The perceptron — one neuron

Before we begin

Every neural network is built from the same brick: a perceptron (one artificial neuron).

Output = activation(weights · inputs + bias)

Module 1 dot products appear again — now with a bias and a non-linear activation.

What you will learn

• Compute a perceptron output step by step.
• Explain linear separability.
• State why XOR needs more than one layer.

Before this lesson

• Module 3 welcome
• Module 1 — Dot products

The formula

For inputs $x_1, …, x_n$ and weights $w_1, …, w_n$:

z = b + w₁x₁ + w₂x₂ + … + wₙxₙ

Then a = activation(z) — e.g. ReLU(z) or sigmoid(z).

• b — bias (shifts the decision boundary)
• w — how much each input matters

This is exactly a dot product plus bias, then a function.

Tiny numeric example

Inputs x = [2, 1], weights w = [0.5, -1], bias b = 0.2

z = 0.2 + 0.5×2 + (-1)×1 = 0.2 + 1 - 1 = 0.2

If activation is ReLU: a = max(0, 0.2) = 0.2
If activation is step (z > 0 → 1 else 0): output = 1

Decision boundary intuition

One neuron with a linear activation before threshold draws a straight line (or hyperplane in higher dimensions) separating “activate” vs “don’t.”

It can learn AND and OR gates with the right weights. It cannot learn XOR alone — no single straight line separates XOR’s 1s from 0s.

That limitation motivated multi-layer networks — hidden neurons build new features, then the output combines them.

Checkpoint

Can one perceptron classify XOR perfectly?

What's next

Lesson 2 — Activation functions