Mathematics is the study of relationships between numbers, expressions, and symbols. It can involve algebra, calculus, geometry, trigonometry, statistics, etc. All of these are represented by symbols. Some are Greek letters such as alpha (α), beta (β), and so on; some are Roman letters like A, B, C; and some have specific shapes.
One such symbol is “approximately.” There are two primary approximately symbols: one with one line (~) and another with two lines (≈). In addition, there are two other approximately-related symbols that look similar to each other: ≃ and ≅. So what distinguishes them from one another?
It turns out that the key difference between them isn’t their appearance, but rather the use to which they’re put. And the distinction matters because of the context in which you’ll find them.
The Approximately Symbol
In mathematics, the approximately symbol means approximately equal to. In essence, this means that a value is close enough to another value, but not exactly equal. This may arise for any number of reasons, including when a result cannot be expressed as an exact number. In many cases, that will mean that it’s an irrational number, such as π, which is 3.1415926…, where the decimal doesn’t repeat itself and never ends.
But even rational numbers (like fractions) might be approximately equal if the resulting decimal goes on forever. For example, the fraction 7/3 is equal to 2.3333…, where the decimal repeats indefinitely. The same could be said for the fraction 8/3, which is 2.6666…
In those situations, the approximately symbol becomes useful, allowing the mathematician to express the concept that the value is close enough to another without needing to provide an infinitely long series of decimals.
The approximately symbol also comes into play during approximation calculations, where a certain value is used to approximate the actual value. That’s especially common in science and engineering, as well as in applied mathematics.
The Approximately Symbols
There are four related symbols, but they do not represent a universal scale of “degrees” of closeness:
• ~
• ≈
• ≃
• ≅
While these all appear similar, they aren’t interchangeable. They all share one thing in common, though: they contain a wavy line that represents the relationship between the values they represent.
Each approximately symbol contains one wavy line. However, depending on the symbol, they also contain different parts of the equals sign, which further defines what the symbol means.
Let’s go over each of them:
~
The tilde is the simplest of the approximately symbols, containing just a single line. It can also represent a relationship, as opposed to being equal to. It can represent a similarity in shape, size, or scale. For instance, if a triangle is roughly the same shape as another triangle, they’re similar. That means their angles match, but the sizes are different. Similarly, two rectangles can be similar to each other, based on how close they are in terms of scale and proportion.
In terms of its use, however, the tilde (~) is primarily used as the single-line approximately symbol.
≈
This symbol uses two wavy lines and is the clearest general-purpose relation for “approximately equal to.” It is a different Unicode character from the tilde. The extra line does not, by itself, promise a measurable greater degree of closeness; the required tolerance comes from the mathematical context.
This symbol also appears in mathematical contexts. For example, when we say that π is approximately 3.14, the approximately symbol is used here. We know that π is actually about 3.1415926535…, but it’s impossible to write it down completely. So, we use the approximately symbol to indicate that we’re using an approximation of the value.
In most contexts, the approximately symbol is the best choice for approximating a value, particularly in mathematical formulas.
≃
The next symbol in the family is ≃, Unicode Asymptotically Equal To. Its use varies by field, but it is commonly chosen for asymptotic or related equivalence. It should not be presented as a generic floating-point comparison operator.
≅
The last related symbol is ≅, Unicode Approximately Equal To. In mathematics it is also widely used for congruence or isomorphism, so its meaning must be read from the field and surrounding notation rather than assumed to mean only “very close numbers.”