Because and therefore symbols
Two triangles of three dots carry the logic of every proof ever written. The because symbol ∵ (two dots above one) introduces a reason; its upside-down twin ∴ (one dot above two) introduces the conclusion — “therefore”. Copy both signs here, learn the one-glance trick that tells them apart, and watch the pair run a complete miniature proof.
Copy the logic signs
The pair at work, ready to copy
Which is which — the distinction
| Sign | Reads as | Its job in a line |
|---|---|---|
| ∴ (U+2234) | “therefore” (also “hence”, “so”) | Stands BEFORE a conclusion — everything above it has earned what follows |
| ∵ (U+2235) | “because” (also “since”) | Stands BEFORE a reason — it hands the reader the cause |
| Telling them apart | Count the top dots | One dot on top = ∴ therefore; two dots on top = ∵ because. The pair are exact mirror images |
| Their neighbours ∶ ∷ | “ratio” and “proportion” | ∶ separates the terms of a ratio (3 ∶ 2); ∷ equates two ratios (1 ∶ 2 ∷ 3 ∶ 6) |
A complete proof in three lines
Watch the pair earn their keep. A triangle’s two known angles are 60° and 70°. ∵ the angles of every triangle sum to 180°, ∴ the third angle is 180° − 130° = 50°. That is the whole rhythm of mathematical writing: facts, a ∵ to supply the reason, a ∴ to land the result. The signs have done this job in print since the 1600s — the therefore sign appears in Johann Rahn’s algebra of 1659, the same book that gave maths the ÷ division sign.
How to type them
| Platform | Method |
|---|---|
| Windows (Word) | Type 2234 then Alt + X for ∴; 2235 then Alt + X for ∵ |
| Mac | No single key — Character Viewer (search “therefore” or “because”) or copy from here |
| HTML / LaTeX | HTML: the “there4” entity name (ampersand + there4;) or code 8756; “becaus” (ampersand + becaus;) or code 8757. LaTeX: \therefore and \because |
| Handwritten notes | Three quick dots — just keep the triangle pointing the right way: conclusion up for ∴, reason up for ∵ |
Explore more
The angles in the little proof belong to the angle symbol page, its totals are measured with the degree symbol, and the whole operator family gathers in the math symbols collection. When a proof is finished, it traditionally closes with the proof symbol — the tombstone mark.
Frequently asked questions
How do you remember which sign is which?
Look at the top row of dots. Therefore ∴ carries ONE dot on top — the single conclusion rising from the two facts beneath it. Because ∵ carries TWO on top — the reasons sitting over the claim they support. Flip one and you have written the other, so the check takes a second.
Do ∵ and ∴ replace the words in normal writing?
In proofs, logic and tight notes, yes — that is what they are for. In essays and emails, write “because” and “therefore” out in full; the signs read as mathematics and would look like shorthand notes that escaped into prose.
Can a line start with ∵?
Yes, inside mathematical working: a fresh line often opens with ∵ to attach the reason for the step above it, exactly as the three-line proof on this page does. The convention is standard in geometry and algebra solutions.
What is the difference between ∶ and a normal colon?
∶ (U+2236, ratio) is a mathematical operator spaced like the other relation signs, used between the terms of a ratio; the everyday colon : (U+003A) is punctuation. In plain text either survives, but typeset maths prefers the true ratio sign.