About Rational Numbers

Date: 3/20/96 at 8:19:13
From: Stuart Henson
Subject: Rational Numbers

Can you explain "rational numbers" to me?  How do you express 
them?

What kind of operations can you perform on them?  Thank you.

Date: 3/20/96 at 18:56:30
From: Doctor Jodi
Subject: Re: Rational Numbers

You're probably familiar with several sets of numbers.  Each set 
acts as as sieve, including some numbers and excluding others.  
Let's start out with a few numbers, let's call them Stuart's set, 
and let's define them like this:

(0,1, 2, 3...)

(I use the ellipsis to mean that you should continue this pattern.)

Okay, good, so we've described all the numbers we want.  What, you 
mean we're missing some?  Negatives, you say?

How can you have negative numbers??? Even Pascal knew that you 
can't subtract anything from 0.

0 - 5, Pascal said, is 0.

Or is it?

Well, around the time of Descartes, instead of thinking of numbers 
as a way to count apples (or even hamsters) or measure lines - for 
which Pascal's example is clearly true - this concept changed.

Signed numbers came to indicate DIRECTION.  

So if 0 is someplace in the hallway of your house, you could say 
that you're walking -5 feet towards your brother's pet snake who's 
just gotten loose.  Does that make sense?

If we want to add negative numbers to our set, we could define 
them like this:

Integers ( ..., -3, -2, -1, 0, 1, 2, 3, ...)

But we could still keep adding numbers. What if we wanted to 
include fractions...

Let's define a fraction as any RATIO of one integer to another.

Does that make sense?

So general, any old fraction looks like this: p  <--some integer
                                              _
                                              q  <--some integer

So what does all of this have to do with RATIONAL numbers?

Well, the set of all rational numbers is the set of all fractions, 
as we've defined them above. Remember, p and q can be the same, or 
p can be larger than q, so we can have fractions like 21/7, 3/3, 
-9/2, 0/49 to include all of the integers and the numbers 
like -9/2 which are sometimes called "mixed numbers" because they 
can also be written as an integer plus a "proper" fraction.  
(School teachers have such a notion of propriety!  Whoever said 
that if you're not between -1 and 1 that you can't be a "proper" 
fraction! What do they know!)

Anyway, so those are the rational numbers, the numbers that can be 
expressed as "well-behaved" ratios (there go those school teachers 
again!).

So, have we finished listing all of the types of numbers?

Nope. It turns out that there are some numbers, that can't be 
expressed as this sort of fractional ratio. They're called 
irrationals. One of the most famous examples comes from the 
Pythagorean theorem.

Are you familiar with the Pythagorean theorem, which says that the 
square of the hypotenuse (the longest side) in a right triangle is 
equal to the square on one shorter side plus the square on the 
other shorter side?  If you need a better explanation, write back! 
There are a LOT of cool proofs for the Pythagorean theorem.

Draw a square. Now draw its diagonal. You now have two right 
triangles. You can erase one of them if you want... Now focus on 
the other triangle. Since you started out with a square, you know 
that the short sides are both equal. Let's say that each of the 
short sides is equal to 1 unit. Now, if we use the Pythagorean 
theorem, we know that 1 squared plus 1 squared is equal to the 
hypotenuse squared.

Ok, 1 squared equals 1. So 1 squared plus 1 squared equals 2.  

So the hypotenuse squared is equal to 2. So the hypotenuse 
itself...

But wait, what sort of number could THAT be?

Let's see... 1 squared is 1, 2 squared is 4, 3 squared is 9, 4 
squared is 16...

Uh oh.  This isn't working.  We're looking for a number squared 
that's equal to 2.  Well, since 1 squared is 1 and 2 squared is 4, 
we know that that number has to be between 1 and 2.

But even if we start squaring numbers like 5/4 (to get 25/16), we 
can't seem to get 2.  (We can, however, get pretty close with this 
method! Try it sometime and see!)

So we have to start a new class of numbers.  We'll call this 
mysterious number the square root of 2.... and the class, we'll 
call IRRATIONAL numbers.

There are still other sorts of numbers... If you'd like to know 
more, write back!

Thanks for your question!

-Doctor Jodi,  The Math Forum