Monday, January 27, 2014

Rational Numbers and the Square Root of 2

I am a mathematician. Please allow me to share with some mathematics which I have found to be both enduringly fascinating and very accessible: I present a discussion on rational numbers and the square root of 2.

Rational numbers---or just rationals---are the result of dividing one integer by another. Examples are 1/2, -1/5, and 23/11; perhaps less obvious examples are 1, 2, and 3, as you can always write 1 = 1/1, 2 = 2/1, and 3 = 3/1. Essentially, rationals are all the numbers that can be written as a fraction.

Generally, we write rationals in reduced or simplest form. 2/4, for example, can be simplified by dividing the top and bottom of the fraction by 2; thus 2/4 = 1/2. Notice that in reduced form, the top and bottom numbers cannot both be even. If they were, just as in the example of 2/4, you could cancel out the common factor of 2 to find a more-reduced form.

Any rational can always be written in reduced form. Rationals have a pleasing property: any basic arithmetic operation (addition, subtraction, multiplication, or division) on two rationals always yields another rational. But is there any operation which might give a different kind of number?

Consider multiplying a rational by itself, or squaring. 2 squared is 4. We might reverse this statement and ask the following: what number squared equals 4? This question defines the square root; the square root of 4 is 2. But what about the square root of 2 itself? Is there any rational which is the square root of 2?

Now we reach the heart of the matter. In fact, no rational could be the square root of 2. To see why, we consider what would happen if it were.

What if the square root of 2 were rational? Then there must be two integers---let's call them p and q---such that "the square root of 2 = the reduced fraction p/q". Equivalently, we can say that "q times the square root of 2 = p". If we square this statement, we say that "q squared times 2 equals p squared".

From this last statement, we note something interesting: p squared must be even. Actually---as an odd number squared is odd---p itself must be even. What does this imply about q? If p is even, then p must equal 2 times something; let's say that "p = 2 times n". Then our statement "q squared times 2 = p squared" can also be written as "q squared times 2 = 4 times n squared". If we divide this equation by 2, we may state that "q squared = 2 times n squared". This statement is familiar. Just as above, we infer that q squared, and thus q, is even.

But now we have found something quite troubling: if the square root of 2 were rational and equaled p/q, then both p and q must be even. If both p and q are even, however, then p/q cannot be a reduced fraction. Since every rational can be written as a reduced fraction, the square root of 2 cannot be a rational number.

Of course, we might wonder what sort of number the square root of 2 is, if it is not rational. This question leads to the irrational numbers. Other questions that could be asked lead to other sorts of numbers such as imaginary numbers, transcendental numbers, quaternions, and the list goes on. This, to me, encapsulates the enduring fascination of mathematics: good questions are always rewarded.


Austin Amaya, PhD
austin.amaya[AT]gmail.com
Reston, Virginia, USA

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