Proportional to (∝) in LaTeX

Q: What is the LaTeX symbol for proportional to?

\propto, which renders the proportionality sign, e.g. $y \propto x$.

Q: What does \propto mean compared with =?

a \propto b means a equals some constant times b; the constant is unspecified, whereas = fixes the exact value.

\propto produces the proportional-to relation, stating that one quantity is a constant multiple of another, as in y proportional to x.

\propto

LaTeX command

\propto

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The command \propto renders the proportionality sign. $y \propto x$ means y equals some fixed constant times x, with the constant of proportionality left unspecified. It is a binary relation, so it carries relational spacing and takes an expression on either side.

It handles direct, inverse and joint proportion. For inverse proportion put the reciprocal on the right: F \propto \frac{1}{r^2} for an inverse-square law, or equivalently $F \propto r^{- 2}$ . In Bayesian statistics it neatly expresses a posterior 'up to a constant', dropping the normalising factor.

\propto is core LaTeX and KaTeX, so no package is needed; a slanted variant \varpropto exists in amssymb but is rarely required. Keep it distinct from \sim, which means 'similar' or 'distributed as' rather than 'a constant multiple of'.

Examples

y \propto x

$y \propto x$

direct proportionality

F \propto \frac{1}{r ^{2}}

$F \propto \frac{1}{r^2}$

an inverse-square law

P (θ ∣ x) \propto P (x ∣ θ) P (θ)

$P(\theta \mid x) \propto P(x \mid \theta)\,P(\theta)$

a Bayesian posterior up to a constant

T^{2} \propto a^{3}

$T^2 \propto a^3$

Kepler's third law

E \propto m

$E \propto m$

a simple direct relation

Tips and common mistakes

\propto is core LaTeX and KaTeX; the slanted variant \varpropto (amssymb) is seldom needed.
Read it as 'equal up to a constant factor'; the constant of proportionality is deliberately unspecified.
For inverse proportion write y \propto \frac{1}{x} or $y \propto x^{- 1}$ , not y \propto -x.
It is a relation, not an operator, so do not stack limits on it.

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FAQ

What is the LaTeX symbol for proportional to?

\propto, which renders the proportionality sign, e.g. $y \propto x$ .

How do I write inverse proportionality?

Put the reciprocal on the right: $y \propto \frac{1}{x}$ or $y \propto x^{- 1}$ .

What does \propto mean compared with =?

$a \propto b$ means a equals some constant times b; the constant is unspecified, whereas = fixes the exact value.

Proportional to (∝) in LaTeX

Examples

Tips and common mistakes

Working from handwritten notes?

FAQ

See also