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Filling holes in the real line

Published online by Cambridge University Press:  01 August 2016

Bob Burn
Bob Burn*
Affiliation:
School of Education, Exeter University, EX1 2LU
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Is there a largest number on your calculator? There is on mine; it is 9·9999999 x 1099. It is larger than any number I have ever wanted to calculate, though it puts some known Mersenne primes out of reach. An upper boundary to the numbers with which one can calculate gives one striking difference between our imagined number line and the numbers on a calculator.

A more serious difference between the numbers on a calculator and the real number line is the existence on the calculator of a smallest number, greater than 0. On my calculator, the smallest positive number is 10-99. So, if I imagine the numbers my calculator will display marked on a number line, and zoom in on the number line to magnify it, the calculator numbers will appear spaced out, rather like integers on a conventional number line. We will see shortly that the calculator numbers are not evenly spread out, but the existence of gaps on the line, in any form, is critical. The calculator numbers are not dense on the line.

Type
Research Article
Information
The Mathematical Gazette , Volume 74 , Issue 469 , October 1990 , pp. 228 - 232
Copyright
Copyright © The Mathematical Association 1990

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