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20 CHAPTER 1. INTRODUCTION
2. Assuming the answer to the previous question is ‘yes’, does x1000 −1
x−1
simplify
to x 999 + · · · + 1? Here the fraction is much shorter than the explicit
polynomial, and we have misled the reader by writing · · · here 8 .
3. Does √ 1 − x √ 1 + x simplify to √ 1 − x 2 ? Assuming the mathematical
validity, which is not trivial [BD02], then the answer is almost certainly
yes, since the second operations involves fewer square roots than the first.
4. Does √ x − 1 √ x + 1 simplify to √ x 2 − 1? This might be thought to be
equivalent to the previous question, but consider what happens when x =
−2. The first one simplifies to √ −3 √ −1 = i √ 3 · i = − √ 3, while the
second simplifies to √ 3. Note that evaluations result in real numbers,
though they proceed via complex ones. Distinguishing this case from the
previous one is a subject of active research [BBDP07, CDJW00].
5. Assume we are working modulo a prime p, i.e. in the field F p . Does
x p − x simplify to 0? As polynomials, the answer is no, but as functions
F p → F p , the answer is yes, by Fermat’s Little Theorem. Note that, as
functions F p 2 → F p 2, say, the answer is no.
In terms of Notation 3, we can say that
but
({(x, x p − x) : x ∈ F p } , F p , F p ) B
= ({(x, 0) : x ∈ F p } , F p , F p ) B
,
({
(x, x p }
− x) : x ∈ F p 2 , Fp 2, F p 2)
≠ ({ }
(x, 0) : x ∈ F
B p 2 , Fp 2, F p 2)
. B
Now we give a few illustrations of what simplification means to different audiences.
Teachers At a course on computer algebra systems for teachers of the French
‘concours’, among the most competitive mathematics examinations in the
western world, there was a round table on the meaning of simplification.
Everyone agreed that the result should be ‘mathematically equivalent’,
though it was less clear, prompted by example 1, exactly what this meant.
The response to example 2 was along the lines of “well, you wouldn’t ask
such a question”. The author wishes he had had examples 3 and 4 to hand
at the time!
The general consensus was that ‘simplification’ meant ’give me the answer
I want’. This answer is not effective, in the sense that it cannot be
converted into a set of rules.
Whether this is an appropriate approach to pedagogy is outside the scope
of this book, but we refer the reader to [BDS09, BDS10].
8 The construct · · · is very common in written mathematics, but has had almost (but see
[SS06]) no analysis in the computer algebra literature.