Square root of -1

From: Petrescu

     Hey Dr. Math - Do you know the square root of -1?

Alex  Petrescu            

Date: Mon, 28 Nov 1994 16:51:23 -0500 (EST)
From: Dr. Sydney
Subject: Re: your mail

Great Question!  Thanks for writing Dr. Math.  The square root of -1 is a
special number we call "i," and it is a member of a set of numbers called
the imaginary numbers.  The imaginary numbers are the set of all numbers
that are the square root of negative numbers.  But these are just
definitions.  Let's try and get a handle on what the square root of -1
really means:

        You know that if you have a positive number, say for instance, 9,
that the square root of that number times itself gives you the original
number.  So 3=sqrt(9) because 3x3=9.

        You have probably also learned that a positive number times a positive
number yields a positive number and that a negative number times a negative
number yields a positive number.  Because of this, it seems to follow that
a number has a square root only if it is positive.  The catch here is that
these rules are based on 8 fundamental axioms which describe only the real
numbers.       

        Other sets of numbers need not obey these rules.  The imaginary
numbers, in particular, do not, because any imaginary number squared will
produce a negative number.  The square root of -1, "i", is in this set of
imaginary numbers because i^2   = -1.  

        The place where you will most likely see imaginary numbers pop up in
your day to day life is when you are solving equations, for instance the
equation:

       x^2 +  1 = 0

This equation holds for x = i or x = -i.

Since i is defined to be the square root of -1, can you figure out what i^2,
i^3, i^4, I^5, etc. is, in terms of 1  and i?  Do you see a pattern?

Another question to consider:  What do all imaginary numbers have in common
(think i!!)

I hope this helps give you a better grasp on the concept of i. If you are at
all confused or have any more questions, please feel free to write back.  

Have a good day!
                                --Sydney


Date: Wed, 9 Nov 1994,

Alex -

Good question!  Lord knows it plagued mathematicians for quite a long 
time. As hard as they tried, they couldn't take the square root of -1, or any
negative number for that matter, because the answer squared should be the
original number, and any number squared is positive.  It was a mess!  Some
got really aggravated and decided that it was a lot saner and less
stressful to just become philosophers.  Others stuck with it, although they
too got grumpy.

In fact, they got so sick of it all that they decided that they needed to
invent a new kind of number... the imaginary number.

The imaginary number (i) is, by definition, the number that is the square
root of -1.   The square of i is therefore -1.

  2
(i  = -1) or (i = square root (-1))

Therefore, the square root of -4 isn't 2, but it is 2 times i.  Whenever
you have to take the square root of a negative number for now on, you can
just get rid of the negative sign by adding i to your answer.

Therefore, the answer to your question is that the square root of -1 is the
imaginary number i.

Phil, Dr. Math

Date: Wed, 9 Nov 1994 18:10:04 -0700 (MST)
From: Petrescu

     Hey Dr Math - the square route of -1 = i

Alex  Petrescu