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### Equation without a Solution

```
Date: 11/14/2001 at 18:25:37
From: Steven and Richard
Subject: Sqrt(x) = -2, order of operations, and other problems with
the square root

What is the solution to the equation sqrt(x) = -2 ?

Some people say that there is no solution, since the square root is
defined to be the side of a square with a given area. We allow for
negative area by allowing these imaginary numbers. If I were to have a
negative side the area would most certainly be imaginary, or would it?

We thought of looking at sqrt(4i^4). Using the laws of exponents
first, the answer is most definitely -2. Using the laws of imaginary
numbers first, the answer is 2. Is there an order of operations for
which law to use first?

Where is the breakdown in the square root? I have ceased calling
the square root a function because 4 = 4i^4 and y = sqrt(x) seems to
be providing me with two real solutions. If we allow negative area,
don't we have to allow negative sides?
```

```
Date: 11/14/2001 at 22:40:10
From: Doctor Peterson
Subject: Re: Sqrt(x) = -2, order of operations, and other problems
with the square root

Hi, Steven.

When you start thinking about negative or imaginary numbers, it's best
to think algebraically rather than geometrically. A square root of a
number is defined as any number whose square is that number. A
positive number has two square roots, one positive and the other
negative. When we write the radical sign (which we're representing as
"sqrt"), that represents only the POSITIVE square root of the number;
so the two square roots of 4 are called sqrt(4) and -sqrt(4). We do
this so that the square root is a function with one value, not
something that somehow takes two values at once.

sqrt(x) = -2

has no solutions, simply because by definition sqrt(x) is always
positive.

This has nothing to do with imaginary numbers, which are the solutions
of equations like

x^2 = -2

Although these look similar, they have no real solutions for entirely
different reasons.

Going back to geometry, if a square could have sides with negative
lengths (which it can't), then the area would still be positive, since
the product of two negative numbers is positive. Only with imaginary
sides (?) could a square have a negative area.

Now, when you move into complex numbers, there is NO square root
FUNCTION. You're exactly right: any definition that chooses a single
square root for every complex number will be inconsistent, in the
sense that

sqrt(ab) = sqrt(a) sqrt(b)

will not always be true. It's only a fortunate convenience that a
square root function over the reals can be defined. This means that
you can't look to the complex numbers for a solution to your equation,
because any equation involving square roots must allow only reals.

You will be interested in these pages that deal with various issues
you have raised:

Square Root of 100 - Dr. Math archives
http://mathforum.org/dr.math/problems/cry.04.15.99.html

What is i? - Dr. Math archives
http://mathforum.org/dr.math/problems/toeti24.html

False Proofs, Classic Fallacies - Dr. Math FAQ
http://mathforum.org/dr.math/faq/faq.false.proof.html

(see the link near the bottom to "1 = 2: A Proof using Complex
Numbers").

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Number Theory

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