- When writing out a solution, ensure that your
argument is complete and accurate, and that, when
the symbols are converted into English, what you
have written makes full sense grammatically and is
- When writing out a solution, pay attention to
the points listed below ((i)-(vii)).
- Good punctuation, particularly the use of
commas and brackets, may remove ambiguities.
The third aspect of doing mathematics is the
written exposition of your work. As a student, you
should always aim to write out your work in a complete,
accurate, and clear fashion. This applies equally to
writing out theory and to solutions of problems.
There are several reasons why good exposition is
important. Firstly, when you come to write down details
of problems, you may find some difficulties which you
had overlooked in your rough working. Secondly, any
tutor or examiner who is going to read your work will
need evidence that you really understood what you were
doing. Thirdly, you should aim to acquire the ability
to communicate mathematics to other people; this will
obviously be a valuable skill if you follow a career
which involves mathematics, and it will also enhance
your ability to write unambiguously and logically in
any occupation. In mathematics, accuracy is much more
important than literary quality!
Although your work may be read only by your tutor,
it is a good idea, when you are writing your solutions,
to imagine that you are writing for a reader who has a
similar level of mathematical knowledge to yourself but
who has not encountered the argument which you are
writing down. Such a reader should be able to
understand what you have written without danger of
ambiguity, to see that it is correct, and to see why it
The English language (in common with other
languages) is prone to ambiguity, and mathematicians
have to be particularly careful to use language (and
symbols, which are part of mathematical language) in an
unambiguous way. Good punctuation can be very helpful;
for example, commas can be used to separate clauses.
Sometimes, mathematicians go beyond normal punctuation
by using brackets to remove ambiguity. The use of
brackets inside algebraic expressions is commonplace
(consider the difference in meaning between (xy)+z and
x(y+z); it is a mathematical convention that the
unbracketed expression xy+z means (xy)+z, so if we mean
x(y+z), we have to include the brackets). However,
brackets can also be used around entire clauses, to
remove ambiguity. For example, the statement:
- x = 0 and y = 1 or z = 2 and t = 3,
is highly ambiguous, but the following are not:
- [x = 0 and y = 1] or [z = 2 and
t = 3],
- x = 0, and [y = 1 or z = 2], and
t = 3.
When you write out your final solution, make sure that
it is fully comprehensible, and that each step follows
logically from previous ones. What you write should be
clear, intelligible and unambiguous, as well as being
logically correct and mathematically accurate. In
particular, when all the symbols are converted into
English, your script should consist of complete
sentences. You should pay particular attention to
points such as the following:
- If you are arguing by contradiction or induction
(see Part2: Proofs
by contradiction and Part 2: Proofs by induction, say
- If you make some assumptions, say so clearly, by
including phrases such as ``Suppose that ...''.
- Be careful to use phrases such as ``if'', ``only
if'', ``if and only if'', correctly (see Implications). If you
use symbols such as , , and
Û, make sure that they go in the
direction which is both logically correct and
relevant to the problem (this may be the opposite of
the direction in your rough working).
- Include phrases such as ``and'', ``or'', ``for
all'', ``for some'' (see Part 2: And & Or and Part 2: For all & There
exists;) ; you cannot expect the reader to know
which you mean if you omit these phrases.
- Make sure that your argument shows correctly
which quantities depend on which other quantities
(see Part 2:
- If the statement which you are writing down
depends on something several lines earlier, make this
clear to the reader, for example by labelling the
earlier statement with (*) and writing ``It follows
from (*) that ...''.
- If your proof involves a division into two or
more cases, make it clear what the different cases
are (see Part 2:
And & Or).
See Part 2: The
formulation of mathematical statements
discussion of several of these points.