Is there a way to add two numbers together on the type level in typescript?

Question

Suppose I had the following types

type One = 1;
type SmallEvens = 0 | 2 | 4 | 6 | 8 | 10;
type Three = 3;

Is there some kind of type Add<T, U> I could define that adds numbers on the type level like so?

type SmallOdds = Add<One, SmallEvens>; // same as `1 | 3 | 5 | 7 | 9 | 11`
type Four = Add<One, Three> // same as `4`

Alternatively, should I be looking at alternative representations of numbers to achieve this effect? However, being able to convert to something that extends number so that I can use it for tuple indexing would be a big plus.

Answer 1

Update 2022

I have written a package which handles your issue (and a pretty much all kinds of type level arithmetic), and which completely avoids the "Type instantiation is excessively deep and possibly infinite" limitation.

https://www.npmjs.com/package/ts-arithmetic

Here are some examples of what you can do with this package:

import { Add, Subtract, Multiply, Divide, Pow, Compare, Mod } from 'ts-arithmetic'
// Check the docs below for more

// Add any two numbers
type AddExample = Add<1024, 256>
// 1280

// Subtract any two numbers
type SubtractExample = Subtract<400, 1000>
// -600

// Multiply any two numbers
type MultiplyExample = Multiply<25, 50>
// 1250

// Divide any two numbers
type DivideExample = Divide<-1, 800>
// -0.00125

// Raise a number to an integer power
type PowExample = Pow<5, 7>
// 78125

// Compare any two numbers (same rules as JavaScript Array.sort())
type CompareExample = Compare<123456, 20>
// 1

// Get the JavaScript mod (i.e. remainder)
type ModExmaple = Mod<87, 7>
// 3