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
other math teachers and we don't all agree.  Please help.  Thanks.

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.