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Integers and Complex Numbers


Date: 02/27/97 at 19:52:17
From: Brian D
Subject: Numbers besides complex

A few weeks ago my math teacher told the class that all numbers are 
contained by the real and imaginary sets of numbers.  I couldn't help 
but be a little curious because this seemed like over-generalization.  
I checked in the math FAQ and came up with some info that was mostly 
way over my head, but I could see that there might be other sets of 
numbers, particularly hyper-reals and octonions.  Do these sets exist 
outside complex numbers, or did I just get confused with all of the 
math jargon?  If they exist inside complex numbers, what sets are they 
in? Could you also explain what these sets are in terms that an 
average math-loving high schooler could understand?  Thanks for taking 
time to read this and providing the Dr. Math service.


Date: 02/28/97 at 09:33:06
From: Doctor Mitteldorf
Subject: Re: Numbers besides complex

Dear Brian,

I think your instincts about this are right.  All numbers are 
abstractions, and how abstract you get is limited only by your 
imagination (no pun intended).

When people had the natural numbers, they could add and multiply but 
they couldn't subtract. By expanding to the integers, they were able 
to subtract. Thus the expression 2-3 has meaning: you just call it -1. 

But this would be a futile endeavor if at the same time you introduced 
a new problem: what does it mean to subtract (-1) from something?  
Do you have to invent a new kind of number ("neganegs") that you get 
if you subtract (-1) from 0?  In fact, you don't: it's not obvious, 
but it's true that you can define 0 - (-1) to be +1, the regular old 
positive integer 1, and you'll still have a system that's just as 
logically self-consistent as the one you had before, but now it's more 
complete.

Continue the story: people had the integers, so they could add, 
subtract and multiply, but they couldn't divide.  What did it mean to 
divide 1 by 2?  You could invent a new kind of number, called a 
fraction. But then you need a self-consistent rule for dividing 
fractions. It turns out that the system of integers and fractions is 
one in which you can add, subtract, multiply and divide.  (You have to 
make an exception for dividing by zero - no way around that.)

Now you can add, subtract, multiply and divide; but you want to solve 
polynomial equations. You want to be able to say that every equation 
you write down, (for example x^2-2x+1 = 0 or x^3 = 5) has a solution.  
You can't do it within the context of the rational numbers (that's 
what you call the integers + fractions.) How about if we add all 
the "in-between numbers"; we admit numbers like 1.4143235... and 
3.1415926... into our number system. That would be the obvious next 
thing to do, but you find it doesn't work! There are still equations 
like x^2+1 = 0 that don't have any solutions.

So now we come to the point: By admitting one more number called i 
into your system (and making it "independent" of the real numbers, so 
that 1+i is just 1+i, its own thing, not reducible to any other form), 
you've now got a closed system in which every polynomial equation has 
a solution. You don't get into a jam when you try to solve equations 
like x^2 + i = 0, or x^17 = 1 + 6i. 

The "complex numbers" form a convenient home for mathematicians who 
want to be sure that whenever they write down a meaningful equation, 
it has a meaningful solution. At this point, you might offer the 
questioner an exercise: what is the square root of i?  Hint: suppose 
it has the form a+bi where a and b are regular old real numbers. Can 
you write equations that tell you what a and b have to be? Can you 
solve them? How can you check your answer?

This doesn't have to be the end of the story. There are all kinds of
other objects that can be constructed. They tend to get progressively
less useful as they get more abstract - after all, what could compare
in usefulness to the counting numbers 1, 2, 3...?  But, on the other 
hand, they keep mathematicians entertained.  And a lot of things that 
were once thought to be pure, useless mathematical imagination have 
turned out to be crucially useful in some branch of science. I'd 
venture to say that the whole branch of physics called quantum
mechanics couldn't have been figured out without complex numbers.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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