Therefore symbol ∴

∴ introduces a conclusion that follows from what came before:

• Math: 2x = 10 ∴ x = 5
• Logic: All mammals breathe air. Whales are mammals. ∴ Whales breathe air.
• Notes and outlines: “Flight delayed 3 hrs ∴ missed connection.”

Its partner ∵ (because) points the other way: x = 5 ∵ 2x = 10.

Both symbols save time in handwritten notes, on whiteboards and in quick study guides. In formal proofs and published writing, many authors prefer the words “therefore” and “because,” or the implication arrow ⇒, so readers don’t have to decode a symbol. In an essay or a business email, always spell out the word.

• ∴ (therefore) has one dot on top and two below, like a triangle standing on its base.
• ∵ (because) flips it: two dots on top, one below.
• ⸫ (one dot over two dots punctuation) looks the same as ∴ but is encoded as a punctuation mark for historical texts, not as a math operator.
• ⁖ (three dot punctuation) is another three-dot mark from historical manuscripts.

For math and logic, use ∴ and ∵, which belong to Unicode’s Mathematical Operators block. Math fonts and equation software handle them correctly, and they match what readers expect to find when they search a document.

• Word: in an equation (Alt+=), type \therefore and press space; \because gives ∵. Outside equations, use the Alt+X code from the table above.
• LaTeX: \therefore and \because, with the amssymb package.
• Google Docs: Insert > Special characters, then search “therefore.”
• Mac: open the Character Viewer with Control+Command+Space and search “therefore.”
• iPhone and Android: no default key. Copy it here once and save a text replacement such as “thrf” → ∴.