Math CentralQuandaries & Queries


Question from Cheryl, a teacher:

Does every number have a square root?

Hi Cheryl,

Negative numbers don't have square roots. If a number is positive its square is positive, if it is negative its square is positive and if it is zero its square is zero. Hence there is no number whose square is negative.

For positive numbers, even positive integers, the answer is not so clear and some of the history is fascinating. In the history of numbers the integers came first, in fact we may come into the world with some integers built in. Research by psychologists has shown that babies, as young as three or four days old, can tell the difference between collections of two and three items.[1] If your number world is the world of integers then not every number has a square root. There is no integer whose square is 2.

The Greeks of Pythagoras' time were comfortable with the square root of 2 because they could construct it. If you have a square figure with side length of 1 unit and the diagonal has length d units then then using Pythagoras' theorem

12 + 12 = d2

Thus d2 = 2 and hence d is the square root of 2. In fact not only is the square root of 2 not an integer it is not even a rational number. Probably the first person who realized that root 2 is irrational was Hippasus, one of the of the disciples of Pythagoras. Legend has it that Hippasus made his discovery at sea and was thrown overboard by fanatic Pythagoreans[2]. Hence if your number world is the world of rational numbers then not every number has a square root. There is no rational number whose square is 2.

This problem is one of the reasons that mathematicians defined the real numbers. If your number world is the world of real numbers then every non-negative number has a square root.

I hope this helps,

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