What ∴ and ∵ mean
The therefore sign ∴ is three dots arranged in a triangle that points up. It introduces a conclusion: x + 3 = 5, ∴ x = 2. Its mirror image, the because sign ∵, points down and introduces a reason: ∵ all three sides are equal, the triangle is equilateral.
An easy way to tell them apart: ∴ stands firmly on two dots, like a conclusion resting on what came before, while ∵ balances on a single dot.
Both signs are common in school geometry proofs, in logic and in quick handwritten notes, where ∴ is much faster than writing "therefore". In formal papers, many authors prefer an arrow such as ⇒ or simply the word itself.
Other uses and look-alikes
Freemasons use ∴ after abbreviations, so you may see it in lodge names and titles. Several characters look similar but mean something else: ⸫ and ⸪ are punctuation marks, ⛬ is a map symbol for a historic site, and ∷ is the proportion sign, as in a : b ∷ c : d. Find more dot patterns on the dot symbol page, and two-dot marks on the colon symbol page.
How to type the therefore symbol
There is no Alt code or Mac Option shortcut for ∴, so copying it is usually fastest. In Microsoft Word, type 2234 and press Alt + X (use 2235 for ∵). On a Mac, open the Character Viewer with Control + Command + Space and search for "therefore". In Google Docs, use Insert > Special characters.
In LaTeX, write \therefore and \because; both need the amssymb package. In HTML, use ∴ for ∴ or, in HTML5, ∵ for ∵. For more signs used in proofs, see the full math symbols list.