# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 49: Church Numerals

In Church encoding, an encoding system using lambda calculus, the Church numerals are the representation of the natural numbers. The distinguishing feature of the Church numerals is that the natural numbers are not treated as a primitive type, as they would typically be, but are simply represented by higher-order functions. Each higher-order function $\boldsymbol{n}$ representing the number $n$ takes two arguments — a function $f$ and a second argument to be passed to $f$ — and returns the $n$-fold composition of $f$.

Examples:

0 is represented as $\boldsymbol{0} = \lambda f . \lambda x . x$, which given any function $f$ returns a function which simply returns $x$ without applying $f$ at all.

1 is represented as $\boldsymbol{1} = \lambda f . \lambda x . f x$, which given any function $f$ returns a function which applies $f$ once to $x$.

2 is represented as $\boldsymbol{2} = \lambda f . \lambda x . f (f x)$, which given any function $f$ returns a function which applies $f$ twice to $x$.

3 is represented as $\boldsymbol{3} = \lambda f . \lambda x . f (f (f x))$, which given any function $f$ returns a function which applies $f$ three times to $x$.