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Chapter 8
Algebra versus Analysis
We have seen in the previous chapter how we can construct an algebraic theory
of mathematical objects such as ‘exp’, ‘log’ and ‘atan’. From an algebraic point
of view, they seem to behave like the mathematical objects we are familiar
with from analysis. Are they the same? If not, what are the differences? This
is perhaps one of the less-discussed topics 1 in computer algebra, and indeed
possibly in mathematics more generally.
Notation 25 Throughout this chapter, we use the notation ? = to denote an
equation that might or might not be true, or partially true, depending on the
interpretations, from algebra or from analysis, that one places on the symbols
either side of ? =.
8.1 Functions and Formulae
This question turns out to be related to the difference between functions and formulae,
which we have also called expressions. Consider x2 −1
x−1
, or (x^2-1)/(x-1)
if we prefer linear expressions. It has potentially many rôles.
formula There are several options here: strings of characters, or a parse tree.
Whichever we choose, (x^2-1)/(x-1) is a different formula from x+1.
∈ Q(x) This is therefore mathematically equivalent to x + 1, and an algebra
system may or may not transform one into the other.
∈ K(x) Of course, it is only convention that chooses Q for the ground field.
It could be any extension of Q, or, more challengingly, a finite field of
positive characteristic.
rule This is what a computer scientist would think of as λx. x2 −1
x−1
: that rule
which, given an x, computes the corresponding value of the formula
1 But see [Dav10].
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