Classifying Numbers

Date: 09/06/2001 at 21:03:22
From: Kelly Wlison
Subject: Algebra - help!

Hi, Dr. Math.

I need help! Can you tell me about complex numbers, real and imaginary 
numbers, rational/irrational numbers? I don't understand any of it.

Date: 09/06/2001 at 22:31:57
From: Doctor Jeremiah
Subject: Re: Algebra - help!

Hi Kelly,

Okay - it's not really that hard.

You know about natural numbers (1,2,3,4,... but no zero).
You know about whole numbers (0,1,2,3,4,... with a zero).
You know about integer numbers (...,-4,-3,-2,-1,0,1,2,3,4,...).

Now imagine that you are Pythagoras. You know you can make fractions 
out of integers. RATIONAL NUMBERS are integers and fractions of 
integers put together - but there aren't any other numbers. Everything 
can be represented as a rational number.

Now imagine that you have just discovered that the square root of two 
cannot be written down as a fraction! What do you do? Well if you are 
a Pythagorean, you hide the knowledge and don't tell anyone that there 
is such a number.

But the knowledge leaked out. Everyone knows that there are numbers 
that cannot be written as fractions of integers (like the square root 
of two, pi, and e), but they aren't rationals, so what do we call 
them? Call them IRRATIONAL NUMBERS (irrational is the opposite of 
rational).

So now we know that there are rationals (which include integers and 
factions made of integers) and irrationals, which are numbers that are 
not rational. Put together, they make the REAL NUMBERS.

But if we can find some numbers that don't fit into the real numbers, 
then what would we call them? Lets call them IMAGINARY NUMBERS 
(imaginary is the opposite of real). Then the set of real numbers and 
non-real (imaginary) numbers put together would be a complex set of 
numbers, so we will call them COMPLEX NUMBERS.

                           complex 
                            /   \
                   imaginary     real
                                /    \
                      irrational    rational
                                     /    \
                             integers      fractions
                                 |             |
                               whole        integers
                                 |
                             naturals

Now you are saying, "Okay, I understand that irrational numbers aren't 
rational, and I know of some examples, but if imaginary numbers aren't 
real numbers, then prove that imaginary numbers exist." Now we need an 
example of a imaginary number. What do you get if you take the square 
root of -1? There is no real number (rational or irrational) that has 
the value of the square root of -1. So we need to make up a number 
(the way we did with pi and e). The square root of -1 will be called 
i.  i is the first imaginary number. So complex numbers contain all 
real numbers and imaginary numbers like i.

Take the square root of -4: the square root of -4 is the square root 
of -1 times 4. After we take the square root we get 2i because the 
square root of 4 is 2 and the square root of -1 is i.

A complete complex number is the sum of a real number and an imaginary
number. For example: 3 + 4i is a complex number. The real part is 3, 
and the imaginary part is 2i.

If you need more help, please write back.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/