The tilde (~) has several distinct meanings in math depending on context. It can indicate similarity in geometry, negation in logic, a distribution in statistics, or an approximation in general usage. Because the same symbol appears across so many branches, understanding the context is the only way to know which meaning applies.
Similarity in Geometry
In geometry, the tilde means “is similar to.” Two shapes are similar when their corresponding angles are equal and their corresponding sides are in the same proportion. Writing △ABC ~ △DEF means triangle ABC is similar to triangle DEF. You can think of similar figures as having the same shape but not necessarily the same size. One can be mapped onto the other through a combination of rotation, reflection, and scaling.
This notation extends beyond triangles. Parallelogram ABCD ~ parallelogram MNOP uses the same symbol to express the same idea: matching angles, proportional sides.
Approximation
In everyday math and science, the tilde is a shorthand for “approximately.” The Sun’s diameter is ~1.4 billion meters means it’s roughly 1.4 billion meters. This is one of the most common uses outside of formal mathematics, and you’ll see it frequently in physics, earth science, and engineering contexts where exact values aren’t necessary or aren’t known.
The tilde used this way is less precise than the formal approximation symbol (≈). In practice, many writers treat them interchangeably, but in strict notation, ≈ means “approximately equal to” while ~ can be looser, sometimes meaning “on the same order of magnitude.”
Negation in Logic
In propositional logic, the tilde placed before a statement means “not.” If p stands for “I race,” then ~p means “I do not race.” The tilde can negate a single variable or an entire compound statement. For example, ~(p ∧ q) means “it is not the case that both p and q are true,” which is different from ~p ∧ q, where only the first part is negated.
So if p is “I race” and q is “I win,” then p ∧ ~q translates to “I race and I do not win,” while ~p ∧ ~q means “I do not race and I do not win.”
Distribution in Statistics
In statistics and probability, the tilde means “is distributed as” or “follows the distribution.” The expression X ~ N(0, 1) is read as “the random variable X follows a standard normal distribution with mean 0 and variance 1.” This is one of the most common notations in any statistics course or textbook.
When written between two variables, X ~ Y means “X has the same distribution as Y.” The tilde here doesn’t say anything about the values being equal, only that the two variables share the same type of probability distribution.
This notation also shows up in statistical programming. In the R language, the tilde defines relationships between variables in model formulas. Writing y ~ x tells R to model y as a function of x.
Equivalence Relations in Set Theory
In set theory, x ~ y means x and y are related by an equivalence relation. An equivalence relation is a way of grouping elements that are “the same” in some defined sense. For the tilde to represent a valid equivalence relation, three properties must hold:
- Reflexive: every element is related to itself (x ~ x)
- Symmetric: if x ~ y, then y ~ x
- Transitive: if x ~ y and y ~ z, then x ~ z
A familiar example: if you define two integers as equivalent when they leave the same remainder after dividing by 5, that’s an equivalence relation. The numbers 3, 8, and 13 would all be equivalent under this rule because they all leave a remainder of 3.
Asymptotic Equivalence
In calculus and computer science, f ~ g means that two functions become essentially identical as their input grows toward infinity. Formally, f ~ g when the ratio f(x)/g(x) approaches 1 as x goes to infinity. This is read as “f is asymptotically equivalent to g” or simply “f tilde g.”
This is a stronger statement than saying two functions grow at the same rate. It says they converge to the same value in a very specific way: their ratio settles to exactly 1. It’s widely used in algorithm analysis and number theory to describe the long-run behavior of functions without worrying about small differences that shrink to irrelevance.
Proportionality
In some formal standards, including the international standard for mathematical notation (ISO 80000-2), a ~ b means “a is proportional to b.” This is an alternative to the more common proportionality symbol (∝). You’re more likely to encounter this in physics and engineering texts that follow ISO conventions than in a typical math class, but it’s worth knowing when you see a tilde between two quantities and none of the other meanings seem to fit.
How to Tell Which Meaning Applies
Context is everything. If you’re in a geometry problem and see triangles on either side, it means similarity. If it’s in front of a logical statement, it means negation. If a random variable is on the left and a distribution name is on the right, it means “follows the distribution of.” And if it appears between two plain numbers or measurements, it almost certainly means “approximately.”
The tilde also sometimes appears above a letter (like x̃) rather than between two expressions. In statistics, x̃ typically denotes the median of a dataset. In other contexts, the overhead tilde simply marks a modified version of a variable, the way a prime mark (x’) or a hat (x̂) might.