Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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.

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/
```
Associated Topics:
High School Calculators, Computers
High School Number Theory
High School Square & Cube Roots
Middle School Square Roots

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search