# Peano Numbers and Arithmetic in JavaScript

Whilst completing the Functional Programming Principles in Scala course a couple of years ago one of the exercises was in the realm of Peano numbers. This subject fascinated me, how we were able to represent non-negative natural numbers without any preformed concepts - relying solely on the logical expressions and recursive algorithms. Peano numbers allow us to represent all natural numbers using a defined zero value and a successor function.

```
const zero = () => false;
const isZero = (a) => a == zero();
const succ = (a) => () => a;
const pred = (a) => a();
```

Following these rules we have created the concept of zero (represented as false within our host language), along with the successor function which wraps and returns this value in a new lambda upon each invocation. We have also supplied accompanying zero comparator and predecessor functions which will assist us in upcoming examples.

## Arithmetic

Using this representation as a basis we are able to expand on the example by describing the four common-place arithmetic operations.

```
const add = (a, b) => isZero(a) ? b : add(pred(a), succ(b));
const sub = (a, b) => isZero(a) || isZero(b) ? a : sub(pred(a), pred(b));
const mul = (a, b) => isZero(a) ? zero() : add(mul(pred(a), b), b);
const div = (a, b) => isZero(a) ? zero() : succ(div(sub(a, b), b));
```

Looking at the definitions above you can see how we have been able to represent these operations using recursion and the functions defined before hand.

## Logic

We are also able to as easily represent logical operations which return the host languages boolean value type.

```
const equal = (a, b) => isZero(a) ? isZero(b) : isZero(b) ? false : equal(pred(a), pred(b));
const less = (a, b) => isZero(a) ? !isZero(b) : isZero(b) ? false : less(pred(a), pred(b));
const greater = (a, b) => ! (equal(a, b) || less(a, b));
```

## Example

Finally, we are able to combine all these operations into succinct examples.

```
const toNumber = (a) => isZero(a) ? 0 : 1 + toNumber(pred(a));
const _0 = zero(),
_1 = succ(_0),
_2 = succ(_1),
_3 = succ(_2),
_4 = succ(_3),
_5 = succ(_4);
toNumber(mul(add(_2, _3), sub(_5, _3))); // 10
toNumber(div(add(_2, _2), _2)); // 2
equal(add(_1, _1), div(_4, _2)); // true
less(div(_4, _2), _4); // true
greater(mul(_4, _2), _5); // true
```