Significance of Irrational Numbers

Date: 08/23/99 at 11:25:34
From: Nkiruka Ukachukwu
Subject: Nonsensical numbers

I can conceptualize numbers like .5, .45, even .34587 and .000082 - 
just so long as there is a finite number of digits following the 
decimal point. But what exactly is the meaning of .3333333... or pi? 
How do we conceptualize numbers with decimals that have no end? Were 
non-terminating decimals created and added to our number system or 
were they discovered as already being a part of our number system? Why 
even acknowledge them? What's the difference between "point three 
repeating" and "point three to the 105th decimal place?"

Date: 08/23/99 at 12:07:37
From: Doctor Ian
Subject: Re: Nonsensical numbers

Hi,

This is a good question. It's easy to get into the habit of just 
following the rules that are taught in mathematics classes without 
ever stopping to think about what's really going on.

One way to conceptualize numbers is as points on a line, the 'real 
line', which contains all the 'real numbers':


     <--------+--------->
              0

There has to be a number at every place on the line.  That is, you 
can't move to a place on the line and find that there is no number 
there.

The integers take some of these places,

     <---+---+---+---+---+---+--->
        -1   0   1   2   3   4

The rational numbers take some of the places in between the integers:

     <---+--+--+--+--+--+--+--->
         1  1     1        2
            -     -
            6     2

In fact, the rational numbers take a lot of places:

     <---+-----+-----+-----+--->
        649   650   651   652
        ---   ---   ---   ---
        673   673   673   673

In fact, given two rational numbers, you can always find another 
rational number in between them. (Can you see why that's true?)

But even so, there are still some spaces left over. In between any two 
rational numbers, there is at least one other number that isn't 
rational. These are numbers like pi, and e, and the square root of 2. 

So, one way to conceptualize these 'nonsensical' numbers is to think 
of them as the numbers that 'fill up' the spaces between the numbers 
that 'make sense'.

For many everyday applications, you're right to think that the 
difference between '0.3333333333333' and '0.333...' is insignificant. 
If I'm sawing boards to build a house, the difference between 33/100 
and 1/3 is unimportant. If I'm hollowing out the barrel of a gun, the 
difference between 3333/10000 and 1/3 is unimportant.

However, with the development of chaos theory, and the realization 
that very small changes in initial conditions can lead to arbitrarily 
large changes in final conditions, it's not always clear when it's 
okay to ignore small differences.

Perhaps more to the point, it's important to remember that 
mathematicians are basically playing a game called 'mathematics', and 
in this game, the difference between finding an exact result and 
finding an approximation, no matter how close, is like the difference 
between hitting or missing the bullseye with an arrow. No matter how 
close the arrow gets to the bullseye, if it's out, it's out.

So, one way to answer your question is to say that whether these 
numbers 'make sense' or whether they are 'worth acknowledging' depends 
on exactly what game you're playing.

I hope this helps. Be sure to write back if I didn't address your 
concerns, or if you just have other questions.  

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

Date: 08/24/99 at 17:38:17
From: Kiki Nwasokwa
Subject: Re: Nonsensical numbers

I realize that repeating decimals are the numbers that "fill the 
spaces" between terminating decimals, but how did people discover that 
they repeated? What does it mean for a number to be repeating? Is it 
kind of like moving toward a wall but moving half your previous 
distance each time? For example, with 4.3333..., is one moving three 
tenths of the way toward 5, then only three hundredths of the way, 
then only three thousands, then only three ten-thousandths, then only 
three hundred-thousandths, then only three millionths, then only...?

Date: 08/25/99 at 22:02:33
From: Doctor Ian
Subject: Re: Nonsensical numbers

Hi Kiki,

Try dividing 1 by 3, and see what happens. After a while, you'll see 
a pattern. That's how repeating decimals were discovered.

Your interpretation is correct - each decimal place that you add 
moves you a little closer toward the theoretical limit (1/3), but you 
never get all the way there until you add an infinite number of 
decimal places.

What you really want to know, I think, is how limits work, which is a 
deep question. Limits are probably the closest thing to magic in 
mathematics. They were used for a long time as a useful problem-
solving technique, even before anyone was able to provide them with a 
sound theoretical basis (in the form of the 'epsilon-delta' proofs 
that you'll see when you take a course in real analysis). Even now, 
they retain a faint air of paradox.

I think you would probably have a lot of fun working through a good 
textbook on real analysis. You might also enjoy Richard Courant's 
book, _What is Mathematics?_.

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