Elements of Style
Modern Algebra, Fall 2011
Anders O.F. Hendrickson
Years of elementary school math taught us incorrectly that the answer to a math problem
is just a single number, “the right answer.” It is time to unlearn those lessons: those days are
over. From here on out, mathematics is about discovering proofs and writing them clearly
and compellingly.
The following rules apply whenever you write a proof. I will refer to them, by number, in
my comments on your homework and tests. You will ﬁnd them summarized on the last page.
Take them to heart; repeat them to yourself before breakfast; bind them to your forehead;
ponder their wisdom with your friends over lunch.
1.
The burden of communication lies on you, not on your reader.
It is your job
to explain your thoughts; it is not your reader’s job to guess them from a few hints.
You are trying to convince a skeptical reader who doesn’t believe you, so you need to
argue with airtight logic in crystal clear language; otherwise he will continue to doubt.
If you didn’t write something on the paper, then (a) you didn’t communicate it, (b)
the reader didn’t learn it, and (c) the grader has to assume you didn’t know it in the
ﬁrst place.
2.
Tell the reader what you’re proving.
The reader doesn’t necessarily know or
remember what “Problem 5c” is. Even a professor grading a stack of papers might
lose track from time to time. Therefore the statement you are proving should be on
the same page as the beginning of your proof. For an exam this won’t be a problem,
of course, but on your homework, recopy the claim you are proving. This has the
additional advantage that when you study for tests by reviewing your homework, you
won’t have to ﬂip back in the textbook to know what you were proving.
3.
Use English words.
Although there will usually be equations or mathematical state
ments in your proofs, use English sentences to connect them and display their logical
relationships. If you look in your textbook, you’ll see that each proof consists mostly
of English words.
4.
Use complete sentences.
If you wrote a history essay in sentence fragments, the
reader would not understand what you meant; likewise in mathematics you must use
complete sentences, with verbs, to convey your logical train of thought.
Some complete sentences can be written purely in mathematical symbols, such as
equations (like
a
3
=
b

1
), inequalities (like
o
(
a
)
<
5), and other relations (like 5
10
or 7
∈
Z
). These statements usually express a relationship between two mathematical
objects
, like numbers (e.g., 7), group elements (e.g.,
g
2
h
), or sets (e.g.,
G
and
R
).
5.
Show the logical connections among your sentences.
Use phrases like “There
fore” or “because” or “if.
..then.
..” or “if and only if” to connect your sentences.