Numbers, Symbols & Algebra

Why start here?

Every machine learning model — from a spam filter to ChatGPT — is, underneath everything, a mathematical function that turns numbers into other numbers. Before we can understand those functions, we need to be comfortable with the language they are written in: algebra.

Don't worry if school math feels far away. We rebuild everything from zero.

Variables: boxes with names

A variable is just a named box that holds a number. Instead of saying "some number", we write $x$. This lets us describe rules that work for any number:

$y = 2x + 1$

Read it as: "take $x$, double it, add one, call the result $y$". If $x = 3$, then $y = 7$. If $x = 10$, then $y = 21$. One line of symbols describes infinitely many calculations — that is the entire superpower of algebra.

In machine learning you will constantly meet:

• $x$ — an input (e.g. the size of a house)
• $y$ — an output (e.g. its price)
• $w$ and $b$ — weights and bias, the numbers a model learns
• $\hat{y}$ — "y-hat", the model's prediction of $y$

The most famous equation in machine learning is barely more complex than the one above:

$\hat{y} = wx + b$

That is a complete machine learning model. Seriously. Learning means finding good values for $w$ and $b$.

The rules of the game

Algebra has a small rulebook you already half-know:

1. Order of operations — parentheses, then powers, then multiplication/division, then addition/subtraction.
2. Balance — an equation stays true if you do the same thing to both sides. From $2x + 1 = 7$ you can subtract 1 from both sides ($2x = 6$), then divide both by 2 ($x = 3$).
3. Distribution — $a(b + c) = ab + ac$.

Powers, roots and logarithms

• Powers repeat multiplication: $2^3 = 2 \cdot 2 \cdot 2 = 8$.
• Roots undo powers: $\sqrt{9} = 3$ because $3^2 = 9$.
• Logarithms answer "what power do I need?": $\log_2 8 = 3$ because $2^3 = 8$.

Logarithms appear everywhere in ML — in loss functions, in information theory, in probabilities — because they turn huge multiplications into manageable additions:

$\log(a \cdot b) = \log(a) + \log(b)$

Sigma notation: loops in math

When a formula repeats over many items, mathematicians use $\Sigma$ (sigma) as a "for loop":

$\sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30$

Read it as: "for $i$ from 1 to 4, add up $i^2$". If you can read a for loop, you can read sigma notation. You will see it in nearly every ML formula, e.g. "sum the error over all training examples".

Key takeaways

• Variables let one formula describe infinitely many situations.
• A machine learning model is a formula with adjustable numbers (parameters) like $w$ and $b$.
• Logs turn multiplication into addition; sigma notation is a for loop.