The artificial neuron

Signals in, a number out.

An artificial neuron receives information and answers with a number. It observes its inputs, x₁, x₂, x₃, the clues it has to decide with. But it does not trust every clue the same way: each input gets a weight, w₁, w₂, w₃. The weight says how much that clue counts. A positive weight favors, a negative weight penalizes.

Learning by correction.

Should you take an umbrella this morning? An 80% chance of rain, 40% cloud cover, 60% humidity: those become the inputs. At first the weights are drawn at random. Then comes training: the neuron lives through thousands of such situations, comparing each prediction to what actually happened. When it is wrong, the error is sent backward and every weight is nudged in the direction that would have reduced it. This is gradient descent, repeated until guesses turn into reliable predictions.

From a score to a confidence.

Once trained, the weights are frozen: this is inference, the trained neuron at work. It multiplies each clue by its weight, sums everything, then adds its prior: the bias, b. That score is z. But a score is not an answer yet: z passes through an activation function, here the sigmoid, which squashes any number into the interval between zero and one. The result, a = 0.74, reads like a probability: yes, at 74% confidence.

The math, plainly.

Everything the neuron computes fits in three lines: a weighted sum, a squashing into the unit interval, and a threshold for the decision.

z = x₁w₁ + x₂w₂ + x₃w₃ + b

a = σ(z) = 1 / (1 + e⁻ᶻ)

a > 0.5 → yes

One brick, a whole network.

This signal does not feed a single neuron: it travels to an entire layer. Every connection carries its own weight, every neuron runs the exact same computation. The last neuron decides: if a exceeds 0.5, take the umbrella; below, leave it. On its own, one neuron can only draw a single straight boundary, which is exactly why networks exist. Stack millions of these bricks in layers, and you get a neural network.