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### Square Root of a Negative Number

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Date: 01/25/97 at 19:04:00
From: Dustin Pedlar
Subject: Square root of negative numbers

Is it possible to find the square root of a negative number and if so,
to what number system do these square roots belong?  Could you please
explain how this number system works?
```

```
Date: 01/25/97 at 20:47:34
From: Doctor Wallace
Subject: Re: Square root of negative numbers

Hi Dustin!

As you no doubt know, a square root is a number that when multiplied
by itself is equal to a given number.  For example, 4 is the square
root of 16, since 4 x 4 = 16.  Note, however, that -4 x -4 = 16, too.
We call 4 the positive square root of 16, and -4 the negative square
root of 16.

Now, you want to know if we can find the square root of a negative
number.  Let's take -16.  We need to find a number, call it x, such
that:
x times x (x^2) = -16

Now, we know that any number times itself must be positive, not
negative.  Therefore, there is no such number x in the set of real
numbers.

A number x is defined, however, in the set of complex numbers.  The
complex numbers are a superset of the real numbers.  That is, the
complex numbers form a bigger set.  The reals are a subset of the
complex.

A complex number has the form a + bi, where a and b are real numbers
and the i is a special number.  The "a" is called the real part; the
"bi" is called the imaginary part.  If we let a equal 0, then we have
an imaginary number.  The set of imaginary numbers is also a subset of
the complex numbers.  If we let b equal 0, then we have a regular real
number.  This is why the reals are a subset of the complex: the reals
are just complex numbers that all have b=0, that is, no imaginary
part.

Now, the number i is defined to be equal to the square root of -1.
This means that i^2 (i squared) is equal to -1.  So now we can find
the square root of -16.

Since -16 = (-1) 16, we can write:

sqr(-16) = sqr(-1) times sqr(16)      (property of square roots)
sqr(-16) = i times 4

This is usually written as 4i.  We can check by squaring 4i.  We get
4 x 4 = 16 times i x i = sqr(-1) times sqr(-1) = -1, giving 16 times
-1 or -16.

There is much more to the complex and imaginary sets of numbers than I
can go into here.  There are entire branches of mathematics (like
complex analysis) which deal with these numbers.  For more
information, consult the math section of your local library or check

http://mathforum.org/dr.math/faq/faq.imag.num.html

-Doctor Wallace,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Exponents
High School Imaginary/Complex Numbers
Middle School Exponents