At school you multiplied with ×. In physics a little dot turns up, ·, and in chemistry and AI there are arrows of every kind. That’s three signs, each with two readings. Here are all six.
· The dot: times
- Say it
- “times”. m·g·h is read “m times g times h”.
- It means
- Between ordinary numbers or letters, the raised dot is the same as ×: multiply.
It’s used because × gets confused with the letter x, and because it takes up less room. Sometimes it isn’t even written: mgh also means m × g × h. When two letters sit side by side in a formula, they’re almost always being multiplied.
Real example · in the lesson Joules, watts or calories? Energy and power without fear
ΔT = Q / (m · c): the heat divided by the mass times the specific heat.
There’s a second use. Between two arrows (vectors), the dot is the dot product: you multiply the numbers in the same position and add up the results. The result is no longer an arrow but a number. It says how much the two point the same way.
Real example · in the lesson How does a computer turn a photo into numbers?
cos θ = a·b / (‖a‖‖b‖): how alike two arrows’ directions are.
→ The arrow: turns into, or tends to
- Say it
- In chemistry, “gives” or “turns into”. In maths, “tends to”.
- It means
- In a reaction, it separates what's there before (left) from what comes out after (right). In maths, it says what value something gets closer to.
In chemistry, a reaction is written like a recipe: ingredients on the left, the arrow, and what comes out on the right. In maths and AI, “x → 0” is read “x tends to zero”: x gets closer and closer to 0, even if it never gets there.
Real example · in the lesson Can you make something from nothing? Conservation, no fear
CH₄ + 2 O₂ → CO₂ + 2 H₂O: methane burns with oxygen and gives carbon dioxide and water.
Real example · in the lesson Logistic regression or neural network? Same neuron
With separable data the optimum doesn’t exist (‖w‖ → ∞): the weights grow without end.
⇒ ⇔ The double arrow: implies
- Say it
- ⇒ is read “implies” or “then”. ⇔ is read “if and only if” or “is equivalent to”.
- It means
- A ⇒ B means: if A is true, B is true too. A ⇔ B means it works both ways.
It’s the short way of writing “therefore”. x > 2 ⇒ x > 1: if x is greater than 2, it’s certainly greater than 1. It doesn’t work backwards: 1.5 is greater than 1 and isn’t greater than 2. When it does work both ways, the arrow has two heads: ⇔.
Real example · in the lesson How much is 7×10²⁷? Huge numbers without fear
N = Π fᵢ ⇒ log N = Σ log fᵢ: if N is a product, its logarithm is a sum.
Try it
Pick a formula and tap each piece to see how it’s read. The whole sentence is underneath.
Challenge
Sources
- The multiplication dot, the dot product, → (tends to), ⇒ and ⇔: ISO 80000-2:2019, Quantities and units — Part 2: Mathematics (iso.org).
- The arrow in chemical equations: IUPAC, Quantities, Units and Symbols in Physical Chemistry (“Green Book”), 3rd ed., 2007 (iupac.org).
How this cheat sheet was made: written with Claude’s help from the sources above and translated from the Spanish original. The examples are copied from the site’s lessons. The drawings are original.