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© Copyright 1998, Jim Loy
Pythagoras (or one of his students) is said to have discovered irrational numbers (see definitions, below). The way this was done, was to show that the square root of 2 could not be expressed as any whole fraction m/n. The story is that they kept this fact a secret, as it was too dangerous. And, supposedly, one of the Pythagoreans let the secret out, and was executed for this crime. This story may not be true, as the Pythagoreans had many enemies, who might have spread this rumor.
Here's the proof. We start by assuming the square root of two, shown as sqr(2) here, is equal to some fraction m/n. We intend to contradict this assumption:
sqr(2) = m/n (a fraction, reduced to its lowest terms) 2 = m^2/n^2 m^2 = 2n^2 (So, m is a multiple of 2, call it 2q) 4q^2 = 2n^2 2q^2 = n^2 (So, n is a multiple of 2)
So, both m and n are multiples of two, which is impossible, because m/n was reduced to its lowest terms. So, we have proved that the square root of two cannot be expressed as a fraction, i.e. it is irrational.
Note: In the third step, above, we are dealing with integers m and n. And we make use of the fact that the square of an even number is even (and the square of an odd number is odd). The fifth step is the same.
Closure is a fairly important principle in algebra. The positive integers are closed under addition, for example. That means that a positive integer added to another positive integer gives us a positive integer. Closure allows us to deduce some simple properties of numbers. Positive integers are not closed under subtraction (you may get zero or a negative integer). Integers are not closed under division. Well, here we see that rational numbers (whole numbers and fractions, some of them improper fractions) are not closed under square roots. This complicates the situation, because it shows us that we have a whole new set of numbers, the irrationals. And, set theory shows us that the irrationals vastly outnumber the rationals.
When I was in grade school, I wondered about the square root of two. I did not know that it was called the square root of two. Here is what I knew: If I walked one block east and one block north, it is shorter to walk diagonally across the block. People called that path a "short cut."
In the diagram, it was obvious that the diagonal distance was less than 2, which was the long distance, by walking on the streets (There were few sidewalks in my part of town). And the diagonal was obviously greater than 1, the length of the side of the square. My first guess was that the diagonal was 1.5.
I already knew that distances and lengths often involved fractions. I had a ruler. And it was marked in fractions of an inch.
Well, if I drew a square, with each side one foot long, and measured the diagonal with my ruler, I got really close to 1 foot, 5 inches (I may have thought that it was exact). I was essentially getting a value for the square root of 2 of about 1.41666... The actual value is 1.4142135... I was close.
But, I wanted to know why. At the time, I could not figure out any general rules here. In high school, I learned about square roots, and the Pythagorean Theorem. And then it became clear. Wouldn't it have been nice if I had discovered the Pythagorean Theorem, all by myself?
Incidentally, my 1 foot square, as a model of a 1 city block, implies that these two objects are similar (proportional). That idea was natural, because I had been subjected to maps with scales that said something like "1 inch=100 miles." My scale drawing was just another map ("one foot=one block").
I received email asking for a proof that the square root of 3 is irrational. The return email address didn't work, so I was not able to answer. But the proof that the square root of 3 is irrational is almost identical to the proof that the square root of 2 is irrational. In fact, the above proof works for all non-squares!
Call this arbitrary non-square a. Then the middle step of the proof becomes "m^2=an^2. So, m is a multiple of a, call it aq." It is not quite that simple, as a might have squares as divisors; then an^2 can be written bc^2n^2, and this is just another an^2 (different a and n) in which a now has no square divisors. Then the rest of the proof is similar to the proof with m^2=2n^2.
So the square root of any non-square is irrational. If a is a square, then m^2 is a multiple of a, but m may not be. You can probably think of an example.
Definitions: An irrational number is one that cannot be expressed as a fraction (proper or improper). A rational number can be expressed as a fraction (proper or improper). A proper fraction is one in which the numerator is less than the denominator. An improper fraction is one in which the numerator is greater than, or equal to, the denominator. The numerator and denominator are, of course, integers. A prime number is an integer greater than one, which is not divisible by any positive integer other than itself and one. A composite number is a positive integer that is not prime and not one. The numbers that divide a composite number are called factors or divisors. n is a factor of n, but is not called a proper factor of n.
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