Calculators and Irrational Numbers

Date: 05/02/2001 at 16:19:28
From: Kathy King
Subject: Calculator calculation of irrationals

I have explained and illustrated the idea of irrational and rational 
numbers to my freshman algebra students. We discussed how the decimal 
representation of a number is a clue to deciding about rational or 
irrational. My students want to conclude that the square root of 11 is 
rational because the decimal values returned by any calculator, when 
reentered and squared, equals 11. I didn't believe this until I tried 
it for many square roots. At 10, the decimal value returned by any 
calculator, when squared, equals 10, etc. How is this done?

I want to be able to explain to my students that this is NOT really 
true, but they are having trouble believing me. Help! I need to 
explain again about perfect square roots and rational numbers and make 
it clear that decimal values for square roots should be ignored when 
determining rationality, but I need to be able to explain the 
calculator problem. Thank you for your expertise.

Kathy King
Mathematics teacher
Tremont High School
Tremont, IL

Date: 05/03/2001 at 13:43:14
From: Doctor Peterson
Subject: Re: Calculator calculation of irrationals

Hi, Kathy.

This could lead to some useful discussion of the difference between 
calculators and math.

You can start by pointing out how silly their claim is that the 
number they see on the calculator is the exact root. You mentioned 
that it works on ANY calculator. That implies that if you square the 
root you find on an 8-digit calculator, you get the original number 
back; and the same happens if you use a 32-digit calculator like the 
one on my computer. But they give different answers! How can they both 
be the right answer?

You can get a clue to what's happening if you find the root on an 
8-digit calculator, and rather than squaring it again on the same 
calculator, copy the number into the 32-digit calculator and square it 
there. You'll find that it doesn't really work out so well.

What's happening is just that the errors the calculator makes in doing 
the square root and keeping only 8 digits (or 32) don't make any 
difference in the 8 (or 32) digits you see after you square it again. 
If you did my suggested experiment, you should find that the first 8 
digits of the answer are right (or it may look like 10.999999..., 
which the calculator would round up); but eventually there will be an 
error. The first calculator's limited display just isn't big enough to 
show the error it introduced; and the second also has an error that it 
can't display.

Let's be a bit more precise about this. Suppose you take the actual 
square root of 11, and introduce an error of about 0.00000005 by 
rounding to 8 digits. When you square it again, you are doing this:

     (sqrt(11) + 0.00000005)^2

which expands to

     11 + 0.0000001 sqrt(11) + 0.000000000000001

or

     11 + 0.00000033 + 0.000000000000001

The last term is invisible on the calculator; the middle term seems 
as if it should be visible. But I've left out an important fact that 
can be fun to explore: the calculator really works with a little more 
precision internally than what it displays, so it didn't really round 
the answer to only 8 digits. If it keeps an extra digit internally, 
then the error after squaring will be just past the end of the 
display, as you've seen.

I'm curious what you meant when you said, "the decimal representation 
of a number is a clue to deciding about rational or irrational." I 
hope that your position in the discussion was that it is NOT a clue. 
Knowing the first 2 billion digits of a number tells me nothing about 
whether the decimal may terminate at the next digit and be rational, 
or continue forever as an irrational number. Irrationality is 
meaningful only with reference to an exact number, not an 
approximation, because rationals and irrationals are intermingled 
densely on the number line: between any two rationals there is an 
irrational, and between any two irrationals there is a rational. And 
that's why calculators are irrelevant to any discussion of 
irrationality, and why students who have come to think that numbers 
are what they see on a calculator can have a hard time believing in 
irrationals.

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