Guidelines on Writing Mathematics

• Mathematics is written in complete sentences. Sentences begin with a capital letter and end with a period or other ending punctuation.



• Don't start a sentence with a mathematical symbol.



• Use mathematical symbols correctly.

For example it is fine to say "Since A=B, we know that..."

It is bad to say "Since A is = B,..."

It is OK to say "Since A is equal to B,..."

• Each statement in a proof should follow logically from previous statements in the proof, hypotheses, definitions, or previously proven theorems.



• Assume you are writing to an audience of your peers. If you think your peers would need an explanation of a step in a proof, then include the details.



• A nice distinction is that theorems have proofs but problems have solutions.



• If you are writing the solution of a problem, please write down the problem unless it is very long to write down. On tests, due to a shortage of time, there is no need to write a problem before writing its solution.



• Be careful with the word "let". You don't want to use the word "let" in an impossible situation. You cannot properly say "Let N be the largest integer." You can say "Suppose N is the largest integer."



• Any statement that is not strictly true is false. "Any number raised to the zero power is 1." is a false statement.



• Whenever you introduce a letter or variable, you should almost certainly say what it is. This will probably involve a phrase such as there exists an integer n such that, for some integer w whose square is, or for all real numbers x.