The technical modes keep showing a sort of pointy E, Σ, with letters stuck all around it. It looks scary, but it’s just a short way of writing “add all this up”. If you can add up a shopping list, you can read it.
Σ Capital sigma: add them all
- Say it
- “sum” or “the sum of”. Σ xᵢ is read “the sum of the x's”.
- It means
- Σ is an order: add up every item in the list that follows.
Σ is the Greek capital S, for “sum”. Instead of writing x₁ + x₂ + x₃ + x₄ + x₅, you write Σ xᵢ. Sometimes it has numbers above and below to say where the list starts and ends: with a 1 below and a 5 above, you add from the first to the fifth.
Real example · in the lesson What is a wall made of? Atoms and elements without fear
Aᵣ(C) = Σᵢ xᵢ · Aᵣ(ⁱC): the average mass of carbon, adding each isotope times the fraction of it there is.
xᵢ The index: the number in the list
- Say it
- xᵢ is read “x sub i” or “the i-th x”.
- It means
- The i at the bottom is a number that keeps changing: 1, 2, 3… It lets you talk about any item in the list without writing them all out.
If you have five test marks, x₁ is the first, x₂ the second, and so on up to x₅. When you write xᵢ, the i is a wildcard: it stands for any of them. That’s why Σ and indices almost always go together. Sometimes j or k is used instead of i, and it means the same.
Real example · in the lesson How does a computer turn a photo into numbers?
‖x‖ = √(Σᵢ xᵢ²): the length of an arrow, adding up the square of each of its numbers.
x̄ The bar on top: the mean
- Say it
- x̄ is read “x bar” or “the mean of x”.
- It means
- A line over a letter is usually its mean: the sum of them all divided by how many there are.
The good old average, the one you use for marks, written in symbols: x̄ = Σ xᵢ / n, where n is how many numbers there are. In energy, P̄ is the average power: what you’d get if you spread all the energy evenly over the whole time.
Real example · in the lesson Joules, watts or calories? Energy and power without fear
P = dE/dt (instantaneous); P̄ = ΔE/Δt (mean).
Π Capital pi: multiply them all
- Say it
- “product” or “the product of”. Π fᵢ is read “the product of the f's”.
- It means
- Π is like Σ, but instead of adding, it multiplies everything in the list.
Π is the Greek capital P, for “product”. Π fᵢ means f₁ × f₂ × f₃ × … It doesn’t come up often, but when it does, now you can read it.
Real example · in the lesson How much is 7×10²⁷? Huge numbers without fear
N = Π fᵢ ⇒ log N = Σ log fᵢ: a Fermi estimate is a multiplication of factors.
Try it
Change the numbers in the list and watch Σ unfold term by term. Try putting in a negative number, then tick the squares box.
Challenge
Sources
- Σ, Π, indices and the mean bar: 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.