The wavy equal sign (≈) means “approximately equal to.” It tells you two values are close to each other but not exactly the same. You’ll see it most often when a number has been rounded, estimated, or simplified, like π ≈ 3.14 (since pi’s actual decimal expansion goes on forever).
But there’s more than one wavy symbol in math, and each one means something different. Here’s how to tell them apart.
The Double Wavy Lines: Approximately Equal (≈)
The most common wavy equal sign is two parallel wavy lines stacked on top of each other: ≈. This is formally called the “approximately equal to” symbol. Under the international standard for mathematical notation (ISO 80000-2), writing a ≈ b means “a is approximately equal to b.”
You’ll run into this symbol whenever exact precision isn’t necessary or isn’t possible. A few everyday examples:
- Rounding numbers: 1/3 ≈ 0.33, because the actual decimal (0.3333…) never ends
- Physical measurements: The speed of light ≈ 300,000 km/s (the precise value is 299,792.458 km/s)
- Estimates and statistics: “The population of Tokyo ≈ 14 million”
The symbol was introduced by British mathematician Alfred Greenhill in 1892, in his book on elliptic functions. It has been the standard notation for approximation ever since.
The Single Wavy Line: Similar or Proportional (~)
A single wavy line, the tilde (~), carries different meanings depending on context. In geometry, it means “is similar to.” Two shapes are similar when they have the same angles and their sides are in the same proportion, like a small triangle that’s a scaled-up version of a larger one. Writing △ABC ~ △DEF means those two triangles are similar.
Outside geometry, the single tilde can also mean “is proportional to.” If a ~ b, it means that as one value changes, the other changes at a consistent rate. Some textbooks use the symbol ∝ for proportionality instead, so you may see either one. The ISO standard lists both ~ and ∝ as valid notations for proportionality.
In casual, non-mathematical writing, people sometimes use the tilde loosely to mean “about” or “around,” as in “~30 minutes.” This isn’t formal notation, but it’s widely understood.
Wavy Line Over an Equal Sign: Congruent (≅)
If you see a tilde sitting directly above a regular equal sign (≅), that’s the congruence symbol. It means “exactly equal in size and shape.” This one shows up almost exclusively in geometry. Two line segments, angles, or triangles are congruent when they match perfectly, not just in proportion (like similarity) but in actual measurement.
Think of it this way: similar shapes look alike but can be different sizes, while congruent shapes are identical copies. The congruence symbol combines the tilde (similarity) with the equal sign (same measure) into one notation.
How to Tell Them Apart at a Glance
- ≈ (two wavy lines): approximately equal to, used for rounded or estimated values
- ~ (one wavy line): similar to (geometry) or proportional to
- ≅ (wavy line over equal sign): congruent, meaning identical in size and shape
The key detail is how many wavy lines you’re looking at and whether there’s a straight equal sign underneath. Two wavy lines alone means “close enough.” One wavy line over a straight line means “exactly the same shape and size.”
How to Type the ≈ Symbol
On a Mac, press Option + X to type ≈. On Windows, there’s no single keyboard shortcut, but you can hold Alt and type 247 on the number pad in many programs. You can also search “approximately equal” in your operating system’s character map or emoji picker. In Google Docs, go to Insert > Special Characters and search for “approximately.”
If you’re writing in a programming or scientific context, most languages and tools accept Unicode character U+2248 for ≈. In LaTeX, the command is \approx.