Approximately Equal in LaTeX

Write approximately equal as \approx in math mode: $\pi \approx 3.14$ gives π ≈ 3.14. Related relations have their own commands: \sim ∼ (asymptotic, similar), \simeq ≃, \cong ≅ (congruent, isomorphic) and \equiv ≡ (identical, congruent modulo n).

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Approximation and equivalence symbols

\approx
$\approx$
Approximately equal (numerical approximation).
\sim
$\sim$
Similar / asymptotically equivalent / distributed as.
\simeq
$≃$
Asymptotically equal, or isomorphic / homotopy equivalent.
\cong
$≅$
Congruent (geometry) or isomorphic (algebra).
\equiv
$\equiv$
Identical to, defined as, or congruent modulo n (the three-line equal sign).
\not\equiv
$≢$
Not congruent / not identical.
\approxeq
$≊$
Approximately equal or equal ≊ (amssymb).
\doteq
$≐$
Approaches the limit; also used for "defined as" by some authors.
\propto
$\propto$
Proportional to (not the same as approximately).

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Approximately equal ≈ examples

Each result below is drawn from the LaTeX under it. Copy the code, edit it on this page, or open it in the full editor.

Numerical value
$π \approx 3.14159$
LaTeX
```
\pi \approx 3.14159
```
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Small-angle approximation
$\sin x \approx x for small x$
LaTeX
```
\sin x \approx x \quad \text{for small } x
```
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Asymptotic equivalence
$n! \sim \sqrt{2 π n} {(\frac{n}{e})}^{n}$
LaTeX
```
n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^{n}
```
Stirling's formula: the ratio of the two sides tends to 1, which is what ∼ means here.
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Congruent triangles
$△ A B C ≅ △ D E F$
LaTeX
```
\triangle ABC \cong \triangle DEF
```
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Congruence modulo n
$17 \equiv 2 (\mod 5)$
LaTeX
```
17 \equiv 2 \pmod{5}
```
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Isomorphic groups
$Z / 2 Z ≅ C_{2}$
LaTeX
```
\mathbb{Z} / 2\mathbb{Z} \cong C_2
```
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Taylor approximation
$e^{x} \approx 1 + x + \frac{x^{2}}{2}$
LaTeX
```
e^{x} \approx 1 + x + \frac{x^2}{2}
```
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Which symbol means what

≈ (\approx) is the everyday "approximately equal": a rounded number or an approximation you can quantify. ∼ (\sim) is used for asymptotic equivalence (the ratio tends to 1), for "of the same order", for similar figures and for "is distributed as" in probability.

≃ (\simeq) and ≅ (\cong) are mostly structural: isomorphism, homotopy equivalence, congruent shapes. ≡ (\equiv) means identically equal, "defined as", or congruence in modular arithmetic. Fields use these differently, so say in the text which meaning you intend.

Congruence modulo n

Write the modulus with \pmod{n}, which adds the parentheses and spacing: a \equiv b \pmod{n}. \bmod is the binary operator for an inline remainder (a \bmod n), and amsmath adds \mod (no parentheses) and \pod.

Approximately in running text

For "about 10 ms" in a sentence, $\approx 10$~ms is the clearest. Some fields write $\sim$10 instead; if you do, use $\mathord{\sim}10$ so no relation space appears before the number.

Common mistakes

~ is a non-breaking space in LaTeX, and = alone is plain equals.
Instead of
```
x ~= 3
```
Write
```
x \approx 3
```
Typed "mod" comes out as italic m·o·d. \pmod sets it upright with the parentheses spaced correctly.
Instead of
```
17 \equiv 2 (mod 5)
```
Write
```
17 \equiv 2 \pmod{5}
```
\approxeq and \thickapprox live in amssymb; \approx itself needs no package.
Instead of
```
\approxeq  (without amssymb)
```
Write
```
\usepackage{amssymb}
```
Relation symbols only work in math mode; outside it LaTeX reports "Missing $ inserted".
Instead of
```
about \approx 3 ms
```
Write
```
about $\approx 3$~ms
```

In Overleaf, MathJax and Word

Overleaf and pdfLaTeX
\approx, \sim, \simeq, \cong, \equiv, \doteq and \pmod are built in; \approxeq and \thickapprox need amssymb.
MathJax and KaTeX
MathJax and KaTeX render all of these, including the amssymb ones, without extra configuration.
Microsoft Word
In a Word equation, \approx and \equiv typed with a trailing space are usually converted by Math AutoCorrect. For a complete formula, write it here and use Copy for Word.