Mathematics

Pure Unicode math per ISO 80000-2

Core Principle: Math is pure Unicode. Write readable, render beautiful.

This standard was designed for Lingenic Compose, but applies anywhere monospace fonts can be used. It uses no special commands for mathematical structure. Authors write mathematics using Unicode characters for everything — symbols, structure, layout. The source is monospace-aligned and readable — exactly as it should be.

Typography follows ISO 80000-2 (Quantities and units — Part 2: Mathematics). Variables and running indices are set in mathematical italic; vectors in bold italic. Mathematical constants whose value never changes — e, i, π — are set upright (roman), as are named functions (sin, exp, ln) and the differential operator d. Digits are upright.

Commands

.begin-math
.end-math

Fractions

.begin-math

  𝑎 + 𝑏
  ⎼⎼⎼⎼⎼
  𝑐 + 𝑑

.end-math

Character	Code point	Name
𝑎	U+1D44E	MATHEMATICAL ITALIC SMALL A
𝑏	U+1D44F	MATHEMATICAL ITALIC SMALL B
𝑐	U+1D450	MATHEMATICAL ITALIC SMALL C
𝑑	U+1D451	MATHEMATICAL ITALIC SMALL D
⎼	U+23BC	HORIZONTAL SCAN LINE-7

Quadratic Formula

.begin-math

          ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
    −𝑏 ± √ 𝑏² − 4𝑎𝑐
𝑥 = ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
          2𝑎

.end-math

Character	Code point	Name
⎽	U+23BD	HORIZONTAL SCAN LINE-9 (vinculum)
⎼	U+23BC	HORIZONTAL SCAN LINE-7 (fraction)
√	U+221A	SQUARE ROOT
±	U+00B1	PLUS-MINUS SIGN
−	U+2212	MINUS SIGN
²	U+00B2	SUPERSCRIPT TWO
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
𝑎	U+1D44E	MATHEMATICAL ITALIC SMALL A
𝑏	U+1D44F	MATHEMATICAL ITALIC SMALL B
𝑐	U+1D450	MATHEMATICAL ITALIC SMALL C

Distance Formula

.begin-math

          ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
𝑑(𝑃,𝑄) = √ (𝑥₂−𝑥₁)² + (𝑦₂−𝑦₁)²

.end-math

Character	Code point	Name
⎽	U+23BD	HORIZONTAL SCAN LINE-9 (vinculum)
√	U+221A	SQUARE ROOT
𝑑	U+1D451	MATHEMATICAL ITALIC SMALL D
𝑃	U+1D443	MATHEMATICAL ITALIC CAPITAL P
𝑄	U+1D444	MATHEMATICAL ITALIC CAPITAL Q
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
𝑦	U+1D466	MATHEMATICAL ITALIC SMALL Y
₁	U+2081	SUBSCRIPT ONE
₂	U+2082	SUBSCRIPT TWO
²	U+00B2	SUPERSCRIPT TWO

Matrices

3×3 Identity Matrix

.begin-math

    ⎡ 1  0  0 ⎤
𝑰 = ⎢ 0  1  0 ⎥
    ⎣ 0  0  1 ⎦

.end-math

Character	Code point	Name
𝑰	U+1D470	MATHEMATICAL ITALIC CAPITAL I
⎡	U+23A1	LEFT SQUARE BRACKET UPPER CORNER
⎢	U+23A2	LEFT SQUARE BRACKET EXTENSION
⎣	U+23A3	LEFT SQUARE BRACKET LOWER CORNER
⎤	U+23A4	RIGHT SQUARE BRACKET UPPER CORNER
⎥	U+23A5	RIGHT SQUARE BRACKET EXTENSION
⎦	U+23A6	RIGHT SQUARE BRACKET LOWER CORNER

2×2 Matrix

.begin-math

    ⎡ 𝑎  𝑏 ⎤
𝑴 = ⎢      ⎥
    ⎣ 𝑐  𝑑 ⎦

.end-math

Character	Code point	Name
𝑴	U+1D474	MATHEMATICAL ITALIC CAPITAL M
𝑎	U+1D44E	MATHEMATICAL ITALIC SMALL A
𝑏	U+1D44F	MATHEMATICAL ITALIC SMALL B
𝑐	U+1D450	MATHEMATICAL ITALIC SMALL C
𝑑	U+1D451	MATHEMATICAL ITALIC SMALL D
⎡	U+23A1	LEFT SQUARE BRACKET UPPER CORNER
⎢	U+23A2	LEFT SQUARE BRACKET EXTENSION
⎣	U+23A3	LEFT SQUARE BRACKET LOWER CORNER
⎤	U+23A4	RIGHT SQUARE BRACKET UPPER CORNER
⎥	U+23A5	RIGHT SQUARE BRACKET EXTENSION
⎦	U+23A6	RIGHT SQUARE BRACKET LOWER CORNER

Column Vector

.begin-math

    ⎡ 𝑥 ⎤
𝒗 = ⎢ 𝑦 ⎥
    ⎣ 𝑧 ⎦

.end-math

Character	Code point	Name
𝒗	U+1D499	MATHEMATICAL BOLD ITALIC SMALL V
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
𝑦	U+1D466	MATHEMATICAL ITALIC SMALL Y
𝑧	U+1D467	MATHEMATICAL ITALIC SMALL Z
⎡	U+23A1	LEFT SQUARE BRACKET UPPER CORNER
⎢	U+23A2	LEFT SQUARE BRACKET EXTENSION
⎣	U+23A3	LEFT SQUARE BRACKET LOWER CORNER
⎤	U+23A4	RIGHT SQUARE BRACKET UPPER CORNER
⎥	U+23A5	RIGHT SQUARE BRACKET EXTENSION
⎦	U+23A6	RIGHT SQUARE BRACKET LOWER CORNER

Cases (Piecewise Functions)

Absolute Value

.begin-math

       ⎧  𝑥   if 𝑥 > 0
|𝑥| =  ⎨  0   if 𝑥 = 0
       ⎩ −𝑥   if 𝑥 < 0

.end-math

Character	Code point	Name
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
⎧	U+23A7	LEFT CURLY BRACKET UPPER HOOK
⎨	U+23A8	LEFT CURLY BRACKET MIDDLE PIECE
⎩	U+23A9	LEFT CURLY BRACKET LOWER HOOK
−	U+2212	MINUS SIGN

Sign Function

.begin-math

          ⎧ −1   if 𝑥 < 0
sign(𝑥) = ⎨  0   if 𝑥 = 0
          ⎩  1   if 𝑥 > 0

.end-math

Character	Code point	Name
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
⎧	U+23A7	LEFT CURLY BRACKET UPPER HOOK
⎨	U+23A8	LEFT CURLY BRACKET MIDDLE PIECE
⎩	U+23A9	LEFT CURLY BRACKET LOWER HOOK
−	U+2212	MINUS SIGN

Summation and Products

Modest-sized operators, closer to Euler. Early printed math (1600s-1700s) used what the printer had in the type case. Giant stretchy ∑ is a 19th-century affectation — printers made them bigger because they could, not because it was clearer. Then textbooks copied textbooks. Traditional math typography treats ∑ like a diva. Compose treats it like any other character that showed up for work.

.begin-math

   𝑛
   ∑  𝑘 = 𝑛(𝑛+1)/2
  𝑘=1

.end-math

Character	Code point	Name
𝑛	U+1D45B	MATHEMATICAL ITALIC SMALL N
∑	U+2211	N-ARY SUMMATION
𝑘	U+1D458	MATHEMATICAL ITALIC SMALL K

.begin-math

   𝑛
   ∏  𝑘 = 𝑛!
  𝑘=1

.end-math

Character	Code point	Name
𝑛	U+1D45B	MATHEMATICAL ITALIC SMALL N
∏	U+220F	N-ARY PRODUCT
𝑘	U+1D458	MATHEMATICAL ITALIC SMALL K

Integrals

.begin-math

 ∞
 ⌠
 ⎮ e⁻ˣ² d𝑥 = √π
 ⌡
 0

.end-math

Character	Code point	Name
∞	U+221E	INFINITY
⌠	U+2320	TOP HALF INTEGRAL
⎮	U+23AE	INTEGRAL EXTENSION
⌡	U+2321	BOTTOM HALF INTEGRAL
e	U+0065	LATIN SMALL LETTER E (upright — Euler's number, ISO 80000-2)
√	U+221A	SQUARE ROOT
π	U+03C0	GREEK SMALL LETTER PI

.begin-math

 𝑏
 ⌠
 ⎮ 𝑓(𝑥) d𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
 ⌡
 𝑎

.end-math

Character	Code point	Name
𝑎	U+1D44E	MATHEMATICAL ITALIC SMALL A
𝑏	U+1D44F	MATHEMATICAL ITALIC SMALL B
𝑓	U+1D453	MATHEMATICAL ITALIC SMALL F
𝐹	U+1D439	MATHEMATICAL ITALIC CAPITAL F
⌠	U+2320	TOP HALF INTEGRAL
⎮	U+23AE	INTEGRAL EXTENSION
⌡	U+2321	BOTTOM HALF INTEGRAL
−	U+2212	MINUS SIGN

Limits

.begin-math

       sin(𝑥)
 lim   ⎼⎼⎼⎼⎼⎼ = 1
 𝑥→0     𝑥

.end-math

Character	Code point	Name
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
⎼	U+23BC	HORIZONTAL SCAN LINE-7
→	U+2192	RIGHTWARDS ARROW

Symbol Reference

Operators

Character	Code point	Name
×	U+00D7	MULTIPLICATION SIGN
÷	U+00F7	DIVISION SIGN
−	U+2212	MINUS SIGN
±	U+00B1	PLUS-MINUS SIGN
·	U+00B7	MIDDLE DOT

Relations

Character	Code point	Name
=	U+003D	EQUALS SIGN
≠	U+2260	NOT EQUAL TO
≤	U+2264	LESS-THAN OR EQUAL TO
≥	U+2265	GREATER-THAN OR EQUAL TO
≈	U+2248	ALMOST EQUAL TO

Greek Letters

Character	Code point	Name
α	U+03B1	GREEK SMALL LETTER ALPHA
β	U+03B2	GREEK SMALL LETTER BETA
γ	U+03B3	GREEK SMALL LETTER GAMMA
δ	U+03B4	GREEK SMALL LETTER DELTA
θ	U+03B8	GREEK SMALL LETTER THETA
λ	U+03BB	GREEK SMALL LETTER LAMDA
π	U+03C0	GREEK SMALL LETTER PI
σ	U+03C3	GREEK SMALL LETTER SIGMA
φ	U+03C6	GREEK SMALL LETTER PHI
ω	U+03C9	GREEK SMALL LETTER OMEGA
Σ	U+03A3	GREEK CAPITAL LETTER SIGMA
Π	U+03A0	GREEK CAPITAL LETTER PI

Variables (Mathematical Italic)

Character	Code point	Name
𝑎	U+1D44E	MATHEMATICAL ITALIC SMALL A
𝑏	U+1D44F	MATHEMATICAL ITALIC SMALL B
𝑐	U+1D450	MATHEMATICAL ITALIC SMALL C
𝑑	U+1D451	MATHEMATICAL ITALIC SMALL D
𝑓	U+1D453	MATHEMATICAL ITALIC SMALL F
𝑖	U+1D456	MATHEMATICAL ITALIC SMALL I
𝑗	U+1D457	MATHEMATICAL ITALIC SMALL J
𝑘	U+1D458	MATHEMATICAL ITALIC SMALL K
𝑚	U+1D45A	MATHEMATICAL ITALIC SMALL M
𝑛	U+1D45B	MATHEMATICAL ITALIC SMALL N
𝑥	U+1D465	MATHEMATICAL ITALIC SMALL X
𝑦	U+1D466	MATHEMATICAL ITALIC SMALL Y
𝑧	U+1D467	MATHEMATICAL ITALIC SMALL Z

Roots

Character	Code point	Name
√	U+221A	SQUARE ROOT
∛	U+221B	CUBE ROOT
∜	U+221C	FOURTH ROOT

Other Symbols

Character	Code point	Name
∞	U+221E	INFINITY
∂	U+2202	PARTIAL DIFFERENTIAL
∇	U+2207	NABLA
∴	U+2234	THEREFORE
∵	U+2235	BECAUSE

Superscripts

Character	Code point	Name
⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹	U+2070–U+2079	SUPERSCRIPT DIGITS
⁺ ⁻ ⁼ ⁽ ⁾	U+207A–U+207E	SUPERSCRIPT OPERATORS
ⁿ ⁱ	U+207F U+2071	SUPERSCRIPT LETTERS

Subscripts

Character	Code point	Name
₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉	U+2080–U+2089	SUBSCRIPT DIGITS
₊ ₋ ₌ ₍ ₎	U+208A–U+208E	SUBSCRIPT OPERATORS
ₐ ₑ ₕ ᵢ ⱼ ₖ ₗ ₘ ₙ ₒ ₚ ᵣ ₛ ₜ ᵤ ᵥ ₓ	U+2090–U+209C	SUBSCRIPT LETTERS

Structure Characters

Horizontal Rules

Character	Code point	Name
⎼	U+23BC	HORIZONTAL SCAN LINE-7 (fraction bar)
⎽	U+23BD	HORIZONTAL SCAN LINE-9 (vinculum for √)

Compose-Terminal font shifts content down to clear the vinculum automatically.

Brackets

Character	Code point	Name
⎡	U+23A1	LEFT SQUARE BRACKET UPPER CORNER
⎢	U+23A2	LEFT SQUARE BRACKET EXTENSION
⎣	U+23A3	LEFT SQUARE BRACKET LOWER CORNER
⎤	U+23A4	RIGHT SQUARE BRACKET UPPER CORNER
⎥	U+23A5	RIGHT SQUARE BRACKET EXTENSION
⎦	U+23A6	RIGHT SQUARE BRACKET LOWER CORNER

Braces

Character	Code point	Name
⎧	U+23A7	LEFT CURLY BRACKET UPPER HOOK
⎨	U+23A8	LEFT CURLY BRACKET MIDDLE PIECE
⎩	U+23A9	LEFT CURLY BRACKET LOWER HOOK
⎫	U+23AB	RIGHT CURLY BRACKET UPPER HOOK
⎬	U+23AC	RIGHT CURLY BRACKET MIDDLE PIECE
⎭	U+23AD	RIGHT CURLY BRACKET LOWER HOOK

Parentheses

Character	Code point	Name
⎛	U+239B	LEFT PARENTHESIS UPPER HOOK
⎜	U+239C	LEFT PARENTHESIS EXTENSION
⎝	U+239D	LEFT PARENTHESIS LOWER HOOK
⎞	U+239E	RIGHT PARENTHESIS UPPER HOOK
⎟	U+239F	RIGHT PARENTHESIS EXTENSION
⎠	U+23A0	RIGHT PARENTHESIS LOWER HOOK

Integrals

Character	Code point	Name
⌠	U+2320	TOP HALF INTEGRAL
⎮	U+23AE	INTEGRAL EXTENSION
⌡	U+2321	BOTTOM HALF INTEGRAL

Summation and Product

Character	Code point	Name
∑	U+2211	N-ARY SUMMATION
∏	U+220F	N-ARY PRODUCT

Expression Evaluation

.set-variable 𝑟 5
Circumference: .calculate 2 × π × 𝑟

Character	Code point	Name
𝑟	U+1D45F	MATHEMATICAL ITALIC SMALL R
×	U+00D7	MULTIPLICATION SIGN
π	U+03C0	GREEK SMALL LETTER PI

Built-in Constants

Character	Value	Code point	Name
π	3.14159265359	U+03C0	GREEK SMALL LETTER PI
e	2.71828182846	U+0065	LATIN SMALL LETTER E (upright — Euler's number, ISO 80000-2)

Functions

sin  cos  tan
asin acos atan
log  exp
abs  floor  ceil  round
min  max
sqrt

trigonometric
inverse trigonometric
natural log, exponential
rounding
minimum, maximum
square root

Design Philosophy

We render math in monospace. Math is a formal technical language — like code. Code moved to monospace decades ago; math just did. For those who prefer Computer Ancient fonts and antique calligraphic rendering, embed LaTeX.

Compose by ｢‍｣ Lingenic. Licensed under Apache 2.0.