The symbol ≈ means "approximately equal to" or "almost equal to". It is used in mathematics, science, and engineering to show that a value is not exact but is a close enough estimate for a given purpose, such as rounding or practical application.
The purpose of approximations
In many situations, an exact value is either unknowable, impractical to use, or simply unnecessary. This is where the approximately equal symbol becomes essential.
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Irrational numbers: Since numbers like πpi
𝜋
or 2the square root of 2 end-root
2√
have an infinite number of non-repeating digits, they can only be expressed exactly in symbolic form. For practical calculations, they must be approximated.
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**Example:**π≈3.14159pi is approximately equal to 3.14159
𝜋≈3.14159
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Scientific and engineering data: Measurements are never perfectly precise due to the limitations of instruments. The ≈ symbol denotes that the recorded value is the best possible measurement.
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Example: The speed of light c≈3.00×108c is approximately equal to 3.00 cross 10 to the eighth power
𝑐≈3.00×108
m/s, accounting for significant figures.
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Simplified models: Scientists and engineers often use simpler models to make complex calculations more manageable. The ≈ symbol indicates that a simpler model is being used in place of a more complex one.
- Example: When calculating gravity, physicists may approximate the Earth as a perfect sphere, even though it's an imperfect spheroid.
History of the symbol
The modern symbol for "almost equal to," ≈ (Unicode U+2248), was formally introduced by British mathematician Alfred Greenhill in 1892. Before and since, different symbols have been used to denote various forms of approximate equality.
- In 1911, the German Committee on Physical Units and Symbols also included ≈ in its standardization proposal.
- Older and related symbols for approximation include:
- Tilde (~): Sometimes used to mean "approximately equal to," but more commonly denotes similarity in geometry or proportionality in other contexts.
- Asymptotically equal (≃): Often used in mathematical analysis to describe the limiting behavior of functions.
- Approximately equal to (≅): Another variant sometimes used, though in some fields it specifically denotes geometric congruence or isomorphism.
Best practices for using ≈ vs. =
Choosing between the equals sign and the approximately equal sign depends on whether the relationship is exact or merely an estimate. A key guideline is: Do not say two things are equal unless they truly are equal.
| Symbol | Name | When to use | Example |
|---|---|---|---|
| = | Equals sign | To assert that two expressions have exactly the same value. | 12=0.5one-half equals 0.5 12=0.5 |
| ≈ | Approximately equal to | To indicate that a value is rounded, an estimate, or functionally equivalent within a certain margin of error. | 2≈1.414the square root of 2 end-root is approximately equal to 1.414 2√≈1.414 |
When performing multi-step calculations with approximations, it is good practice to use the ≈ symbol only for the first instance of an estimate. Subsequent steps can use the = sign to avoid over-complicating the notation, as long as no new rounding errors are introduced.
Other contexts and notations
The need for symbols representing approximate equality varies by discipline, leading to different conventions and interpretations.
- Logic: The tilde (~) is sometimes used to mean "not".
- Physics: Some physicists use the single tilde (~) for an even looser approximation, such as "on the order of" or "of the same magnitude".
- Engineering: Engineers must be careful about the degree of accuracy. The use of ≈ must be accompanied by an understanding of the acceptable margin of error for a specific application.
- Computer Science: In some programming languages,
==denotes a simple equality test, while===might indicate a stricter test for value and type. However, floating-point comparisons often require special functions due to the inherent approximations in representing decimal numbers.