# Triple bar
The triple bar (≡) is a mathematical symbol of three stacked horizontal lines used to state that two expressions are identically equal, that one is defined as the other, or that two integers are congruent modulo a number. In Unicode it is encoded at U+2261 under the name IDENTICAL TO, with the HTML entity `≡`.[1](https://www.fileformat.info/info/unicode/char/2261/index.htm) Its meaning is fixed by context rather than by the symbol alone, which is why writers are advised to make the intended reading clear from the surrounding text.[2](https://www.mathwords.com/t/triple_bar.htm)
| Key fact | Detail |
|---|---|
| Symbol and encoding | ≡, Unicode U+2261 "IDENTICAL TO", HTML entity `≡`; negated form ≢ is U+2262[1](https://www.fileformat.info/info/unicode/char/2261/index.htm) • [3](https://www.unicode.org/versions/Unicode17.0.0/UnicodeStandard-17.0.pdf) |
| Identity use | Marks an equation true for every value of the variable, e.g. \( (x+1)^2 \equiv x^2 + 2x + 1 \)[2](https://www.mathwords.com/t/triple_bar.htm) |
| Definition use | Introduces a new symbol or expression, e.g. \( f(x) \equiv 3x + 7 \)[2](https://www.mathwords.com/t/triple_bar.htm) |
| Modular congruence | \( a \equiv b \pmod{n} \) means \( n \) divides \( a - b \); the two sides leave the same remainder on division by \( n \)[2](https://www.mathwords.com/t/triple_bar.htm) |
| Logical equivalence | In logic, \( A \equiv B \) means the two formulas take the same truth value for every assignment[4](https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/AllLectures2015Typed.pdf) |
| Origin in number theory | The congruence symbol ≡ was introduced in print in 1801 by Carl Friedrich Gauss in *Disquisitiones arithmeticae*[5](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/nth/) |
## Identity
When the triple bar marks an identity, the equation holds for every possible value of the variable, not just for particular solutions. The example \( (x+1)^2 \equiv x^2 + 2x + 1 \) is true for all \( x \), so it states a universal fact rather than a condition to be solved.[2](https://www.mathwords.com/t/triple_bar.htm) [Trigonometry](https://www.edgepedia.org/trigonometry) is a common setting: an identity such as \( \sin^2\theta + \cos^2\theta \equiv 1 \) is written with the triple bar to distinguish it from a conditional equation in \( \theta \).[2](https://www.mathwords.com/t/triple_bar.htm)
Practice varies.
## Definition
The symbol also signals that the left side is being defined as the right side, as in \( f(x) \equiv 3x + 7 \).[2](https://www.mathwords.com/t/triple_bar.htm)
## Modular congruence
In number theory, \( a \equiv b \pmod{n} \) means that \( n \) divides \( a - b \), or equivalently that \( a \) and \( b \) leave the same remainder when divided by \( n \).[2](https://www.mathwords.com/t/triple_bar.htm) A worked check shows the mechanics: to test \( 17 \equiv 5 \pmod{4} \), compute the difference \( 17 - 5 = 12 \); since \( 4 \) divides \( 12 \) with no remainder, the congruence holds.[2](https://www.mathwords.com/t/triple_bar.htm)
This usage goes back to the symbol's origin. [Carl Friedrich Gauss](https://www.edgepedia.org/carl-friedrich-gauss) introduced ≡ for congruence in the 1801 *Disquisitiones arithmeticae*, and it has remained the standard notation of modular arithmetic since.[5](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/nth/) Modular congruence appears throughout number theory, cryptography including RSA encryption, and computer science.[2](https://www.mathwords.com/t/triple_bar.htm)
## Logical equivalence
In logic, ≡ records that two formulas are logically equivalent: in a truth table, the left-hand and right-hand sides share the same truth values for all possibilities. A standard example is the double negation law \( \neg(\neg A) \equiv A \), proved by observing that the first and last truth-table columns match; [De Morgan's laws](https://www.edgepedia.org/de-morgans-laws) are stated the same way, \( \neg(A \lor B) \equiv \neg A \land \neg B \).[4](https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/AllLectures2015Typed.pdf) Formally, for formulas \( A \) and \( B \), one writes \( A \equiv B \) if and only if \( A \Leftrightarrow B \) is a tautology; the equivalence sign is not a new connective but a statement that two formulas always have the same logical value, so it can be used much like the equality symbol.[6](https://www3.cs.stonybrook.edu/~cse371/21lec2a.pdf) Texts also express logical equivalence with a double arrow (⟺ or ↔), and some use other decorated equals signs such as \( \cong \).[7](https://www.techtarget.com/whatis/definition/logical-equivalence) • [8](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_%28Fields%29/02%3A_Logic_and_Quantifiers/2.03%3A_Logical_Equivalences)
## Related symbols and usage notes
The negated form ≢ (U+2262, NOT IDENTICAL TO) can be composed in Unicode from U+2261 plus U+20D2, a vertical line overlay, without a precomposed character being required.[3](https://www.unicode.org/versions/Unicode17.0.0/UnicodeStandard-17.0.pdf) The horizontal triple bar should not be confused with the vertical triple bars: U+2980 TRIPLE VERTICAL BAR DELIMITER, used as a fence around expressions, and U+2AF4 TRIPLE VERTICAL BAR BINARY RELATION, an operator placed between two elements; Unicode's own naming note remarks that U+2980 substitutes BAR for LINE.[9](https://www.unicode.org/reports/tr25)
Because the symbol is overloaded, style guidance for general documentation lists ≡ simply as "identical to" alongside the plain equals sign and its other variants, leaving the specialized reading to the author's field.[10](https://developers.google.com/style/mathematical-notation) In practice, outside a few areas of mathematics, ordinary writing rarely needs the triple bar, and the equals sign suffices in most cases.[11](https://math.stackexchange.com/questions/1505026/when-do-you-make-use-of-%E2%89%A1-when-verifying-trigonometric-identities/1505096)
## References
1. [Unicode Character 'IDENTICAL TO' (U+2261), fileformat.info](https://www.fileformat.info/info/unicode/char/2261/index.htm)
2. [Triple Bar — Definition, Formula & Examples, Mathwords](https://www.mathwords.com/t/triple_bar.htm)
3. [The Unicode Standard, Version 17.0 — Core Specification](https://www.unicode.org/versions/Unicode17.0.0/UnicodeStandard-17.0.pdf)
4. [Math 135 lecture notes, University of Waterloo (definition of ≡ in logic)](https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/AllLectures2015Typed.pdf)
5. [Earliest Uses of Symbols of Number Theory, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/nth/)
6. [CSE 371 Logic lecture notes, Stony Brook University](https://www3.cs.stonybrook.edu/~cse371/21lec2a.pdf)
7. [What is logical equivalence and why is it important?, TechTarget](https://www.techtarget.com/whatis/definition/logical-equivalence)
8. [Logical Equivalences, Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_%28Fields%29/02%3A_Logic_and_Quantifiers/2.03%3A_Logical_Equivalences)
9. [Unicode Technical Report #25: Unicode Support for Mathematics](https://www.unicode.org/reports/tr25)
10. [Mathematical notation, Google developer documentation style guide](https://developers.google.com/style/mathematical-notation)
11. [When do you make use of ≡ when verifying trigonometric identities?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/1505026/when-do-you-make-use-of-%E2%89%A1-when-verifying-trigonometric-identities/1505096)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Logical calculi and logical syntax › Propositional logic*
*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license