Why Can't the Square Root of a Negative Number Be a Real Number?

Date: 12/07/2003 at 14:56:08
From: Stephanie
Subject: square roots

Can you explain why you can't find the square root of a negative 
number on the real number line?

Date: 12/07/2003 at 19:28:07
From: Doctor Schwa
Subject: Re: square roots

Hi Stephanie,

The square root of a number is something that, multiplied by itself,
gives the number you're taking the square root of.

Suppose the square root is positive.  Then, if you multiply it by
itself, you will get a positive result since a positive times a
positive makes a positive.  So, the number you're taking the square
root of must be positive.

How about if the square root is negative?  Then, if you multiply it by
itself, the number you're taking the square root of still must be
positive, because a negative times a negative is also a positive.

How about if the square root is zero?  Then the number you're taking
the square root of is zero since 0 * 0 = 0.

That covers the whole number line, so any square root you can find on
the real number line must be the square root of a positive number or
zero.  It's impossible for it to be the square root of a negative number.

This is why in the context of real numbers we say that you cannot take
the square root of a negative number.  There is no real number that
you can square and get a negative result. 

Does that help?

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/