Contents - Student subdomain for University of Bath

Chapter 1
Introduction
Computer algebra, the use of computers to do algebra rather than simply arithmetic,
is almost as old as computing itself, with the first theses [Kah53, Nol53]
dating back to 1953. Indeed it was anticipated from the time of Babbage, when
Ada Augusta, Countess of Lovelace, wrote
We may say most aptly that the Analytical Engine weaves algebraical
patterns just as the Jacquard loom weaves flowers and leaves.
[Ada43]
In fact, it is not even algebra for which we need software packages, computers
by themselves can’t actually do arithmetic: only a limited subset of it. If we ask
Excel 1 to compute e π√ 163 −262537412640768744, we will be told that the answer
is 256. More mysteriously, if we go back and look at the formula in the cell, we
see that it is now e π√ 163 −262537412640768800. In fact, 262537412640768744 is
too large a whole number (or integer, as mathematicians say) for Excel to handle,
and it has converted it into floating-point (what Excel terms “scientific”)
notation. Excel, or any other software using the IEEE standard [IEE85] representation
for floating-point numbers, can only store them to a given accuracy,
about 2 16 decimal places. 3 In fact, it requires twice this precision to show that
e π√ 163 ≠ 262537412640768744. Since e π√ 163 = (−1) √ −163 , it follows from deep
results of transcendental number theory [Bak75], that not only is e π√ 163 not
an integer, it is not a fraction (or rational number), nor even the solution of a
polynomial equation with integer coefficients: essentially it is a ‘new’ number.
1 Or any similar software package.
2 We say ‘about’ since the internal representation is binary, rather than decimal.
3 In fact, Excel is more complicated even than this, as the calculations in this table show.
i 1 2 3 4 . . . 10 11 12 . . . 15 16
a 10 i 10 100 1000 1. . . 0 . . . 1. . . 0 10 11 10 12 . . . 10 15 10 16
b a-1 9 99 999 9999 9. . . 9 . . . 9. . . 9 10 12 . . . 10 15 10 16
c a-b 1 1 1 1 1 . . . 1 1 . . . 1 0
We can see that the printing changes at 12 decimal digits, but that actual accuracy is not lost
until we subtract 1 from 10 16 .
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