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### Simplifying and Working with Imaginary Numbers

```Date: 04/11/2008 at 11:05:07
From: L
Subject: simplifying imaginary square roots

I gave a question on a test that said to simplify the square root of
50 over the square root of -5.  The answer I accepted was -i times the
square root of 10 but one of my students claims he is right with the
answer of i times the square root of 10.  I have spoken with many

```

```
Date: 04/11/2008 at 11:37:35
From: Doctor Ian
Subject: Re: simplifying imaginary square roots

Hi L,

There doesn't seem to be universal agreement on whether

sqrt(4)

is equal to just 2, or equal to both 2 and -2.  This site,

MathWorld - Square Root
http://mathworld.wolfram.com/SquareRoot.html

uses "principal square root" to refer only to the positive one; I
think that makes sense, but it still leaves open the question of
convention, i.e., when someone writes "sqrt(x)", does this default to
the principal square root, or not?

I generally go for the more inclusive definition--especially once
imaginary numbers are being considered, since the actual definition of
i is not

i = sqrt(-1)

but rather

i^2 = -1

However, as you've found, not everyone agrees.

Having said that, it seems to me that the student's reasoning is as
important as the answer he gave.  Did he explain how he got his answer?

I would simplify the expression this way.  If I use psr(x) to refer to
the principal square root, then

sqrt(50)
--------
sqrt(-5)

sqrt(5) * sqrt(10)
= ------------------
sqrt(5) * sqrt(-1)

sqrt(10)
= --------
sqrt(-1)

+/- psr(10)
= -----------
+/- psr(-1)

psr(10)    -psr(10)    psr(10)    -psr(10)
= ------- OR -------- OR ------- OR --------
i          i          -i          -i

which is to say,

psr(10)
+/- -------
i

So personally, I'd be inclined to mark it wrong if EITHER answer was
omitted.  :^D

Does this help at all?

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

```

```
Date: 04/11/2008 at 12:43:53
From: Doctor Peterson
Subject: Re: Thank you (simplifying imaginary square roots)

Hi, L.

I'd like to add a little to what Dr. Ian said.

A lot depends on what the student has been taught--what does your text
do with square roots of negative numbers?  Although it is common to
take sqrt(x) to mean the principal root when talking about real
numbers only (roots of positive numbers), there is no such thing when
working with complex numbers, so if (as usual) you want any expression
to have only one value, you really have to refuse to define sqrt(-5).
(The alternative is to take Dr. Ian's position and take sqrt as a
multiple-valued function, which is not generally done at an elementary
level.)

The reason for this is that if you take the square root of a negative
number as having a single value, then no matter what rule you choose
to select that value, there will be some cases where it is not true
that sqrt(ab) = sqrt(a)sqrt(b), which is a highly desirable rule to
have.

That means that problems involving such a root are technically
invalid.

However, it seems to be common for texts to define

sqrt(-x) = i sqrt(x)

when x>0, so that they are allowed to write such roots, and then tell
students that when they see such an expression, they must IMMEDIATELY
translate it to the form on the right, BEFORE applying any rules such
as the invalid product rule above, or the equivalent quotient rule.
That makes sqrt(-5) essentially no more than a shorthand for i
sqrt(5), rather than an expression you can manipulate by the usual
rules.

So, under these rules, your problem works this way:

sqrt(50)   sqrt(25)*sqrt(2)   5 sqrt(2) * i sqrt(5)   5 sqrt(10)
-------- = ---------------- = --------------------- = ---------- i
sqrt(-5)      i sqrt(5)       i sqrt(5) * i sqrt(5)       -5

= -i sqrt(10)

You could also (with care) do it this way:

sqrt(50)   sqrt(50)*sqrt(-5)   sqrt(2)*sqrt(25)*sqrt(5)*i
-------- = ----------------- = -------------------------- = ...
sqrt(-5)   sqrt(-5)*sqrt(-5)              -5

But you CAN'T do this:

sqrt(50)   sqrt(50)*sqrt(-5)   sqrt(-250)   sqrt(25)*sqrt(-10)
-------- = ----------------- = ---------- = ------------------ =
sqrt(-5)   sqrt(-5)*sqrt(-5)    sqrt(25)    sqrt(25)

= sqrt(-10) = i sqrt(10)

The error is in the simplification in the denominators where an
invalid rule is invoked.

So only your answer is correct if you have taught that you can treat
sqrt(-5) as i sqrt(5).  If you haven't dealt with this issue at all,
then the question itself is really invalid.  The student's answer is
not really valid (giving only the one answer as correct), but he can't
be blamed for it if he hasn't been taught to be careful with this kind
of problem.

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

```

```
Date: 04/11/2008 at 15:01:33
From: L
Subject: Thank you (simplifying imaginary square roots)

Excellent answer.  The student was taught that the sqrt(-5) is i times
the sqrt(5), so thank you.
```
Associated Topics:
High School Functions
High School Imaginary/Complex Numbers
High School Square & Cube Roots

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