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Never begin a sentence with a mathematical symbol. Certain mathematical symbols (A) have an English meaning, Sentences begin with capital letters, but mathematical symbols are case sensitive.
Bad: A is a subset of B.
Good: The set A is a subset of B.
End any sentence with a period, even if the sentence ends with a mathematical expression. If there is English anywhere in the sentence, it needs a period. The only time you can not have English in your sentence is if it is part of a series of equations.
Separate mathematical symbols with words.
Bad: Because x^2 - 1 = 0, x = 1 or x = -1.
Good: Because x^2 - 1 = 0, it follows that x = 1 or x = -1.
Do not use mathematical symbols in place of plain English. Mathematical symbols should appear only in mathematical expressions.
Bad: Since x and y are =, their square are equal
Good: Since x and y are equal, their squares are equal.
Bad: ∃ a number larger than x.
Good: There exists a number larger than x.
Use the active, plural, voice. You and the reader are travelling together through this proof. The actors in your proof are the readers and the mathematical objects.
Bad: The value x = 3 is obtained by factoring.
Good: By factoring, we get the value x = 3.
Bad: Since both sides can be divided by y, we know...
Good: Since y divides both sides, we know..
Watch out for “it” and “any”, they are ambiguous.
Bad: Since x < y and 0 < x, we see that it is positive.
Good: Since x < y and 0 < x, we see that y is positive.
For ‘any’, prefer the more explicit ‘every’.
"Since", "because", "as", "for", and "so" are your new best friends. They all mean “I am going to give the premise and the conclusion in one sentence."
"Thus", "hence", "therefore", and "consequently" are all good ways of starting a sentence saying “Based on what I just said, I’m going to conclude some new stuff.”
Many suggestions and examples taken from The Book of Proof by Richard Hammock
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