Associated Topics || Dr. Math Home || Search Dr. Math

```
Date: 07/12/2000 at 17:07:26
From: Amy
Subject: i(imaginary)

Hi,

This question appeared in a math problem in school:

i * sqrt(-98) - sqrt(98)

This is how it is meant to be solved (according to our book):

i * sqrt(-98) - sqrt(98)
= i * i * sqrt(98) - sqrt(98)
= -1 * sqrt(98) - sqrt(98)
= -2 sqrt(98)   < the answer in the book

However...I solved it another way and got 0 as an answer (not listed
as an answer in the book):

i = sqrt(-1)  so...
i * sqrt(-98) - sqrt(98)
= sqrt(-1)* sqrt(-98) - sqrt(98)
= sqrt(98) - sqrt(98)
= 0

Can you multiply square roots of negative numbers? You can multiply
radicals when they're positive.  It looks as if it should work, but
the answer in the book is the first one (more simplified but you get
the idea).

Thanks for the help.
Amy
```

```
Date: 07/12/2000 at 22:44:20
From: Doctor Peterson
Subject: Re: i(imaginary)

Hi, Amy.

What you did is not quite legal, but there's an interesting twist:
what they did isn't quite legal either. Before we look at why, I'll
simplify the problem to make it obviously wrong. Watch carefully
(nothing up my sleeve):

i = sqrt(-1)
i*i = sqrt(-1)*sqrt(-1)
-1 = sqrt(-1*-1)
-1 = sqrt(1)
-1 = 1

Think maybe there's something wrong here?

The only questionable step is where I did just what you did, assuming
that

sqrt(a)*sqrt(b) = sqrt(a*b)

This is valid when a and b are positive, because the square root of a
is defined as the positive number whose square is a; since the left
side is positive, and its square is ab, it is, in fact, the square
root of ab.

But when we move into the complex numbers, we can no longer so easily
define the square root as a function. There is no such thing as a
"positive complex number." Rather, when we say i is sqrt(-1), we
really mean i is one of two numbers whose square is -1, and there's
no way to distinguish them. As a result, we can no longer be sure of
the sign; we can only say that sqrt(a)*sqrt(b) is _a_ square root of
ab, but not which one. All we can really say, in my "proof" that
-1 = 1, is that -1 is equal to + or - 1, which of course is true.

i * sqrt(-98) - sqrt(98)

I've just told you that we can't uniquely define the square root for
all complex numbers; sqrt(-98) can equal either +-i sqrt(98). So in
answer is not complete. Of course, you can choose to define the
square root of a negative number to be a positive multiple of i; but
if your book hasn't done that, it doesn't have a right to ask this
question.

You may enjoy this page, which you can get to from our FAQ on "false
proofs"; it has a nice explanation of the fallacy I've described:

1=2: A Proof using Complex Numbers - University of Toronto
http://www.math.toronto.edu/mathnet/falseProofs/second1eq2.html

- Doctor Peterson, The Math Forum
http://www.mathforum.org/dr.math/
```
Associated Topics:
High School Imaginary/Complex Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search