What is the difference between ~ and ≈?

Both indicate approximation, but $\approx$ is the more standard and widely recognized symbol for "approximately equal to" in most textbooks. The tilde $\sim$ is more casual for approximation and has additional formal meanings (equivalence relations, distributions, asymptotic behavior) that $\approx$ does not carry. In homework, $\approx$ is usually the safer choice for "approximately equal."

Tilde — Definition, Formula & Examples

Tilde is the wavy symbol ~ used in mathematics to indicate that two values are approximately equal, that two things are equivalent, or that quantities are proportional. When you write

a \sim b

, you typically mean "

a

is approximately

b

" or "

a

is related to

b

in a defined way."

The tilde symbol ( $\sim$ ) serves multiple roles depending on context. In estimation, $a \sim b$ denotes that $a$ approximates $b$ . In set theory and abstract algebra, $a \sim b$ indicates that $a$ and $b$ belong to the same equivalence class under a specified equivalence relation. In asymptotic analysis, $f(x) \sim g(x)$ as $x \to \infty$ means $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1$ . In statistics, $X \sim N(\mu, \sigma^2)$ states that the random variable $X$ follows a given distribution.

How It Works

The tilde's meaning depends on the branch of math you are working in. In everyday calculations and science classes,

\sim

is a shorthand for "approximately equal to" — a quicker, less formal alternative to the symbol

\approx

. In more advanced courses,

\sim

describes precise relationships: an equivalence relation in algebra, an asymptotic relationship in calculus, or a probability distribution in statistics. To read it correctly, always check the surrounding context. If you see

\pi \sim 3.14

, it means roughly equal; if you see

X \sim \text{Uniform}(0,1)

, it means

X

is distributed as a uniform random variable on

[0,1]

Worked Example

Problem: A city's population is 1,283,947. Express this using the tilde for a quick approximation.

Identify the precision needed: For a rough estimate, rounding to the nearest hundred thousand is practical.

Round the value: 1,283,947 rounded to the nearest hundred thousand is 1,300,000.

1{,}283{,}947 \approx 1{,}300{,}000

Write using the tilde: Using tilde notation to express the approximation concisely:

\text{Population} \sim 1.3 \text{ million}

Answer: The population is ~ 1.3 million.

Another Example

Problem: Show that

\sqrt{50}

is approximately 7.07 using tilde notation, and verify.

Compute the exact value: Simplify the radical:

\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}

\sqrt{50} = 5\sqrt{2}

Use a known approximation: Since

\sqrt{2} \sim 1.414

, multiply by 5.

5 \times 1.414 = 7.07

State the result: Write the approximation with the tilde symbol.

\sqrt{50} \sim 7.07

Answer:

\sqrt{50} \sim 7.07

Why It Matters

The tilde appears frequently in AP Statistics, where

X \sim N(0,1)

is standard notation for describing how a random variable is distributed. In physics and engineering, using

\sim

for quick order-of-magnitude estimates helps you check whether an answer is reasonable before committing to a full calculation. Recognizing the tilde's different meanings across contexts is essential once you move beyond arithmetic into algebra, analysis, and data science.

Common Mistakes

Mistake: Treating ~ as meaning exactly equal

Correction: The tilde indicates approximation or a defined relation, not strict equality. If you need exact equality, use the equals sign

=

Mistake: Using ~ and ≈ interchangeably in formal proofs or statistics

Correction: In statistics,

X \sim N(0,1)

means "

X

follows a normal distribution" — replacing

\sim

with

\approx

would be incorrect and confusing. Always match the symbol to its intended meaning in context.

Related Terms

Accuracy — Approximation quality the tilde implies
Measurement — Tilde used when reporting measured estimates
Inequality — Another relational symbol comparing values
Braces — Notation symbol used in math expressions
Brackets — Notation symbol often paired with tilde usage
Greek Alphabet — Source of many math symbols alongside tilde
Mean — Often approximated and reported with tilde
Arithmetic Mean — Value frequently expressed as an approximation