At school, two things were either equal or they weren’t. Science has more shades: nearly equal, the same order, one hugely bigger than the other… Each shade has its own sign, and almost all of them look like the equals sign or the less-than sign.
≈ ≈: approximately equal
- Say it
- “approximately equal to” or “roughly”.
- It means
- The two sides aren't exactly equal, but they're so close that, for what you're doing, it makes no difference.
It’s used when you round or when the exact result has too many digits. 250 / 18.015 gives 13.8773… and you write ≈ 13.9.
Real example · in the lesson What is a wall made of? Atoms and elements without fear
A glass with 250 g of water is 250 / 18.015 ≈ 13.9 mol.
~ ~: on the order of
- Say it
- “on the order of” or “about”. ~10⁵ is read “on the order of a hundred thousand”.
- It means
- In front of a number, ~ says only its size matters: how many zeros it has, not the exact digits.
It’s even vaguer than ≈. If something is ~10⁵, it could be 50,000 or 300,000: what matters is that it isn’t 1,000 or 10 million.
Real example · in the lesson What is a wall made of? Atoms and elements without fear
R_atom/R_nucleus ~ 10⁵: the atom is about a hundred thousand times wider than its nucleus.
≤ ≥ ≤ and ≥: less or equal, greater or equal
- Say it
- “less than or equal to” and “greater than or equal to”.
- It means
- They're the less-than and greater-than signs with a little equals line: as well as smaller (or bigger), being equal also counts.
x < 10 leaves 10 out; x ≤ 10 lets it in. It’s the difference between “under 18” and “18 or under”.
Real example · in the lesson How much is 7×10²⁷? Huge numbers without fear
Mantissa a, with 1 ≤ |a| < 10: it can be 1, but never 10.
≠ ≠: not equal to
- Say it
- “not equal to”.
- It means
- A crossed-out equals sign: the two sides aren't equal.
It’s mostly used to rule out a case that would cause trouble, like dividing by zero: “for every x ≠ 0”.
Real example · in the lesson How much is 7×10²⁷? Huge numbers without fear
Every real x ≠ 0 can be written uniquely as x = a·10ⁿ.
≪ ≫ ≪ and ≫: much, much less and much, much greater
- Say it
- “much less than” and “much greater than”.
- It means
- One side is so small (or so big) next to the other that it can be ignored.
There’s no fixed figure: it depends on how precise you need to be. As a rule of thumb, think ten or a hundred times or more. If v ≪ c, the speed is so small compared with the speed of light that the everyday formulas work.
Real example · in the lesson Joules, watts or calories? Energy and power without fear
If t ≪ τ, the system is almost adiabatic: it doesn’t have time to lose heat.
≡ ≡: is defined as
- Say it
- “is defined as” or “is identical to”.
- It means
- An equals sign with three lines. It isn't a result you calculate, but an agreement: this equals that because it was decided so.
It’s used for definitions. An electronvolt isn’t measured: it’s defined as the energy an electron gains crossing one volt, and its value in joules is exact by definition.
Real example · in the lesson Joules, watts or calories? Energy and power without fear
1 eV ≡ 1.602 176 634×10⁻¹⁹ J.
|x| |x|: the absolute value
- Say it
- “the absolute value of x” or “mod x”.
- It means
- Two bars around a number take away its sign: |−7| = 7 and |7| = 7. It's the distance from zero.
It’s used to talk about how far apart two things are, whichever is bigger: |a − b| is the distance between a and b. That’s why it shows up next to ≈ and tolerances.
Real example · in the lesson Why does the computer say 0.1 + 0.2 isn't 0.3?
fl(x) = x(1 + δ), |δ| ≤ u: the relative rounding error, up or down, is never more than u.
Try it
Type two numbers and a margin for ≈. The widget tells you which signs hold. Try a = 1 and b = 500.
Challenge
Sources
- ≈, ~, ≤, ≥, ≠, ≪, ≫, ≡ and |x|: ISO 80000-2:2019, Quantities and units — Part 2: Mathematics (iso.org).
How this cheat sheet was made: written with Claude’s help from the source above and translated from the Spanish original. The examples are copied from the site’s lessons. The drawings are original.