QED

QED

An abbreviated Latin phrase used to indicate the end of a proof, especially if it may not be immediately obvious that the proof is complete.

QED stands for the Latin phrase quod erat demonstrandum, which means "That which was to be proven."

Example

Problem: Prove that the sum of two even integers is even.

Step 1: Let the two even integers be expressed as 2a and 2b, where a and b are integers.

Step 2: Add them together and factor out 2.

2a + 2b = 2(a + b)

Step 3: Since a + b is an integer, the sum 2(a + b) is even by definition. This completes the proof, so we write QED (or place the symbol ∎) at the end.

Answer: The sum of two even integers is always even. QED.

Why It Matters

QED gives the reader a clear signal that a proof has reached its logical conclusion. Without it, a reader may wonder whether the argument is finished or whether additional steps follow. In modern mathematics, a filled square symbol (∎), sometimes called a "tombstone" or "Halmos symbol," often replaces the letters QED, but the purpose is identical.

Common Mistakes

Mistake: Writing QED before the proof is actually complete, as if it can substitute for a missing final logical step.

Correction: QED is a marker, not an argument. Every logical step must be present before you place QED at the end. It declares completion — it does not create it.

Related Terms

Proof — The logical argument that QED concludes
Theorem — A statement that a proof (ending with QED) establishes
Conjecture — An unproven statement that lacks a QED
Axiom — An accepted truth that needs no proof