ORIGIN OF THE CONCEPT OF IRRATIONAL NUMBERS
The origin of the concept of an irrational number lies in the following fact discovered by the ancient Greeks.
Fact. In a square with sides of unit length there is no rational number equal to the length of its diagonal.
Proof. Consider a square with sides of length 1 and a diagonal of length a. See Fig. 1. According to the Pythagorean Theorem
a2 = 12 + 12 = 2 .
But there is no rational number whose square is equal to 2. We prove this as follows. A rational number is a number that can be written as the quotient of two integers. Thus we wish to prove that there is no quotient p/q whose square is equal to 2 where p and q are positive integers. Suppose there are positive integers p and q such that (p/q)2 = 2 . We can assume that p and q have no common factor since otherwise we could simplify the fraction. If (p/q)2 = 2, then p2 = 2q2. Thus p2 is an even number since twice any number is an even number i.e. the factor 2 in 2q2 reveals it as even. But if p2 is an even number then p must also be an even number. (Why? The square of any odd number is also odd and if p were odd its square would be odd). Since p is even let b = p/2. Then p2 = 4b. Thus 2q2 = 4b and q2 = 2b. Thus q2 is even and q must also be even. Thus, based on our original suppositions, we have deduced that both p and q are even. Thus both must be divisible by 2. But this contradicts the supposition that p and q have no common factor. This contradiction proves that there are no positive integers p and q such that (p/q)2 = 2 .
Prove. The square of any odd number is odd.
Proof. The odd numbers are given by the algorithm
2n + 1 where n = 0, 1, 2, ....
The square of an odd number is given by
(2n + 1)2 = 4 n2 + 4n + 1
which must be odd since 4 n2 + 4n is even.
Q. What is the significance of the fact that there is no rational number that gives the length of the diagonal of a square with unit sides?
A. A rational number, defined as a quotient of two integers, is simply a devise invented by man for designating continuous quantities. Man came up with this idea as a technique for naming any continuous quantity. The significance of the above fact is simply that this devise of using the quotient of two integers for naming continuous quantities is incapable of naming some quantities. Those quantities that it is unable to name are the irrational numbers. Th rational numbers plus the irrational numbers constitute the real numbers.