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Zero and Imaginary Numbers

Date: 07/18/2001 at 23:35:24
From: Angela Cherie Aninao
Subject: Zero and imaginary numbers

My teacher said that zero divided by an imaginary number is zero. Is 
this true? How can the answer be in the real number line when the 
divisor can't be found in the real number line?

Date: 07/19/2001 at 10:26:32
From: Doctor Rick
Subject: Re: Zero and imaginary numbers

Hi, Angela.

Your problem is with 0/i = 0. You'd have the same difficulty with 
0*i = 0, I assume. These are pretty straightforward: zero times 
anything, and zero divided by anything except zero, is zero. That 
doesn't change when you go from real numbers to complex numbers. 

But what do you think about i*i = -1? Here *both* numbers are 
imaginary, and the answer is real!

Let's forget about complex numbers for a moment. Hmm ... I also notice 
that -1 times -1 is 1. How can the answer be positive when neither 
factor can be found in the positive numbers?

Do you see how this is similar to i*i = -1?

There is nothing wrong with the result of an operation on a "new" kind 
of number being an "old" kind of number. We invent the "new" numbers 
so that we can solve problems that couldn't be solved using the "old" 
numbers alone. But the "new" numbers aren't a group off by themselves 
that have nothing to do with the "old" numbers. Rather, we *extend* 
the number system by adding the "new" numbers to it. A whole number 
(positive integer, the first kind you learned about) is a member of 
the integers just as much as a negative number (which was once "new" 
for you) is an integer. Likewise, a real number is a member of the 
complex numbers just as much as an imaginary  number (your current 
"new" type) is.

What I'm getting at is that the result of an operation with complex 
numbers must be complex, but the result of an operation with imaginary 
numbers need not be imaginary. We say that the real numbers - the 
"old" system - are "closed under multiplication" because the product 
of two real numbers is always real. (Division is multiplication by a 
reciprocal, so the real numbers are also closed under division, except 
for zero which has no reciprocal.) Likewise the complex numbers - the 
"new" system - are closed under multiplication. But the pure imaginary 
numbers (like 2i) are not a closed system, nor are the non-real 
complex numbers (like 2+3i): they simply extend one closed system (the 
reals) to make another (the complex numbers).

If you found that one real number times another real number gave a 
non-real answer, that would be a problem, because the real numbers 
should be closed under multiplication. But there is nothing wrong with 
a multiplication of imaginary numbers giving a non-imaginary number; 
the answer just has to be complex. Zero (or -1) is a complex number as 
well as a real number, so we don't have a problem.

- Doctor Rick, The Math Forum   
Associated Topics:
High School Number Theory

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