Tilde - Mathematics

Mathematics

In mathematics, the tilde operator (Unicode U+223C), sometimes called "twiddle", is often used to denote an equivalence relation between two objects. Thus "x ~ y" means "x is equivalent to y". It is a weaker statement than stating that x equals y. The expression "x ~ y" is sometimes read aloud as "x twiddles y", perhaps as an analogue to the verbal expression of "x = y".

In the 1800s x ~ y could also mean | x − y | (the absolute value of x − y).

The tilde can indicate approximate equality in a variety of ways. It can be used to denote the asymptotic equality of two functions. For example, f (x) ~ g(x), means that lim_{x → ∞} f( x) ∕ g(x) = 1. A tilde is also used to indicate "approximately equal to" (e.g. 1.902 ~= 2). This usage probably developed as a typed alternative to the libra symbol used for the same purpose in written mathematics, which is an equal sign with the upper bar replaced by a bar with an upward hump, bump, or loop in the middle (♎) or, sometimes, a tilde (≃). The symbol "≈" is also used for this purpose. Similarly, a tilde can be used on its own between two expressions (e.g. a ~ 0.1) to state that the two are of the same order of magnitude.

A tilde in front of a single quantity can mean "approximately" or "about".

In statistics and probability theory, ⟨~⟩ means "is distributed as". See random variable. A tilde placed on top of a variable is sometimes used to represent the median of that variable.

A tilde can also be used to represent geometric similarity, for example: ∆ABC ~ ∆DEF (meaning "triangle ABC is similar to triangle DEF"). A triple tilde (≋) is often used to show congruence, an equivalence relation in geometry.

All of the above usages are in-line tildes, not raised.

The tilde is also used as a modifier for symbols. The symbol "" is often pronounced "eff twiddle" or, particularly in American English, "eff wiggle". This can be used to denote the Fourier transform of f, or a lift of f, and can have a variety of other meanings depending on the context.

A tilde placed below a letter in mathematics can represent a vector quantity.