ligo/gitlab-pages/docs/language-basics/math-numbers-tez.md

id	title
math-numbers-tez	Math, Numbers & Tez

LIGO offers three built-in numerical types: int, nat and tez. Values of type int are integers; values of type nat are natural numbers (integral numbers greater than or equal to zero); values of type tez are units of measure of Tezos tokens.

• Integer literals are the same found in mainstream programming languages, for example, 10, -6 and 0, but there is only one canonical zero: 0 (so, for instance, -0 and 00 are invalid).

• Natural numbers are written as digits follwed by the suffix n, like so: 12n, 0n, and the same restriction on zero as integers applies: 0n is the only way to specify the natural zero.

• Tezos tokens can be specified using literals of three kinds:

  • units of millionth of tez, using the suffix mutez after a natural literal, like 10000mutez or 0mutez;
  • units of tez, using the suffix tz or tez, like 3tz or 3tez;
  • decimal amounts of tz or tez, like 12.3tz or 12.4tez.

Note that large integral values can be expressed using underscores to separate groups of digits, like 1_000mutez or 0.000_004tez.

Addition

Addition in LIGO is accomplished by means of the + infix operator. Some type constraints apply, for example you cannot add a value of type tez to a value of type nat.

In the following example you can find a series of arithmetic operations, including various numerical types. However, some bits remain in comments as they would otherwise not compile, for example, adding a value of type int to a value of type tez is invalid. Note that adding an integer to a natural number produces an integer.

// int + int yields int
const a : int = 5 + 10

// nat + int yields int
const b : int = 5n + 10

// tez + tez yields tez
const c : tez = 5mutez + 0.000_010tez

//tez + int or tez + nat is invalid
// const d : tez = 5mutez + 10n

// two nats yield a nat
const e : nat = 5n + 10n

// nat + int yields an int: invalid
// const f : nat = 5n + 10;

const g : int = 1_000_000

Pro tip: you can use underscores for readability when defining large numbers:

const sum : tez = 100_000mutez

// int + int yields int
let a : int = 5 + 10

// nat + int yields int
let b : int = 5n + 10

// tez + tez yields tez
let c : tez = 5mutez + 0.000_010tez

// tez + int or tez + nat is invalid
// let d : tez = 5mutez + 10n

// two nats yield a nat
let e : nat = 5n + 10n

// nat + int yields an int: invalid
// let f : nat = 5n + 10

let g : int = 1_000_000

Pro tip: you can use underscores for readability when defining large numbers:

let sum : tez = 100_000mutez

// int + int yields int
let a : int = 5 + 10;

// nat + int yields int
let b : int = 5n + 10;

// tez + tez yields tez
let c : tez = 5mutez + 0.000_010tez;

// tez + int or tez + nat is invalid:
// let d : tez = 5mutez + 10n;

// two nats yield a nat
let e : nat = 5n + 10n;

// nat + int yields an int: invalid
// let f : nat = 5n + 10;

let g : int = 1_000_000;

Pro tip: you can use underscores for readability when defining large numbers:

let sum : tex = 100_000mutez;

Subtraction

Subtraction looks as follows.

⚠️ Even when subtracting two nats, the result is an int

const a : int = 5 - 10

// Subtraction of two nats yields an int
const b : int = 5n - 2n

// Therefore the following is invalid
// const c : nat = 5n - 2n

const d : tez = 5mutez - 1mutez

let a : int = 5 - 10

// Subtraction of two nats yields an int
let b : int = 5n - 2n

// Therefore the following is invalid
// let c : nat = 5n - 2n

let d : tez = 5mutez - 1mutez

let a : int = 5 - 10;

// Subtraction of two nats yields an int
let b : int = 5n - 2n;

// Therefore the following is invalid
// let c : nat = 5n - 2n;

let d : tez = 5mutez - 1mutez;

Multiplication

You can multiply values of the same type, such as:

const a : int = 5 * 5
const b : nat = 5n * 5n

// You can also multiply `nat` and `tez`
const c : tez = 5n * 5mutez

let a : int = 5 * 5
let b : nat = 5n * 5n

// You can also multiply `nat` and `tez`
let c : tez = 5n * 5mutez

let a : int = 5 * 5;
let b : nat = 5n * 5n;

// You can also multiply `nat` and `tez`
let c : tez = 5n * 5mutez;

Euclidean Division

In LIGO you can divide int, nat, and tez. Here is how:

⚠️ Division of two tez values results into a nat

const a : int = 10 / 3
const b : nat = 10n / 3n
const c : nat = 10mutez / 3mutez

let a : int = 10 / 3
let b : nat = 10n / 3n
let c : nat = 10mutez / 3mutez

let a : int = 10 / 3;
let b : nat = 10n / 3n;
let c : nat = 10mutez / 3mutez;

LIGO also allows you to compute the remainder of the Euclidean division. In LIGO, it is a natural number.

const a : int = 120
const b : int = 9
const rem1 : nat = a mod b  // 3
const c : nat = 120n
const rem2 : nat = c mod b  // 3
const d : nat = 9n
const rem3 : nat = c mod d  // 3
const rem4 : nat = a mod d  // 3

let a : int = 120
let b : int = 9
let rem1 : nat = a mod b  // 3
let c : nat = 120n
let rem2 : nat = c mod b  // 3
let d : nat = 9n
let rem3 : nat = c mod d  // 3
let rem4 : nat = a mod d  // 3

let a : int = 120;
let b : int = 9;
let rem1 : nat = a mod b;  // 3
let c : nat = 120n;
let rem2 : nat = c mod b;  // 3
let d : nat = 9n;
let rem3 : nat = c mod d;  // 3
let rem4 : nat = a mod d;  // 3

From int to nat and back

You can cast an int to a nat and vice versa. Here is how:

const a : int = int (1n)
const b : nat = abs (1)

let a : int = int (1n)
let b : nat = abs (1)

let a : int = int (1n);
let b : nat = abs (1);

Checking a nat

You can check if a value is a nat by using a predefined cast function which accepts an int and returns an optional nat: if the result is not None, then the provided integer was indeed a natural number, and not otherwise.

const is_a_nat : option (nat) = is_nat (1)

let is_a_nat : nat option = Michelson.is_nat (1)

let is_a_nat : option (nat) = Michelson.is_nat (1);