Even, Odd, Periodicity

Single-variable calculus from first principles

Spotting a symmetry in a function is a genuine shortcut: it halves the work of understanding a graph, integrating it, or storing it. There are two symmetries worth knowing by name, even and odd, plus the idea of a function that repeats.

A function is even if flipping the input's sign changes nothing: f(−x) = f(x). The graph looks the same on the left and right of the y-axis, a perfect mirror. The standard example is x²: squaring kills the sign, so (−3)² = 3².

A function is odd if flipping the input flips the output too: f(−x) = −f(x). The graph has rotational symmetry: spin it 180° about the origin and it lands on itself. The standard example is x³, since (−2)³ = −8 = −(2³).

Where this lives in MLThe activation tanh is odd, which keeps activations centred around zero and helps gradients flow. The same even/odd structure runs through signal processing, where Fourier cosine series capture even parts and sine series the odd. Periodicity is the backbone of positional encodings in transformers, where sine and cosine of different frequencies tag each position in a sequence.

▶ Even, Odd, Periodicity

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