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A More Formal Definition of the Imaginary Unit i

Date: 01/12/2004 at 05:29:39
From: Daniel Russell
Subject: Contradiction in Simple Imaginary Number Algebraic Equation

Although I have read the few pages on your site that deal with 
problematic imaginary number proofs, I feel that I am still not
satisfied why the following is not valid:

1.     1/i    = sqrt(1)/sqrt(-1)
2.  1/i * i/i = sqrt(1/-1)
3.    i / -1  = sqrt(-1)
4.        -i  = i

I read somewhere on this site that we are not allowed to combine roots
unless the arguments of the roots are both positive.  But this rule is
blatantly ignored all the time!  For instance, it is common practice
to do all the steps on the left side of the equality sign in my proof.

Another example is the idea that an imaginary number such as
sqrt(-4) = sqrt(4) * sqrt(-1) = 2i.

Why is this allowed, yet the contradictory proof above is not?  What
exactly are the rules of imaginary numbers?  When can we do operations
with imaginary numbers, like multiply, divide, etc, and with what
kinds of other operands can we do these operations?

Date: 01/12/2004 at 06:14:43
From: Doctor Pete
Subject: Re: Contradiction in Simple Imaginary Number Algebraic Equation

Hi Daniel,

You bring up a very important point when dealing with complex numbers,
and the only way to give you a satisfactory answer is to say that the
"usual" definition of

     i = Sqrt[-1]

is really not a formal one, and does lead to a great deal of
confusion.  The actual definition is much more clear on this issue,
and is well-understood by mathematicians.

A complex number is an ordered pair (a,b) of real numbers, with the
following rule for addition:

     (a,b) + (c,d) = (a + c, b + d)

and the following rule for multiplication:

     (a,b) * (c,d) = (ac - bd, ad + bc),

where addition and multiplication on the left-hand side are performed
in the domain of the complex numbers, and addition and multiplication
on the right-hand side is naturally done on reals.

Thus defined, it is obvious that the complex numbers contain the real
numbers as a proper subset, since for all real numbers a,

     a = (a,0).

(To verify this, all one needs to do is show that the rules for
addition and multiplication in the complex numbers correspond to
regular real addition and multiplication for numbers of the form (a,0).)

Furthermore, let us define

     i = (0,1).

Then the computation

     i^2 = (0,1) * (0,1) = (0*0 - 1*1, 0*1 + 0*1) = (-1,0) = -1

shows that i has the property that its square is equal to -1.  We can
then proceed to observe that if a and b are real numbers, we have

     a + bi = (a,0) + (b,0)(0,1) = (a,b)

so our definition of complex numbers as ordered pairs is consistent
with our "usual" notation.

Set up in this manner, there is absolutely no ambiguity whatsoever
with the imaginary unit i, since we have not talked about square roots
at all!

The reason for the "paradox" in your original question is that there
are two complex numbers, (0,1) and (0,-1), whose squares are equal to
-1.  In fact, there are exactly two other complex numbers whose
squares equal to any given complex number, with the exception of zero.

Thus, we call the square root function a "one-to-two" function, just
like the square function is a "two-to-one function."  Formally
speaking, the function

     f(z) = z^2

maps two values in the domain to a single value in the range, and the

     g(z) = z^(1/2)

maps every value in the domain to two values in the range.  You may
also be interested to know that there are certain complex functions
which are "many-to-one" and "one-to-many," such as

     f(z) = e^z
     g(z) = log(z),

the complex exponential and logarithm functions.  So we must be very
careful when making a statement like

     log(-1) = Pi*i,

because in truth, the logarithm of -1 is actually an infinite set of 
values, each differing from the others by a multiple of 2*Pi*i.

I know my discussion does not pinpoint the fallacy in your question,
and so sidesteps the issue entirely with a formalized definition, but
the reason for this is that I want to re-conceptualize your notion of
complex numbers as having anything to do with square roots of negative
numbers, and instead encourage you to think of them as ordered pairs
of reals, which obey particular rules for multiplication and addition. 
Metaphorically speaking, I don't want to patch the leaky boat you've
found yourself sailing in, but rather give you a new vessel--a sturdy,
streamlined one.

Undoubtedly you will have more questions about our discussion, as I
know you'll do your own exploration on this fascinating subject--feel
free to write back!

- Doctor Pete, The Math Forum 

Date: 01/12/2004 at 07:04:22
From: Daniel Russell
Subject: Thank you (Contradiction in Simple Imaginary Number Algebraic

Thank you very much!  That answer was far better than I expected, and
exactly what I needed.  I hope it will make its way onto the site
because just about every student I know has been taught incorrectly
that i = sqrt(-1).
Associated Topics:
College Definitions
College Imaginary/Complex Numbers

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