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One wave or two, and the difference survives only in print.
One wave is the tilde operator: asymptotic equivalence in analysis, similarity in geometry, and 'is distributed as' in statistics, decided by what sits either side of it. Two stacked waves is approximation. In handwriting the difference disappears entirely, which is why a reader who learnt the double form first tends to treat a single wave as a careless version of it — exactly backwards, since they are separate operators with separate meanings.
One wave marks equivalence or distribution; two stacked waves mark approximation.
| Tilde operator | Approximately equal to sign | |
|---|---|---|
| Strokes | One | Two, stacked |
| Means | Asymptotic equivalence, geometric similarity, or 'is distributed as' | Approximately equal in value |
| Decided by | What sits on either side of it | Nothing — the meaning is fixed |
| Relationship (from Symbolwise's records) | Commonly confused with Approximately equal to sign — One wave marks equivalence or distribution; two stacked waves mark approximation. | Commonly confused with Tilde operator — One wave is equivalence, similarity or distribution; two waves is approximation. |
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics
International Organization for Standardization · retrieved 2026-07-31
Paywalled and NOT READ for this batch. Registered so the standard can be named as existing and as covering mathematical signs; it supports no meaning in any record. See docs/paywalled-standards.md.
NIST/SEMATECH e-Handbook of Statistical Methods
National Institute of Standards and Technology · retrieved 2026-08-02
A freely published reference from a national metrology institute. Authoritative for standard statistical practice and the notation it uses. It records convention rather than mandating it, and no record cites it for a claim that notation is prescribed.
The tilde on a keyboard is a punctuation mark and a diacritic, encoded at U+007E. The mathematical tilde operator is a separate character at U+223C, drawn a little longer and set at a different height. Mathematical text typed with the keyboard character is technically the wrong character and is universally understood, which is why almost everybody does it — and why a typeset paper and a forum post render the same statement slightly differently.
Alpha is a Greek letter and stands for a quantity — an angle, a coefficient, a significance level. The proportionality sign is an operator and stands between two quantities to say they keep a constant ratio. The drawings differ in that the proportionality sign's right-hand stroke runs out in an open curve rather than closing into the letter's tail.
Look at the left-hand edge. A straight flat back is an AND gate: the output is true only when every input is. A curved concave back is an OR gate: true when any input is. Everything else about the shapes is similar, which is why the back edge is worth checking deliberately rather than at a glance.
Look at what follows. A prefix never stands alone — it is glued to a unit symbol, as in kW or kg. A constant stands by itself in an equation. Careful physical writing subscripts the constant as k_B for exactly this reason, and the kelvin's capital K is a third claimant to the same letter.