Therefore Symbol (∴): Meaning, Uses and How to Type It

The therefore symbol ∴ is three dots arranged in a triangle that points up. It is shorthand for "therefore" or "so" and introduces a conclusion: x + 2 = 5, ∴ x = 3. Its mirror image ∵, with the triangle pointing down, means "because" and introduces a reason. Both are common in school algebra and geometry, in handwritten notes, and in step-by-step solutions.

Below: both characters to copy, the codes for every platform, how the symbols are used, and how to write them in LaTeX.

Therefore Symbol (∴): Copy and Paste

Copy the symbols from the box below, or take the code you need from the table:

1∴ ∵
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Therefore Symbol Definition

∴ is a logical connective written as a symbol: "A ∴ B" reads "A, therefore B" and claims that B follows from A. ∵ is its converse in reading order: "B ∵ A" reads "B, because A." The two are often paired in a written argument, with ∵ marking the given facts and ∴ marking what you conclude from them.

The three-dot sign goes back to the 17th century and appears in Johann Rahn's 1659 algebra book Teutsche Algebra and its 1668 English translation. English textbooks later settled on ∴ for "therefore" and ∵ for "because," which is the convention used today.

In formal mathematical writing the symbols are rare: published proofs usually spell out "therefore," "hence" or "so," and logicians use ⊢ or ⇒ for entailment and implication. ∴ and ∵ are at home in teaching, worked solutions, lecture notes and exams, where they save space.

Uses of the Therefore and Because Signs

• Algebra steps: 2x = 10 ∴ x = 5. Each ∴ links a line to the conclusion drawn from it.
• Geometry proofs: "∠A + ∠B = 180° ∵ co-interior angles; ∴ AB ∥ CD." Two-column proofs often put the reasons after ∵.
• Logic and argument diagrams: in introductory logic, an argument is written as premises followed by a line beginning with ∴ for the conclusion.
• Freemasonry: the three-dot triangle is used after abbreviations in Masonic texts, for example "R∴W∴" for "Right Worshipful."

How to Type the Therefore Symbol

• Windows: ∴ and ∵ have no Alt code. In Word, type 2234 or 2235 and press Alt + X. In other programs, copy the character from Character Map (search "therefore") or from this page.
• Mac: open the Character Viewer with Control + Command + Space and search "therefore" or "because." With the Unicode Hex Input source enabled, hold Option and type 2234.
• Word equations: press Alt + = to insert an equation, then type \therefore or \because and a space.
• Google Docs: Insert → Special characters, then search "therefore." The equation editor also accepts \therefore.
• HTML: ∴ for ∴ and ∵ for ∵, or the numeric references ∴ and ∵.
• Phones: neither iOS nor Gboard has the symbols on the keyboard; copy them or add a text replacement such as "thf" → ∴.

Therefore Symbol in LaTeX

Both commands come from the amssymb package and work in math mode. Because they are relation symbols, LaTeX spaces them like =. In running text, wrap them in dollar signs.

1\documentclass{article}
2\usepackage{amsmath, amssymb}
3\begin{document}
4\begin{align*}
5  2x + 3 &= 11 \\
6  2x &= 8 \\
7  \therefore\ x &= 4
8\end{align*}
9
10$\angle A = \angle B$ $\because$ base angles of an isosceles triangle.
11
12All men are mortal; Socrates is a man; $\therefore$ Socrates is mortal.
13\end{document}
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• Without amssymb, \therefore gives "Undefined control sequence." The mathabx and MnSymbol packages also define both commands, but they change many other symbols, so amssymb is the usual choice.
• The \ (backslash space) after \therefore in the example adds a normal space before x, since math mode ignores typed spaces.
• With XeLaTeX or LuaLaTeX and unicode-math, you can type ∴ directly in math.

For a full table of logic and relation symbols, see the math symbols lesson; for typesetting proofs themselves, see theorems and proofs in LaTeX. Every logic symbol with its command is listed on the logic symbol list.

Related symbols and guides

• Therefore symbol page — copy the symbol and its LaTeX command
• Logic symbols reference — every symbol in the category with its LaTeX command
• Mathematical expressions in LaTeX — how to typeset these symbols in a document
• For all (∀) and there exists (∃)
• And symbol (& ∧)

Therefore Symbol FAQ

What does ∴ mean in math?

∴ means "therefore." It introduces a conclusion that follows from the previous line, as in 3x = 12 ∴ x = 4.

What is the because symbol?

The because symbol is ∵, three dots in a triangle pointing down (U+2235). It introduces a reason, as in "x = 4 ∵ 3x = 12." In LaTeX it is \because with amssymb.

How do I type the therefore symbol in Word?

Type 2234 and press Alt + X, or insert an equation with Alt + = and type \therefore followed by a space. You can also use Insert → Symbol and search the Mathematical Operators subset.

Is there an Alt code for ∴?

No. Windows Alt codes only cover the old 256-character code pages, which do not include ∴ or ∵. Use Alt + X in Word, Character Map, or copy and paste.

Why does \therefore not work in LaTeX?

\therefore and \because are defined by amssymb, not by LaTeX itself. Add \usepackage{amssymb} to the preamble and use them in math mode.