Roots or Radicals

Because , we say that 3 is the square root of 9, written

Thus is defined to the be the number whose square is 9. Now, you may recall that as well. Thus, it would appear that every positive number has two square roots – a positive number and the same number with a minus sign. By convention, the square root symbol, , without an explicit sign is reserved for the positive square root. To indicate the negative square root, you need to write an explicit minus sign, as in but

In symbols, then, for any positive number b, the symbol has the meaning

and

Remarks

(i) Some numbers, such as 1, 4, 9, 16, 25, etc., are the squares of whole numbers, so their square roots are whole numbers:

, , , , etc

For this reason, the numbers 1, 4, 9, 16, 25, etc., are said to be perfect squares. Numbers which are not perfect squares still have square roots, but their square roots are not whole numbers. In fact (to the horror of those ancient Greeks who first discovered this), all simple whole numbers which aren’t perfect squares not only don’t have whole number square roots, but their decimal parts go on for an infinite number of digits without coming to any end. They are an example of what mathematicians call irrational numbers. So, for example, your calculator will tell you that

= 24 2 1.414213562

But, if you were to carefully multiply (1.414213562)= 1.414213562 x 1.414213562 without any error, you would get an answer which is not exactly 2. (When I tried this, I got 1.999999998944727844 – close, but not exactly 2.) That’s because 1.414213562 gives only the first 9 decimal places of the actual value of , and so is only just quite a good approximation to the exact value of the square root of 2. We will use the symbol to represent the exact value of the square root of 2, even if we can never write this number down exactly in decimal form.

(ii) Recall the rules for multiplying two signed numbers. If both numbers have the same sign, then the result is positive. This means that we can never find an ordinary (“real”) number which can be multiplied by itself or squared to give a negative result. But this means that negative numbers have no square roots. So, for the number system we ordinarily use in basic technical applications (the so-called real number system, quantities such as , , etc. must be considered to be undefined or non-existent. A common error is to assume that square roots of negative numbers are just negative square roots. That is, for example, that

But you can easily see that this cannot be correct by simply checking:

Since squaring –3 gives +9, then –3 cannot be the square root of –9. The same sort of thing will be true for all negative numbers. Mathematicians have developed the so-called complex number system in which negative numbers do have meaningful square roots, but that very useful topic is far beyond the scope of this text. For now, if you are solving a problem and in the process, the square root of a negative number arises, you must first check to ensure you haven’t made an arithmetic error someplace. If no error can be found, then the occurrence of the square root of a negative number must mean that the problem actually has no real solution.