Re: Dividing complex numbers

In article <dbaf88e0-dc86-4c52-8628-a478e154c7db@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
jehugaleahsa@xxxxxxxxx <jehugaleahsa@xxxxxxxxx> wrote:
Hello:

I have been reading about basic operations on complex numbers.

However, now that I have been away from complex numbers for so long, I
have a really simple question.

You divide two complex numbers by taking the conjugate of the
denominator and multiplying the dividend and denominator by it.

Now, I assume this is valid because a complex number divided by itself
is equal to 1 + 0i.

How is it that we can say it is 1 + 0i? Is it an axiom? Otherwise, I
feel like we are solving division by using division.

Division is multiplication by the multiplicative inverse.

That is: if c is a nonzero complex number, then c^{-1} is defined to
be the unique complex number (if one exists) such that c*c^{-1}=1+0i.

Then, if c and d are complex numbers and d is nonzero, then "c/d"
really means the multiplication c*(d^{-1}).

You can verify directly that if a+b*i is a complex number with a,b
real numbers, not both zero, then (a/(a^2+b^2)) - (b/(a^2+b^2))*i is a
multiplicative inverse of a+b*i.

As it happens, this means that (a+b*i)^{-1} is equal to the number

(1/(a^2+b^2))(a-b*i)

(the first factor is just a real number, so there should be no problem
there).

So: if you are dividing x+yi by a+bi (a,b,x,y real numbers), then:

(x+yi)/(a+bi) = (x+yi)*(a+bi)^{-1} (the definition)
= (x+yi)*(1/(a^2+b^2))(a-bi) (the value of (a+bi)^{-1})
= (x+yi)(a-bi)(1/(a^2+b^2)) (multiplication commutative).

As it happens, the number a^2+b^2 is, of course, the norm of a+bi, and
is equal to (a+bi)(a-bi). So we can replace (a^2+b^2) by (a+bi)(a-bi),
and then what you have looks like

(x+yi)/(a+bi) = (x+yi)(a-bi)/[(a+bi)*(a-bi)].

You aren't really ->defining<- multiplication that way; that's one way
to actually COMPUTE it. Don't confuse the algorithm to do something
with the definition of the something.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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