About Math Symbols
What are math symbols?
Math symbols are characters like +, ≤ and √ that stand for values, operations and relations.
Mathematical symbols like these are separate Unicode characters, so a pasted ≤ is real text that works in email, documents and chat. Many live in the Mathematical Operators block, U+2200 to U+22FF, and a few older ones, such as ±, × and ÷, come from the Latin-1 character set.
What do math symbols mean?
The chart gives the meaning of common math symbols by area, with LaTeX commands where they exist. Where authors differ, as with ⊂ and ℕ, it notes both readings.
Math symbols by area, with their names, meanings and LaTeX commands
| Symbol | Name | Meaning | LaTeX |
|---|
| Arithmetic |
|---|
| + | Plus | Addition, as in 2 + 3 = 5 | + |
| − | Minus | Subtraction, as in 5 − 3 = 2 | - |
| × | Multiplication sign | Multiplication, as in 3 × 4 = 12 | \times |
| ÷ | Division sign (obelus) | Division, as in 12 ÷ 4 = 3 | \div |
| ± | Plus-minus sign | Plus or minus, as in 5 ± 0.2 for any value from 4.8 to 5.2 | \pm |
| ⋅ | Multiplication dot | Multiplication, as in 3 ⋅ 4 | \cdot |
| √ | Square root | The nonnegative value whose square is the number, as in √9 = 3 | \sqrt{x} |
| ∛ | Cube root | As in ∛8 = 2 | \sqrt[3]{x} |
| % | Percent | Parts per hundred, as in 25% = 0.25 | \% |
| ∞ | Infinity | A quantity without bound | \infty |
| Equality and inequality |
|---|
| = | Equals | Both sides have the same value | = |
| ≠ | Not equal to | As in 2 + 2 ≠ 5 | \neq |
| ≈ | Approximately equal to | As in √2 ≈ 1.41 | \approx |
| ≡ | Identical to | Identity, or congruence modulo n, as in 17 ≡ 2 (mod 5) | \equiv |
| ≅ | Congruent to | Figures with the same shape and size | \cong |
| ∼ | Similar to | Similar figures, as in △ABC ∼ △DEF | \sim |
| ∝ | Proportional to | As in y ∝ x | \propto |
| < | Less than | As in 3 < 5 | < |
| > | Greater than | As in 5 > 3 | > |
| ≤ | Less than or equal to | As in x ≤ 10 | \leq |
| ≥ | Greater than or equal to | As in x ≥ 18 | \geq |
| ≪ | Much less than | One value is far smaller than the other | \ll |
| Algebra |
|---|
| ∣ | Divides | As in 3 ∣ 12 | \mid |
| ⌊x⌋ | Floor | Rounds down, as in ⌊2.7⌋ = 2 | \lfloor x \rfloor |
| ⌈x⌉ | Ceiling | Rounds up, as in ⌈2.1⌉ = 3 | \lceil x \rceil |
| ‖ | Norm | The length of a vector, as in ‖v‖ | \Vert |
| ⋯ | Ellipsis | Terms left out, as in 1 + 2 + ⋯ + n | \cdots |
| Geometry |
|---|
| ∠ | Angle | As in ∠ABC | \angle |
| ⊥ | Perpendicular to | As in AB ⊥ CD | \perp |
| ∥ | Parallel to | As in AB ∥ CD | \parallel |
| △ | Triangle | As in △ABC | \triangle |
| ° | Degree | An angle measure, as in 90° | ^\circ |
| Calculus |
|---|
| ∫ | Integral | As in ∫ f(x) dx | \int |
| ∮ | Contour integral | An integral around a closed path | \oint |
| ∂ | Partial derivative | As in ∂f/∂x | \partial |
| ∇ | Nabla | The gradient, as in ∇f | \nabla |
| ∑ | Summation | The sum of a list of terms | \sum |
| ∏ | Product | The product of a list of terms | \prod |
| ′ | Prime | A derivative, as in f′(x) | f'(x) |
| ∘ | Composition | As in (f ∘ g)(x) = f(g(x)) | \circ |
| → | Tends to | As in x → 0 | \to |
| ↦ | Maps to | As in x ↦ x² | \mapsto |
| Set theory |
|---|
| ∈ | Element of | As in 3 ∈ ℕ | \in |
| ∉ | Not an element of | As in −1 ∉ ℕ | \notin |
| ⊂ | Subset of | As in A ⊂ B (some authors mean a proper subset) | \subset |
| ⊆ | Subset of or equal to | As in A ⊆ B | \subseteq |
| ∪ | Union | Everything in A or B | \cup |
| ∩ | Intersection | Everything in both A and B | \cap |
| ∖ | Set difference | Everything in A but not in B | \setminus |
| ∅ | Empty set | The set with no elements | \emptyset |
| ℕ | Natural numbers | 1, 2, 3 and so on (some authors include 0) | \mathbb{N} |
| ℤ | Integers | …, −1, 0, 1, … | \mathbb{Z} |
| ℚ | Rational numbers | Fractions of integers | \mathbb{Q} |
| ℝ | Real numbers | Every point on the number line | \mathbb{R} |
| ℂ | Complex numbers | Numbers of the form a + bi | \mathbb{C} |
| Logic |
|---|
| ¬ | Not | Negation, as in ¬P | \neg |
| ∧ | And | As in P ∧ Q | \land |
| ∨ | Or | As in P ∨ Q | \lor |
| ⇒ | Implies | As in P ⇒ Q | \Rightarrow |
| ⇔ | If and only if | As in P ⇔ Q | \Leftrightarrow |
| ∀ | For all | As in ∀x | \forall |
| ∃ | There exists | As in ∃x | \exists |
| ∴ | Therefore | Placed before a conclusion | \therefore |
| ∵ | Because | Placed before a reason | \because |
| ∎ | End of proof | Also called QED or the tombstone | None |
How to type math symbols
Keyboards have keys for +, =, < and >, but other symbols need a shortcut or copying.
- Windows Alt codes. Hold Alt and type the number on the numeric keypad. Codes with a leading 0, such as Alt+0177 for ±, Alt+0215 for × and Alt+0247 for ÷, use the Windows-1252 set, so they work on English and Western European setups.
- Alt codes without the 0. Alt+251 for √, Alt+243 for ≤ and Alt+242 for ≥ come from the US DOS code page, so they work on US English Windows, while other setups can type different characters.
- Mac shortcuts (US layout). Option+= types ≠, Option+Comma types ≤, Option+Period types ≥ and Option+X types ≈. Option+V gives √, Option+5 gives ∞ and Option+Shift+= gives ±.
- Word for Windows. Type a code point such as 2264 and press Alt+X for ≤. Or press Alt+= to start an equation, then type
\infty and a space for ∞.
- LaTeX. Symbols are commands in math mode, such as
\leq, \neq and \infty. Click a symbol above to see its command and whether it needs a package such as amssymb.
- HTML. Most have a named entity, such as
≤ for ≤ and ≠ for ≠. Any symbol also works as a numeric reference, such as ≤ for ≤.
What is the difference between ≈ and ≅?
≈ means approximately equal, as in π ≈ 3.14, while ≅ usually means congruent in geometry.
Their Unicode names are easy to mix up, because ≈ is ALMOST EQUAL TO and ≅ is APPROXIMATELY EQUAL TO. Go by the meaning, not the name, and use ≈ for rounded numbers.
≡ means identical to, or congruent in modular arithmetic, as in 17 ≡ 2 (mod 5). ∼ means similar to, and in statistics, X ∼ N(0, 1) reads as X is distributed as N(0, 1).
Which math symbols have look-alikes?
Several symbols look like keyboard characters or letters but have different code points.
- − minus sign (U+2212). The proper minus for typeset math. Code needs the keyboard hyphen-minus (U+002D), because JavaScript and Python read only that one as minus, so a pasted −5 is a syntax error.
- × multiplication sign (U+00D7). In typeset math, a letter x between numbers reads as a variable, so write 3 × 4 with the sign.
- ⋅ dot operator (U+22C5). The multiplication dot, as in 3 ⋅ 4. The middle dot · (U+00B7) is a punctuation mark that looks almost the same.
- ′ and ″ primes (U+2032 and U+2033). For derivatives, feet and inches, as in f′(x) and 5′ 11″. The apostrophe and quotation mark are different characters.
- ∣ divides (U+2223). For divisibility, as in 3 ∣ 12. The keyboard vertical bar | is a separate character.
- ∅ empty set (U+2205). Not the Latin letter Ø (U+00D8), which looks similar.
How do you write a fraction as a symbol?
Unicode has ready-made characters for common fractions such as ½, ⅓ and ¾.
For any other fraction, put the fraction slash ⁄ (U+2044) between superscript and subscript digits, as in ⁵⁄₁₂, or write 5/12. Our superscript generator makes both the raised and the lowered digits.
On English and Western European Windows, only ¼, ½ and ¾ have standard Alt codes, Alt+0188, Alt+0189 and Alt+0190. In LaTeX, write \frac{1}{2} for one half.
Common mistakes
A symbol that's right in a document can be wrong in code, in a course's notation or in LaTeX.
- Pasting ≤, ≥ or ≠ into code. JavaScript and Python need the two-character operators
<=, >= and !=, and Excel formulas use <> for not equal.
- Mixing up ⊂ and ⊆. Some books use ⊂ only for a proper subset, so check which convention your course or paper follows.
- Forgetting amssymb. In pdfLaTeX,
\therefore and \varnothing need \usepackage{amssymb} in the preamble. Without that line, LaTeX reports an undefined control sequence error, and any PDF it still produces leaves the symbols out.
Common Use Cases
People copy math symbols into places that have no equation editor.
- Homework, worksheets and emails. Inequalities like 0 ≤ x ≤ 1, or an answer of ≈ 4.2, in plain text.
- Spreadsheets and reports. Labels such as ±2% or ≥ 18 where no formula is needed.
- Web pages. The HTML entity, such as
≤, or the character itself in a UTF-8 page.
- Chat and social posts. ∞, ∴ and ≠ as quick shorthand.
For Greek letters like π, θ and Σ, see the Greek alphabet page. Exponents like x² are on the squared symbols page, and the text symbols page links every symbol collection on the site.
Frequently Asked Questions
What is the square root symbol?
The square root symbol is √, U+221A in Unicode. Type it with Alt+251 on US English Windows or Option+V on a US-layout Mac.
How do you type the less than or equal to sign?
Press Alt+243 on US English Windows or Option+Comma on a US-layout Mac to type ≤. For the greater than or equal to sign ≥, use Alt+242 or Option+Period.
What does the therefore symbol (∴) mean?
∴ means therefore and comes before the conclusion of an argument. The upside-down ∵ means because.
What is the approximately equal symbol?
The approximately equal symbol is ≈, which Unicode names ALMOST EQUAL TO. Type it with Alt+247 on US English Windows or Option+X on a US-layout Mac.
What does the plus-minus sign (±) mean?
± means plus or minus, so x = ±3 means x is 3 or −3. Type it with Alt+0177 on English and Western European Windows, or Option+Shift+= on a US-layout Mac.
How do you type the infinity symbol?
Press Alt+236 on US English Windows or Option+5 on a US-layout Mac to type ∞. In LaTeX, write \infty.
What is the division symbol called?
÷ is called the division sign or the obelus. Type it with Alt+0247 on English and Western European Windows, or Option+Slash on a US-layout Mac.
