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 socalled real number system,
quantities such as , ,
etc. must be considered to be undefined or nonexistent. 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 socalled
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.
