Part I · Maths · Chapter 2

Algebra, functions and graphs

An AI model is a function: numbers go in, numbers come out. This chapter rebuilds the school algebra you need to read every formula in the rest of the book.

Letters that stand for numbers

Start with something you already do. A coffee costs 3 dollars. Two coffees cost 3 × 2 = 6 dollars. Five coffees cost 3 × 5 = 15 dollars. You are repeating one rule with different numbers.

Algebra writes the rule once, with a letter in place of the number that changes:

The letter is a variable: a name for a number we do not want to fix yet. Mathematicians usually drop the × sign between a number and a letter, so 3n means 3 times n.

The order of operations

In 2 + 3 × 4 the multiplication happens first, so the answer is 2 + 12 = 14, not 20. The order is: brackets first, then powers, then multiply and divide, then add and subtract. When in doubt, add brackets: (2 + 3) × 4 = 20.

Powers, roots and negatives

A power is repeated multiplication. 5² ("five squared") is 5 × 5 = 25. 2³ is 2 × 2 × 2 = 8. A square root undoes squaring: √25 = 5 because 5 × 5 = 25.

Negative numbers follow two rules worth memorising. A negative times a negative is positive: (−3) × (−4) = 12. So any number squared is never negative: (−3)² = 9. That fact matters later, because squared errors are always zero or more.

Fractions, decimals and percentages

Money questions are full of these. 7% means 7 per hundred, so 7% = 7 / 100 = 0.07. To take 7% of 2,000, multiply: 2,000 × 0.07 = 140. To grow 2,000 by 7%, multiply by 1.07: 2,000 × 1.07 = 2,140. Remember "multiply by 1 + rate"; chapter 3 builds compound growth on it.

Solving equations

An equation says two things are equal. Solving it means finding the value of the variable that makes it true.

Worked example. Solve 2x + 3 = 11.

1. Subtract 3 from both sides: 2x = 8.
2. Divide both sides by 2: x = 4.
3. Check by putting it back: 2 × 4 + 3 = 11. Correct.

The one rule: whatever you do to one side, do to the other, and the balance stays true. That is all of school algebra in one sentence.

Functions: machines with one output

A function is a rule that takes an input and gives exactly one output. Picture a machine: drop in a number, get a number out.

Take the machine "double it, then add one". Feed it 3: double is 6, add one is 7. Feed it 0: you get 1. Feed it −2: you get −3. We write the machine as

So f(3) = 7, f(0) = 1 and f(−2) = −3. A table of inputs and outputs is a perfectly good way to describe a function:

"Exactly one output" is the important part. The same input always gives the same output. (Language models break this on purpose by sampling; chapter 16 explains how.)

Graphs

A graph draws a function. The horizontal axis is the input x; the vertical axis is the output y. Each row of the table becomes a point: (0, 1), (1, 3), (2, 5) and so on. Join the points and you see the function's shape at a glance.

For f(x) = 2x + 1 the points lie on a straight line. Functions whose graphs are straight lines are called linear.

Straight lines: slope and intercept

Every straight line can be written as

Slope is "rise over run": how far the line goes up for each step to the right. Worked example with the line through (1, 3) and (3, 7):

• Run: from x = 1 to x = 3 is 3 − 1 = 2.
• Rise: from y = 3 to y = 7 is 7 − 3 = 4.
• Slope: rise / run = 4 / 2 = 2.
• Intercept: at x = 1, y = 3, so 3 = 2 × 1 + b, giving b = 1. The line is y = 2x + 1, our machine from before.

A positive slope goes up to the right. A negative slope goes down. A slope of zero is flat. Slope is the single most important idea in Part I: chapter 5 extends it to curves, and chapter 6 uses it to train models.

Functions of several inputs

Real functions often take several inputs. A flat's rent might be estimated as

For 60 square metres and 2 bedrooms: 400 + 25 × 60 + 150 × 2 = 400 + 1,500 + 300 = 2,200.

The numbers 25 and 150 are weights: how much each input counts. 400 is a starting amount, which machine learning calls the bias. With inputs x₁, x₂ and weights w₁, w₂ it reads

Hold this shape in mind. A single neuron in a neural network computes exactly this, then bends the result with an activation function (chapter 12).

Adding many things: the Σ sign

With 384 inputs, writing every term is hopeless. The Greek capital sigma, Σ, means "add these up". Worked example: the weights are 2, 3 and 1; the inputs are 4, 0 and 5.

Term by term: 2 × 4 = 8, 3 × 0 = 0, 1 × 5 = 5. Total 8 + 0 + 5 = 13. In symbols:

This "multiply pairs and add" is called the dot product. It is the most common calculation in all of AI, and chapter 4 gives it a picture.

Curves and composition

Not every function is a straight line. g(x) = x² makes a U-shaped curve (a parabola): g(−2) = 4, g(0) = 0, g(3) = 9. Its lowest point is at x = 0. Finding the lowest point of a curve is what training a model means, so the parabola will come back many times.

Functions can be chained: feed one machine's output into another. If f(x) = 2x + 1 and g(x) = x², then g(f(3)) means: first f(3) = 7, then g(7) = 49. This is called composition. A neural network is a long composition: layer after layer, each feeding the next. The chain rule tells us how to find slopes through such chains.

In AI

A model is a function whose rule contains adjustable numbers called parameters. The simplest model is a line, y = wx + b, with two parameters w and b. A large language model is also a function: its input is a list of token ids and its output is a probability for every possible next token. It just has billions of parameters instead of two. Learning means adjusting the parameters until the function's outputs match the data, which is the subject of Part II.

Plain functions with no learned parameters are the safe choice for money figures. The rule can be read and checked, and the same input always gives the same output. A months-of-cover calculator is one line of algebra: months_covered = cash ÷ monthly_expenses. With 12,000 in cash and 3,000 a month of expenses it returns 4 months, and it refuses expenses of zero, because dividing by zero has no answer.

A language model should not do this arithmetic in its head. A well-built system lets the model call such a function as a tool and quote the result (chapter 28 explains why).