Subdividing Numbers, Number Line

Date: 20 Dec 1994 09:44:27 -0500
From: Megin Charner
Subject: For Sydney

Dear Sydney "Math Rocks" Foster,

        You are the greatest mathematician in the world. Thanks for 
teaching me what you did, and I have another question!

Here it is:

        I know about negative numbers and positive numbers but are there 
any other numbers? Please tell me. What is a number line?

Merry XMas?

From,
            Sam

Date: 21 Dec 1994 14:21:40 -0500
From: Molly Foster
Subject: Re: For Sydney

Dear Sam,

     Hello again, my young mathematician buddy!  I'm so glad you 
wrote to Dr. Math again!  You asked another great question. 

     Mathematicians like to classify numbers into many different 
groups so they can study numbers with similar properties more carefully.  
One way they classify numbers (really, we are talking about numbers 
called REAL numbers here, that means all the numbers you are probably 
used to seeing like 28, -8.23298938593582, and pi.  Numbers like the 
square root of -5 are not "real"--if a number is the square root of a negative 
number, we call it "imaginary.") is by dividing them into 2 groups--the 
positive numbers and the negative numbers.  

     But there are also many other ways to subdivide the real numbers.  
There are 2 big important subdivisions of the real numbers.

     Every real number is either a RATIONAL or IRRATIONAL number.  
These groups themselves can also be subdivided further!  The set of 
integers lies within the set of rational numbers.  Integers are just positive 
or negative whole numbers. So, for instance, 28, 43434235,and -3030 
are integers.  But numbers like 1/2 and 4.00032 are not.  

     We can subdivide even further!  The set of whole numbers lies within 
the set of integers (whole numbers are positive integers and 0 (I think 0 
is included, though I'm not definite on that one)).  

     Those are the major subdivisions that mathematicians like to look 
at.  Each of the above groups of numbers has special properties unique
to it.  This idea of having sets within sets within sets, etc. is quite
prevalent in mathematics.  There is a whole area in math called set theory 
that deals with all kinds of sets. Group theory, ring theory, and field theory
use many of the same ideas, too.  It's pretty amazing that these ideas pop
up in so many areas of math.  That's part of what I think makes math so 
neat!  

     On to your other question about the number line.  A number line
is a tool for students of math that helps them visualize numbers.  Do you 
use a number line at school?  For some people it is more helpful than for 
others.  A number line is simply a horizontal line with vertical markings
that represent numbers.  Maybe your mom, dad, or sister could draw you 
a little number line so you can see what it is.  When adding numbers, you 
can use a number line to count.  For instance, say you were adding 3 + 5:
Then, you would count 3 up from the 0 mark on the number line (so you 
would be at the 3 mark), and then you would count 5 up from the 3 mark.  
That would put you at the 8 mark, so you would then know 3 + 5 = 8.  
You can also use number lines to help you subtract numbers.  Can you 
figure out how?

        To relate the number line to what I was talking about in the beginning 
of this letter, consider this...  Most number lines have markings for every 
INTEGER.  That means, that most number lines will have vertical 
markings for 0, 1, 2, 3, ... and -1, -2, -3, ...  But there are TONS of 
numbers not marked on the number line.  If you started counting whole numbers 
right now, you would never run out of numbers, right?  That is because 
there are an infinite number of whole numbers.  

     Do you know what the word "infinite" means? It isn't really a number, 
it's more like a concept...it's just really really big.  Anyway, in math 
lingo there are "countably infinite" whole numbers. As it turns out, 
there are even more numbers in the interval between 0 and 1 than there 
are whole numbers.  Mathematicians say there are "uncountably infinite" 
numbers in the interval from 0 to 1 to show how big this is.  

     It's kind of a strange concept that one infinity could be more than 
another infinity, isn't it?  In the interval between 0 and 1 there are lots 
and lots of rational and irrational numbers.  

     Wow, I hope this doesn't overwhelm you!  Don't worry about it if
most of this seems really complicated or confusing.  I just learned about 
most of this stuff this year, and I'm a sophomore in college!  

     I must tell you one more neat thing relating to all of this... everything 
I've said above can be proven from the 8 basic axioms for real numbers.  
For instance the idea that there are an infinite number of whole numbers 
can be proven (mathematicians say it can be proven RIGOROUSLY) 
based on just 8 axioms.  In fact, all of higher math dealing with real 
numbers like calculus can be proven with these 8 axioms.  That is pretty 
amazing, don't you think?

     Well, I hope this helps.  Write back if you have any more questions,
okay?  I hope you have a good holiday!

Sydney, Dr. "Math Rocks" Foster