Do I Need to State Every Definition I Use, or Can I Assume the Reader Knows Them?

You are writing a proof. You are about to unpack "n is even" into "n = 2k for some integer k." Do you need to write out the full definition of even — "an integer is even if it is twice some integer" — or can you just skip to the algebra? And what about all the other definitions in the background: integer, multiplication, equality?

The answer is a working compromise, and it is worth spelling out because it decides whether your proof reads like a tight argument or a bureaucratic form.

The rule in one line

State the definition you are actively using at the moment you use it. Cite standard definitions by name ("by definition," "by the definition of even"). Do not restate background definitions that you are not invoking as part of the argument.

The words "actively using" carry the weight. A definition is active if the step depends on the precise wording of the definition. A definition is background if it is part of the shared vocabulary and no step hinges on a subtle reading of it.

Why the rule strikes this balance: a proof is a conversation with a reader who knows the prerequisites. Writing out every definition would be like restating the rules of English grammar before every paragraph — technically complete, practically unreadable. But the reader cannot follow a step that depends on a precise definition unless that definition is visible. So the visible definitions are the ones the step depends on, and the invisible ones are the ones already in the reader's head.

What counts as "actively using" a definition

A definition is actively used when the next step of the proof is the algebraic form the definition guarantees. Three clear signals:

Unpacking a hypothesis. "Assume n is even. By definition, n = 2k for some integer k." The move from "even" to "2k" is exactly the definition, and writing "by definition" makes that visible.
Repacking into a conclusion. "We have m+n = 2(a+b) where a+b is an integer, so by definition m+n is even." The move from the algebraic form back to "even" is again exactly the definition.
Invoking a term you have not used yet in this proof. The first time "divides" appears in a divisibility proof, write the definition inline: "a \mid b means there exists an integer q with b = aq." Subsequent uses can rely on that opening statement.

What you do not need to state

Background definitions that are in the reader's vocabulary but not hinged upon by any step. For a typical high-school or early-college proof:

"Integer", "real number", "rational" — shared vocabulary. The reader knows what these mean.
Arithmetic operations (addition, multiplication, exponentiation) and their standard rules (commutative, associative, distributive). You will of course use these, but you do not announce each use.
Equality, inequality, and their elementary properties.
Logical connectives: and, or, not, implies.

If your proof is for an audience that is not assumed to know these — say, you are writing a foundations article that is about to redefine "integer" — the rule reverses for that context. Audience awareness is the point.

The "state it the first time, cite it after" pattern

In a proof with multiple divisibility steps, you do not need to restate the definition of "divides" at each step. Write it once, at the first use, and cite it by name on subsequent uses.

The clean version states the definition of "divides" once, on the line that first uses it, and cites it by name thereafter. The verbose version restates the unpacking on every line. Both are correct; the clean one respects the reader's time.

Two cases where you should state more

There are two situations where the rule tilts towards writing more, not less.

Case A: the definition you are using is non-standard or has multiple conventions. Example: some textbooks include 0 in \mathbb{N}; others do not. If your proof hinges on whether 0 is natural, state which convention you are using at the start: "In this proof, \mathbb{N} = \{1, 2, 3, \dots\}." Otherwise the reader has no way to check the argument.

Case B: the definition has a subtle quantifier structure. "a \mid b means there exists an integer q with b = aq" has an existential quantifier hidden inside it, and proofs about divisibility usually manipulate that q explicitly. Writing the definition out, including the "there exists an integer q" part, is safer than a terse "by the definition of divides." The quantifier is the thing the next step acts on.

Two cases where you should state less

Case C: algebraic manipulations with no subtlety. "(2k)^2 = 4k^2" does not need a justification like "by the definition of squaring and the commutative law of multiplication." The arithmetic is the shared background. Writing it out adds noise without adding information.

Case D: the word is being used in its everyday mathematical sense, and the proof does not hinge on its formal definition. "Let x be a real number" is fine on its own; you do not need to append "where real number means an element of \mathbb{R}, the complete ordered field, etc." The reader is not going to use the Dedekind-cut definition — they are just using x as a real number.

The reader-model test

The cleanest way to decide is to imagine a reader at the expected level and ask two questions:

Can this reader follow the next step without the definition being explicit? If yes, do not state it. If no, state it.
If I skip stating the definition, does the next step look like magic? If it looks like magic, the definition was load-bearing — state it.

Apply this at each critical step and the right amount of definition-stating will settle itself. You will not end up writing a dictionary at the top of every proof, nor will you leave the reader stranded at a crucial unpacking step.

Where stating definitions becomes non-optional

A few places where stating the definition is not really a choice:

The opening of any definition-heavy proof. A proof that will repeatedly use "divides," "prime," or "even" benefits from a short block at the start listing the active definitions. Your reader can return to them as needed.
Any time the definition differs from common usage. Convention mismatches cause more confused readers than any other issue.
When the proof is graded or reviewed. Graders cannot award credit for reasoning they cannot see. Spelling out "by definition of..." at each unpacking step is a cheap way to signal that your step is legal rather than a leap of intuition.

The short summary

State definitions actively used: the ones whose exact wording the next step depends on.
Cite standard definitions by name after the first statement.
Do not restate background vocabulary (integer, real number, arithmetic rules).
State more when the convention is non-standard or the definition has tricky quantifiers.
State less when the reader's default reading will match what you mean.

Getting this right is the difference between a proof that feels like a conversation and one that feels like a form-filling exercise. Neither extreme — too much nor too little — does the job.