An Ancient Mathematical Crisis

PythagoreanCat

[When Alexandria Jones and her family visited an excavation in southern Italy, they learned several tidbits about the ancient school of mathematics and philosophy founded by Pythagoras. Here is Alex’s favorite story.]

It hit the Pythagorean Brotherhood like an earthquake, a crisis of faith which shook the foundations of their universe. Some say Pythagoras himself made the dread discovery, others blame Hippasus of Metapontum.

Something certainly did happen with Hippasus. The Brotherhood sent him into exile for insubordination, or for breaking the rule of secrecy — or was it for proving the unthinkable? According to legend, Hippasus drowned at sea, but was it a mere shipwreck or the wrath of the gods? Some say the irate Pythagoreans threw him overboard…

Once Upon a Time

Trying to learn about the secretive Pythagoreans is like wading into a swamp of myth and hearsay. But one thing we can’t doubt: The group ran into a serious problem when someone tried to compare the diagonal of a square with its side.

You see, to Pythagoras, number really meant natural number. The only kind of numbers that he recognized were the kind you count with: 1, 2, 3, 4, etc. The Greeks didn’t use fractions as we understand them. Instead, they used ratios to compare two whole numbers. The ratio 2/3 meant “2 compared to 3.”

Think of comparing pizzas to teenagers. “If it takes 2 pizzas to feed 3 kids, how many will we need for a party of 18?”, when translated into ratios, looks like this:

\frac{2}{3}  = \frac{x}{18}

Ratios are great for solving story problems.

According to Pythagoras, everything in the universe is made out of numbers. Which means the relationship between any two things — from musical notes to the sun and moon — can be described with a ratio.

Then There Was a Problem

But when the Pythagoreans looked at a simple square and its diagonal, they hit a philosophical snag. Of course, they knew by their own Pythagorean Theorem that if the side of the square was s units long, then the diagonal was s \sqrt{2} units.*

But what was the ratio between the diagonal and the side?

Aristotle wrote about the problem almost 200 years later. He explained it something like this:

  • Try to find two whole numbers such that \frac{s\sqrt{2}}{s}  = \frac{x}{y}  .

First, the length s cancels out:

\sqrt{2}  = \frac{x}{y}

We can easily put any ratio into lowest terms, so let’s assume that our fraction x/y is in lowest terms. This is important, as you will see.

*[The Pythagoreans would not have explained it with algebra, but that was what they meant. We are using algebra because it is so much more efficient than writing everything out in words, like: “The square constructed on the hypotenuse of a right triangle is equal in area to the sum of the two squares constructed on the legs of that triangle.”]

An Algebra 1 Student Can Do This

Next, square both sides of the equation:

2  = \frac{ {x}^{2} }{ {y}^{2} }

Cross-multiply by the denominator:

2 {y}^{2}  = {x}^{2}

The product 2 {y}^{2} is obviously an even number. Only even numbers are even when squared, so we know that x must also be an even number.

And since our fraction x/y was in lowest terms, it naturally follows that y  has to be an odd number.

So Far, So Good…

If x is even, then it must be double some other number: x  = 2n . We can substitute:

2 {y}^{2}  = {\left(2n \right)}^{2}

And squaring the term inside the parentheses:

2 {y}^{2}  = 4{n}^{2}

Dividing both sides by 2:

{y}^{2}  = 2{n}^{2}

The product 2 {n}^{2} is obviously an even number. Only even numbers are even when squared, so we know that y  must also be an even number.

Reductio ad Absurdum

But y  cannot be both odd and even — it’s impossible!

Therefore, our original assumption must be wrong. There is no ratio \sqrt{2}  = \frac{x}{y} .

That means we can never calculate the exact value of \sqrt{2} . Today we call numbers like \sqrt{2} irrational (not ratio-nal). Another famous irrational number is the circle constant \pi .

To the Pythagoreans, however, there had to be a ratio. If there were no ratio, then \sqrt{2} couldn’t be made of whole numbers, and everything was made of numbers. Wasn’t it?

It was enough to drive a mathematical cultist crazy.

To Be Continued…

Read all the posts from the March/April 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.


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10 thoughts on “An Ancient Mathematical Crisis

  1. This is a great story, and a deservedly famous proof, one of the best proofs ever in my opinion.

    I used to think my students understood it. Then one year I did this proof in class, asked them to use the same method to prove that the square root of 3 is irrational for homework, which they did … and then asked them to use the same method to prove that the square root of 4 is irrational on the test … and most of them did. I think if a student can’t find the flaw in using the same proof to prove that the square root of four is irrational, then they don’t really understand the proof. Your explanation here is very good, so I’d give your students above average chances of figuring it out.

  2. I find it tough to justify that we can just assume a/b is reduced thereby creating the setup that one of them has to be odd. It is a little too convenient that the only thing proving root two to be irrational here is that it contradicts the assumption. We do not use the assumption to generate the contradiction, only test ourselves in the end against our assumption.

    To me it is seems a little too much like “If your Toyota wasn’t a spaceship then it would be purple. It has four doors, the color on all doors is the same, this color is the color of the car, and this color is blue, which contradicts our assumption. Therefore: Your Toyota is a spaceship”

    I did try to prove that root 4 was irrational just as a test and I unfortunately got to the conclusion that b=a/2, which is what one would expect.

    In university I did used to be confident in a form of this proof, which did rely on proving that a number was both even and odd. I’ll try to find the more pleasing version to me.

    By the way, my favorite is the proof that there is no largest prime

  3. “I find it tough to justify that we can just assume a/b is reduced thereby creating the setup that one of them has to be odd. “

    Can’t any fraction be reduced to lowest terms without loss of generality? If diagonal/side can be written as a ratio of natural numbers, then it must have a lowest-terms equivalent. We just assume from the beginning that a/b (or x/y, as I used in my article) is that lowest-terms equivalent.

    In your proof that there is no largest prime, you may want to point out that the number constructed will not always be prime itself. 31 is prime, but if your list of primes had started out as 2 and 7, you would have constructed the number 15. That is not prime, which would tell you there is at least one prime number less than 15 that is not in your set.

  4. Assuming something and then proving it later is called Mathematical Induction, and it is widely used in mathematics. It is a little hard to grasp sometimes, but when you do grasp it it makes sense.

  5. I thought mathematical induction was a specific type of proof:

    * If we can show that whenever something is true for n, it is also true for n+1,
    * and if we can show that our something is true for n=1,
    * then it must be true for all natural numbers.

    But assuming something and then proving it later — whatever that technique might be called — is certainly a useful trick for some problems, as long as one does actually come back and prove it. Usually my students want to just assume and leave it at that.

  6. Irrational numbers are also rational because we can speculate about objects with irrational sizes, for example, we can speculate about circumference with irrational length or about square with a diagonal of irrational length, but they are ideal, because in reality does not exist object with irrational size.
    Length of the real circumference will never be equal to the length of a perfect circumference, and length of diagonal of a real square will never be equal to the length of diagonal of a ideal square.
    Therefore, there will not be a problem of squaring the circle and a problem of incommensurability of the side and diagonal of the real square.
    Incommensurability problem is just for ideal square.
    Therefore irrational numbers should be renamed in ideal numbers, and rational numbers should be renamed in real numbers.
    The set of real and ideal numbers will together constitute the set of rational numbers (for mathematics needs), but only real numbers (ex-rational) will be needed in practice.

  7. Terminology grows from history, and it’s not easy to change after it’s established. The words “rational” and “irrational” seem pretty well settled to me. Students always have trouble remembering them at first, but the connection to “ratio” provides an opportunity to review that fundamental concept.

    I’m not sure your suggested terms are an improvement. There is no perfect “2” in the real world, either — all numbers are “ideal” in that sense. Real world measurement is always an approximation, even if you use an electron microscope. But math lives in the ideal world of imagination, where both 2 and its square root are equally real.

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