See “Perfect Numbers,” on page 43.)
If God promised to whisper one secret into your ear, it is a good bet that most mathematicians would ask for the simple algorithm for the primes (i.e., a simple formula that would identify all the primes and only the primes, a formula that is both necessary and sufficient for the identification of primes.) For it is likely that such an algorithm would take us to the heart of the heart of numbers.
There is such an algorithm if we drop the word “simple.” If you take a number, find the factorial of one less than that number, add one to the result, divide by the number, and find that the remainder is 0, the number is prime. Otherwise it is not prime. For example, start with 5; 4 × 3 × 2 × 1 = 24; 24 + 1 = 25; 25/5 = 1 with no remainder. So 5 is prime. The problem is that it takes virtually as long to do this than to test for primality by trial and error.
As mathematician Tobias Dantzig points out in Number, while the above is called the Wilson Index, it was Leibnitz who proved that the condition is necessary, and Lagrange who proved, a hundred years later, that it is sufficient.
Incidentally, you may have been wondering why 1 is not considered a prime. (It is considered a unique number, neither prime nor composite.) The reason is this: A crucial mathematical theorem, the Fundamental Theorem of Arithmetic, states that every positive integer (1 excepted) is the product of one, and only one, set of primes or 1 and a prime. For example, 11 = 1 × 11 and 36 = 2 × 2 × 3 × 3. If 1 were considered a prime, then, for example, 11 would equal 1 × 11 or 1 × 1 × 11 or 1 × 1 × 1 × 11, and so on.
You will remember that a rational number is one that can be expressed as a fraction (i.e., a “ratio”). All integers are rational because they can be expressed as themselves over 1 (e.g., 7 = 7/1). In some cases, it is easy to tell that a number is not rational. The square root of any integer that is not a whole number is not rational (i.e., the square root is not rational.) The square roots of 2, 3, 5, 6, 7, and 8 are all nonrational. The square root of 9 is rational because it is the whole number 3.