Complete example of an arithmetic system built inside TypeScript's type system for this article: https://itnext.io/implementing-arithmetic-within-typescripts-type-system-a1ef140a6f6f

Arithmetic.ts

// Utility types
type Length<T extends any[]> = T extends { length: infer L } ? L : never;

type BuildTuple<L extends number, T extends any[] = []> =
    T extends { length: L }
        ? T
        : BuildTuple<L, [...T, any]>;

type MultiAdd<N extends number, A extends number, I extends number> =
    I extends 0
        ? A
        : Add<N, A> extends number
            ? MultiAdd<N, Add<N, A>, Subtract<I, 1>>
            : never;

type AtTerminus<A extends number, B extends number> =
    A extends 0
        ? true
        : (B extends 0 ? true : false);

type EQ<A, B> =
    A extends B
        ? (B extends A ? true : false)
        : false;

type LT<A extends number, B extends number> =
    AtTerminus<A, B> extends true
        ? EQ<A, B> extends true
            ? false
            : (A extends 0 ? true : false)
        : LT<Subtract<A, 1>, Subtract<B, 1>>;

type MultiSub<N extends number, D extends number, Q extends number> =
    LT<N, D> extends true
        ? Q
        : Add<Q, 1> extends number
            ? MultiSub<Subtract<N, D>, D, Add<Q, 1>>
            : never;

type IsPositive<N extends number> =
    `${N}` extends `-${number}`
        ? false
        : true;

type IsWhole<N extends number> =
    `${N}` extends `${number}.${number}`
        ? false
        : true;

type IsValid<N extends number> =
    IsPositive<N> extends true
        ? (IsWhole<N> extends true ? true : false)
        : false;

type AreValid<A extends number, B extends number> =
    IsValid<A> extends true
        ? (IsValid<B> extends true ? true : false)
        : false;

// Arithmetical types
type Add<A extends number, B extends number> =
    Length<[...BuildTuple<A>, ...BuildTuple<B>]>;

type Subtract<A extends number, B extends number> =
    BuildTuple<A> extends [...(infer U), ...BuildTuple<B>]
        ? Length<U>
        : never;

type Multiply<A extends number, B extends number> =
    MultiAdd<A, 0, B>;

type Divide<A extends number, B extends number> =
    MultiSub<A, B, 0>;

type Modulo<A extends number, B extends number> =
    LT<A, B> extends true
        ? A
        : Modulo<Subtract<A, B>, B>;

// Safeguarded arithmetical types
type SafeAdd<A extends number, B extends number> =
    AreValid<A, B> extends true
        ? Add<A, B>
        : never;

type SafeSubtract<A extends number, B extends number> =
    AreValid<A, B> extends true
        ? Subtract<A, B>
        : never;

type SafeMultiply<A extends number, B extends number> =
    AreValid<A, B> extends true
        ? Multiply<A, B>
        : never;

type SafeDivide<A extends number, B extends number> =
    AreValid<A, B> extends true
        ? Divide<A, B>
        : never;

type SafeModulo<A extends number, B extends number> =
    AreValid<A, B> extends true
        ? Modulo<A, B>
        : never;

@ryandabler Very cool!

However, this errors on 4.4.4 (sadly).

See playground link.

@ryandabler Very cool!

However, this errors on 4.4.4 (sadly).

Thanks for pointing that out! I didn't have those errors when I originally wrote this, so I'm not sure when they popped up.

It's all fixed now, though.

Tuple representation, unfortunately, has a limit of 10000. In TS 4.8 there's support for converting strings to numbers in compile-time. After that we can easily do operations on numbers of arbitrary length by convert them first to strings. An example.