How We Write Numbers

You have probably heard at some point in your life that we use a "base ten" notation system for our numbers, without necessarily understanding what this actually means.

The point of a notation system for numbers is to keep track of how many items have been counted (as in a census, inventory report, etc.).  Therefore, the first question is how people count.  The answer is that people generally count on their fingers (you may be thinking only of small children at this point, but try listing all seven dwarves from Snow White without using your fingers to keep track of how many you've gotten (and writing them down is cheating)).  Counting on your fingers is a convenient way of keeping track of relatively small numbers, but what happens when they get bigger?  This is when you need to get creative.

One solution would be to gather several of your friends together and have someone new take over counting each time the previous person runs out of fingers, but then you have to add them all up at the end (essentially counting again), and it would take five people just to count to forty-one, which is not an outlandishly big number.  I'd like a better solution.

Suppose that instead of having a friend continue the count when I run out of fingers, I instead ask my friend to count how many times I run out of fingers.  Then forty-one is simply four of my friend's fingers and one of mine.  This essentially what we do in base ten notation.

There are ten symbols we use to write our numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Each of those symbols alone represents a number, but it helps to think of numbers in terms of "how many."  Here, the symbol "3" means "three fingers" (similarly for the other symbols).  The pair of symbols "41" (in that order) means "four sets of fingers and one leftover finger" (also known as forty-one, since we have ten fingers).  This shorthand notation is significantly easier than having to write down forty-one symbols to represent the forty-one items you counted.

If I have a second friend counting how many times my first friend runs out of fingers, we can make some even bigger numbers.  The string of symbols "123" means "one set of ten sets of ten fingers (one hundred fingers) and two sets of ten fingers (twenty) and three more fingers," or one hundred and twenty-three fingers.

You may have noticed that in general we are using the symbol "10" to mean "one set of all my fingers, with no leftovers" and begun to wonder why we should always count with ten fingers.  Suppose, for instance, that one hand is busy with something else and therefore can't help with our counting?  This would result in a base five system (which did develop in some cultures).

In a base five notation system, you would only need five symbols: 0, 1, 2, 3, and 4.  The symbol "10" would mean "one set of five fingers with no leftovers."  If you wanted to write ten, it would look like "20" because you would need two sets of five with no leftovers.  The number directly before this, nine, would look like "14."  And forty-one would have to be written as "131" (one set of five fives (twenty-five) plus three sets of fives (fifteen) plus one).  Your arithmetic would look a bit different, but it would work just as well (try a few calculations and see how it turns out).

For any whole number n bigger than or equal to two, you can create a base n notation system.  Currently, most of the human race uses base ten, most likely simply because we have ten fingers.  Most human cultures have used base five, ten, or twenty (I'm sure you can think of justifications for these).  One notable exception is the ancient Babylonians, who used base sixty.  As far as I know, no one is quite sure how they decided on this (though it has been pointed out that sixty is the smallest number divisible by 2, 3, 4, 5, and 6, which makes for fewer repeating decimals for fractions), but you can see its influence in how we measure time and angles.

Death by Proof By Contradiction

In my first post, I talked a bit about Pythagoras and his school.  One important feature of Pythagoras's belief system was that he believed that the natural world was composed entirely of positive whole numbers and ratios of such numbers (called "rational numbers" because they can be expressed as ratios).  This belief was as fundamental to the Pythagoreans as the notion of a benevolent creator is to modern Christianity.

Because of their belief that numbers defined the natural world, Pythagoras and his students put a lot of time and energy into studying the properties of numbers and operations on numbers.  (They also attributed various properties, such as masculine and feminine, to different numbers, but that's not important to this story).  Their work was based in part on real-world observations ("square" numbers derive their name from the Pythagorean observation that certain numbers of stones can be used to make perfect squares in the sand) and they wanted to use their work to understand the workings of the real world.

One day, a student by the name of Hippasus was attempting to discover the rational number that would measure the diagonal of a square whose sides were equal to 1 unit ("unit" can be any unit of measurement you like, the argument will still work the same).  The Pythagoreans knew that a square's diagonal cut the square into two identical right triangles, with both legs equal to 1 in this case and the diagonal being the unknown.

Hippasus assumed, as any good student of Pythagoras would, that the measurement of the diagonal would be a rational number of units.  We'll call it p/q units, where p and q are both positive real numbers that share no common factors with each other (this is allowed because if they shared common factors, we could cancel them and get a rational number u/v, which equals p/q but doesn't have common factors floating around to complicate things, and we would use that one).

By the Pythagorean Theorem, Hippasus was then able to say that 1²+1²=p²/q².

This simplifies (through some basic arithmetic) to 1+1= p²/q², or 2=p²/q².

Multiplying both sides by q² gives 2q²=p².

Recall that p and q are both whole numbers. This means that p² and q² are also whole numbers, since any product of whole numbers is a whole number. Then 2q² is an even whole number, since twice any whole number is even. That tells us that p² is even.

A product of two odd numbers is odd, so p itself can't be odd. It must be even.

Since p is even, let's write it as 2k, where k is also a whole number.

Then p²=(2k)² (by squaring both sides of the equation p=2k). Distributing the exponent on the right side tells us that p²=4k².

But we said earlier that  2q²=p², so we know have 2q²=4k².

Divide both sides by two, and we get  q²=2k².  Now by the same reasoning we used before to show that p was even, we can see that q is even.  But we had said (without breaking any rules of math) that p and q have no common factors, and now we have shown that they are both divisible by two.  The only assumption we made without justification was that our hypotenuse consisted of a rational number of units (or in modern parlance, that the square root of two is rational), and so this assumption must be wrong.

By essentially this same method (I updated notation and language to make it more accessible to modern readers) Hippasus had proven that one of the central tenets of Pythagoras's belief system was false by demonstrating that the diagonal of a square was incommensurable with its sides (sharing no common measure; this is how they would have phrased our statement that the diagonal is irrational), thus showing that there is some part of the natural world which cannot be described as a ratio of whole numbers under any measuring system.

Remember that this had not been Hippasus's intention.  He was simply trying to find the correct ratio, and ended up proving that no such ratio exists (it is not uncommon in mathematics for investigations in one direction lead to astonishing unexpected discoveries).  Unfortunately, that did not soften the response from the other Pythagoreans.  Like many people throughout history, they did not take kindly to having their belief system uprooted with a few strokes of a pen.  Accounts vary about how they treated him.  The kindest claim that they kicked him out, erected a tombstone for him, and never spoke to him again, while other sources claim they took him out to sea and threw him overboard.  In any case, he was punished for his discovery.