What Does 'QED' Actually Mean, and Do I Still Need to Write It?

You finish a proof, and the textbook ends with three mysterious letters: QED. Your teacher uses a small black square \blacksquare. A friend draws \square. Everyone seems to agree the proof is over — but what do the marks actually mean, and is any of them required?

The short answer: QED means "which was to be shown," it has been the standard end-of-proof mark in Western mathematics for about four centuries, and it is no longer mandatory. What is mandatory is some clear signal to your reader that the proof has ended and the claim is established. The symbol is the signal, whatever form you pick.

What QED stands for

QED is the abbreviation of the Latin phrase quod erat demonstrandum — literally "that which was to be demonstrated." It is the translation of a Greek phrase, hoper edei deixai, that Euclid used at the end of his geometric proofs in the Elements around 300 BCE. European mathematicians adopted the Latin version when translating and extending Greek mathematics in the Renaissance, and for centuries "QED" was the standard way to announce that a proof was complete.

The phrase has two halves worth noticing:

Which was to be — the thing you set out to show at the top of the proof.
Demonstrated — not merely claimed, but demonstrated: made visible through an argument that the reader can verify step by step.

So when you write QED, you are saying: "The thing I promised you I would prove, I have just proved. This proof is now complete."

What symbols are in use today

The tradition has several close relatives, all doing the same job — this proof ends here:

QED — the traditional abbreviation in words. Still used in older textbooks and in some formal writing.
\blacksquare — the Halmos tombstone, introduced by the American mathematician Paul Halmos in the 1950s. Halmos borrowed the symbol from newspaper and magazine layout, where a small filled square marks the end of an article.
\square — the hollow version of the same symbol. Some authors prefer the open square as slightly lighter on the page.
\Box — another variant used in some typesetting systems.
"The end" — plain English. Sometimes appropriate in informal writing.
Nothing at all — some modern textbooks simply leave blank space after the last line of the proof, trusting the reader to see that it has ended.

All of these are legitimate. The one you pick is a stylistic choice, not a mathematical one.

Why the mark matters at all: a reader who is following a proof carefully needs to know when the argument has ended. Without an end-mark, the reader might keep looking for one more step that confirms the conclusion — and a proof that leaves the reader wondering "is it done?" has failed part of its job. A clear end-mark tells the reader: the argument is complete; you may now stop verifying and accept the conclusion.

A concrete example

Claim. For every integer n, the number n^2 + n is even.

Proof. Consider two cases.

Case 1: n is even. Then n = 2k for some integer k, and

n^2 + n = 4k^2 + 2k = 2(2k^2 + k),

which is even because 2k^2 + k is an integer.

Case 2: n is odd. Then n = 2k + 1 for some integer k, and

n^2 + n = (2k+1)^2 + (2k+1) = (2k+1)(2k+2) = 2(2k+1)(k+1),

which is even because (2k+1)(k+1) is an integer.

In either case, n^2 + n is even. \blacksquare

The black square at the end is the QED. The reader now knows the proof has ended and the claim is established for every integer n.

How the mark changes depending on context

Four equivalent ways to signal the end of a proof. The mathematics is identical in each case; the choice is purely a stylistic one based on the audience and the typography.

Practical guidance

School and exam proofs. Use either the letters QED or the symbol \blacksquare at the end. Both are universally recognised, and the mark signals to the grader that you believe the proof is complete. The grader then verifies that you are right.
Informal notes and problem sets. Either mark works; the tombstone is slightly more visual. Some students use "So done" or a double line beneath the last equation. All are acceptable if the end is clear.
Typeset papers and textbooks. The norm is \blacksquare or \square, handled automatically by LaTeX via \qedsymbol inside a proof environment. You do not usually write "QED" by hand in a typeset document unless your style guide specifically asks for it.
Reading other people's proofs. Any of the symbols above means the same thing: the argument is complete from here backwards. After a QED-mark, the author is claiming the proof is finished; your job is to verify, not to look for continuation.

When the mark is missing

If a proof ends with no mark at all, it is not automatically invalid. Some mathematicians — especially in informal writing — rely on context: the next paragraph's first sentence ("From this it follows that..." or "We have therefore established...") serves as the implicit end-of-proof signal. Others use a section break, a horizontal rule, or simply a paragraph of blank space.

The only real failure is a proof that ends mid-thought — where the reader cannot tell whether there were more steps to come. That happens when the final line does not match the stated conclusion and no end-mark resolves the ambiguity. The mark, or a clear alternative signal, is the fix.

The short summary

QED = Latin quod erat demonstrandum = "which was to be demonstrated."
It marks the end of the proof. The symbol \blacksquare (or \square) is the modern equivalent.
You do not need to write the letters; you do need to make the end of the proof unambiguous.
The reader needs a signal — a symbol, a phrase, or a clear new paragraph — that the argument has closed.

A good proof ends with confidence: the last line matches the claim, the end-mark sits neatly on the right, and the reader can now move on without wondering whether more was supposed to come.