The Sigmoid Function: From 'e' to Neural Networks

The sigmoid function is used in neural networks and logistic regression to map values to a probability between 0 and 1. Its equation, σ(x) = 1 / (1 + e⁻ˣ), contains the mathematical constant 'e', which arises naturally from the concept of continuous compounding growth. The value of 'e' is the limit of (1 + 1/n)ⁿ as 'n' approaches infinity, representing the maximum possible growth from a continuously compounding process. A key property of the function eˣ in calculus is that its rate of change is equal to its current value, a unique characteristic that is fundamental to its role in many mathematical models.