End of proof symbol
Every proof needs a full stop with authority, and maths gives it a monument: the end of proof symbol ∎, a small square nicknamed the tombstone. Placed at the end of the final line, it says what the Latin abbreviation Q.E.D. used to say — “that which was to be demonstrated” — without a single letter. Copy ∎ and its open twin □ here, learn the formatting rules around them, and find the LaTeX recipes that produce them.
Copy the proof marks
Endings and LaTeX, ready to copy
How the mark is used — formatting
| Rule | How it works |
|---|---|
| Placement | Flush right at the end of the proof’s final line — the stone closes the argument, it never opens one |
| Filled or open | A house-style choice: ∎ (filled) and □ (open) mean exactly the same; textbooks and journals each settle on one |
| After a display | If a proof ends with a displayed equation, the stone rides on the equation’s own line rather than dropping to a line of its own (LaTeX: \qedhere) |
| Never mid-proof | The stone means finished — intermediate steps take ∴ (“therefore”) instead; one proof, one stone, at the very end |
The letters came first. Euclid closed his demonstrations with a Greek phrase meaning “just what was to be shown”; Latin scholarship carried it as quod erat demonstrandum — Q.E.D. The little square is a twentieth-century shortcut to the same words, popularised by the mathematician Paul Halmos from 1950 onward and jokingly called the tombstone (Halmos himself called it a “tombstone” lifted from magazine typography). Today the stone has largely replaced the letters it abbreviates.
The LaTeX recipes (and typing everywhere else)
If you arrived hunting the end of proof symbol latex recipe, here it is in full: load the amssymb package and use \blacksquare for the filled stone or \square for the open one — or better, write the proof inside the amsthm proof environment, which places an open square for you automatically at the end. \qedhere moves the stone onto a displayed equation’s line when a proof ends in a display.
| Platform | Method |
|---|---|
| LaTeX | \blacksquare (∎) and \square (□) from amssymb; automatic with amsthm’s proof environment |
| Windows (Word) | Type 220E then Alt + X for ∎; 25A1 then Alt + X for □ |
| Mac | No single key — Character Viewer (search “end of proof” or “white square”) or copy from here |
| HTML | Decimal codes: 8718 for ∎ and 9633 for □ — neither sign has a named entity in common use |
Explore more
The step just before the stone is usually marked by the because symbol pair (∵ and ∴), geometry proofs lean on the angle symbol, and the whole family gathers in the math symbols collection.
Frequently asked questions
What does Q.E.D. stand for?
Quod erat demonstrandum — Latin for “that which was to be demonstrated”. Greek geometers ended proofs with the equivalent phrase as far back as Euclid. The tombstone ∎ is simply Q.E.D. drawn as a symbol: same claim, no letters.
Why is it called a tombstone?
Because the squat little square looks like a headstone standing at the end of the argument — the proof is, after all, finished and laid to rest. Paul Halmos, who popularised the mark from 1950, used the nickname himself; some writers call it “the Halmos” in his honour.
Filled ∎ or open □ — which is correct?
Both. Unicode names ∎ (U+220E) END OF PROOF outright, while □ (U+25A1, white square) is the form the popular amsthm LaTeX package prints by default. Pick the one your teacher, journal or house style uses, and stay consistent through the document.
Can I use ■ (black square) instead?
You can, and plenty of documents do — ■ (U+25A0) is the text-geometric black square, visually near-identical. Strictly, though, ∎ U+220E is the character defined for this job, and it is the one that survives copy-paste into mathematical contexts with its meaning intact.