Part 0 · Getting Started

0.2 Reading math notation

Σ, functions, derivatives and vectors without fear

10-minute readPrerequisite: none▶ Video coming soon

What you will learn

Read aloud the main symbols used on this site
Read a formula as “a short sentence describing a calculation”
Picture where Σ, functions, derivatives and vectors are used in automated driving and AI

Build intuition

A formula is shorthand for a procedure that would take many words to describe. For example, the sentence “add up the distances driven over three days and divide by the number of days” can be written like this:

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$

A row of symbols can look intimidating, but once you know how to read them and what they mean, all this says is “add, then divide”. When you meet a formula, try these three things:

Read it aloud: if you do not know how to read a symbol, check the quick-reference table at the end of this lesson
Plug in small numbers: try values you can calculate by hand, such as n = 3
Keep going even if you do not understand everything: the lessons are written so that the explanations and the “Connection to automated driving” block carry the main story

Check with math

① Σ (sigma): add everything up

$x_i$ means “the i-th x”, and the small letter i at the lower right is called a subscript (or index). If you drove 120 km, 80 km and 100 km over three days, then $x_1 = 120,\ x_2 = 80,\ x_3 = 100$.

The large $\Sigma$ (capital sigma) means “add up”. Starting from $i = 1$ at the bottom, you increase i one step at a time up to the number at the top, adding as you go.

$$\sum_{i=1}^{3} x_i = x_1 + x_2 + x_3 = 120 + 80 + 100 = 300$$

Dividing this by the number of days, $n = 3$, gives the average $\bar{x}$ (read “x bar”) = 100 km. The average formula at the top of this lesson is exactly this calculation.

② Functions f(x): a box that turns inputs into outputs

$f(x)$ (read “f of x”) is “a box that, given x, returns exactly one answer according to a fixed rule”. For example, the distance a car travels from the moment the brakes are applied until it stops — the braking distance $s$ — can be approximated as a function of the speed $v$:

$$s(v) = \frac{v^2}{2a}$$

Here $a$ is the braking deceleration. With $a = 5\ \mathrm{m/s^2}$ (roughly a firm brake application), $s(v) = v^2/10$. At 72 km/h (20 m/s), $s(20) = 400/10 = 40$ m. (This does not include the distance covered before the driver or system starts braking.)

The $e^{x}$ (read “e to the x”, also written $\exp(x)$) and $\ln x$ (the natural logarithm) used in lesson 1.8 are also functions. $\ln x$ answers “to what power must e be raised to get x?”, so it does exactly the opposite of $e^{x}$.

③ Derivatives: how much does the output change if you change the input a little?

A derivative describes how much the output changes when the input changes slightly — in other words, the slope of the graph. Differentiating the braking-distance formula gives:

$$s(v) = \frac{v^2}{10} \quad\Longrightarrow\quad \frac{ds}{dv} = \frac{2v}{10} = \frac{v}{5}$$

$\frac{ds}{dv}$ is read “d s d v”. At 20 m/s the slope is 20 / 5 = 4, which means “if the speed increases by 1 m/s, the braking distance grows by about 4 m”. Indeed, $s(21) = 44.1$ m — about 4 m more.

Braking distance grows with the square of speed. The faster you go, the steeper the curve (the larger the derivative), so the same 1 m/s difference makes a bigger difference in stopping distance. The dotted line is the tangent at v = 20 (slope 4).

You do not need to memorize differentiation rules. On this site it is enough to read a derivative as “the slope” or “the rate of change when you nudge the input”. When there are several inputs, the slope obtained by moving just one of them is called a partial derivative, written $\frac{\partial L}{\partial w}$, and all of them lined up together form the gradient $\nabla L$ (read “nabla L” or “grad L”). The gradient is the star of how AI learns (2.3 Gradient descent).

④ Vectors: numbers bundled into a list

A vector lets you treat several numbers as a single symbol. It is written in bold, as $\mathbf{x}$. For example, the state of the car ahead — “distance 30 m, relative speed −2 m/s, lateral position 0.5 m” — can be bundled into one vector:

$$\mathbf{x} = (x_1,\ x_2,\ x_3) = (30,\ -2,\ 0.5)$$

Multiplying the matching elements of two vectors and adding the results gives the dot product (inner product). With Σ it can be written compactly:

$$\mathbf{w} \cdot \mathbf{x} = \sum_{i=1}^{n} w_i x_i$$

Example: if $\mathbf{w} = (0.5,\ -1,\ 2)$ and $\mathbf{x} = (2,\ 1,\ 3)$, then $0.5 \times 2 + (-1) \times 1 + 2 \times 3 = 6$. Each individual calculation inside a neural network (3.1) is in fact this dot product. Numbers arranged in a grid of rows and columns are called a matrix.

⑤ Quick-reference table of symbols

Symbol	How to read it	Meaning	Main lessons
$\sum$	sigma (capital)	Add everything up	1.4, 2.2
$f(x)$	f of x	Function: give it x, get one answer	Throughout
$e^{x},\ \exp(x)$	e to the x	Exponential function (e ≈ 2.718)	1.3, 1.8
$\ln x$	natural log of x	Natural logarithm: the power of e that gives x	1.8, 2.4
$\frac{ds}{dv}$	d s d v	Derivative: the slope	2.3, 3.2
$\frac{\partial L}{\partial w}$	partial L partial w	Partial derivative: the slope when only one input moves	2.3, 3.2
$\nabla L$	nabla L / grad L	Gradient: the partial derivatives lined up as a vector	2.3
$\mathbf{x}$	vector x	A list of numbers	2.2, 3.1
$P(A)$	P of A	Probability that A happens	Part 1
$P(B \mid A)$	P of B given A	Conditional probability	1.2
$E[X]$	E of X	Expected value (mean)	1.4
$\sigma$	sigma (lowercase)	Standard deviation (spread)	1.4
$\lambda$	lambda	Rate (e.g. accidents per km)	1.3, 1.8
$\approx$	approximately equal to	Nearly equal	Throughout

Capital $\Sigma$ (sum) and lowercase $\sigma$ (standard deviation) are both read “sigma”, but they mean completely different things.

Summary

A formula is shorthand for a procedure. Read it aloud and plug in small numbers
$\Sigma$ means “add everything up”, $f(x)$ is “a box from input to output”, a derivative is “a slope”, and a vector is “a list of numbers”
If you lose track of a symbol, come back to the table in this lesson (all symbols are listed in the appendix “Notation”)

Quick quiz

Q1. What is the value of $\sum_{i=1}^{4} i$?

10. You add i itself from 1 to 4: 1 + 2 + 3 + 4 = 10.

Q2. With braking distance $s(v) = v^2/10$, if the speed rises from 10 m/s to 20 m/s, how many times longer is the braking distance?

Four times (10 m → 40 m). Doubling the speed multiplies the distance by 2² = 4, because of the square.

Q3. Using the same formula, what is the slope $ds/dv$ at v = 10 m/s?

10 / 5 = 2. Going 1 m/s faster adds about 2 m. That is half the slope at 20 m/s (4), which shows that each extra 1 m/s matters more the faster you go.

Q4. What is the dot product $\mathbf{w} \cdot \mathbf{x}$ of $\mathbf{w} = (1,\ 2)$ and $\mathbf{x} = (3,\ -1)$?

1. 1 × 3 + 2 × (−1) = 3 − 2 = 1.

Symbol	How to read it	Meaning	Main lessons
\(\sum\)	sigma (capital)	Add everything up	1.4, 2.2
\(f(x)\)	f of x	Function: give it x, get one answer	Throughout
\(e^{x},\ \exp(x)\)	e to the x	Exponential function (e ≈ 2.718)	1.3, 1.8
\(\ln x\)	natural log of x	Natural logarithm: the power of e that gives x	1.8, 2.4
\(\frac{ds}{dv}\)	d s d v	Derivative: the slope	2.3, 3.2
\(\frac{\partial L}{\partial w}\)	partial L partial w	Partial derivative: the slope when only one input moves	2.3, 3.2
\(\nabla L\)	nabla L / grad L	Gradient: the partial derivatives lined up as a vector	2.3
\(\mathbf{x}\)	vector x	A list of numbers	2.2, 3.1
\(P(A)\)	P of A	Probability that A happens	Part 1
\(P(B \mid A)\)	P of B given A	Conditional probability	1.2
\(E[X]\)	E of X	Expected value (mean)	1.4
\(\sigma\)	sigma (lowercase)	Standard deviation (spread)	1.4
\(\lambda\)	lambda	Rate (e.g. accidents per km)	1.3, 1.8
\(\approx\)	approximately equal to	Nearly equal	Throughout