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# 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 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

## 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

## 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

## 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

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

For cases when you need both the quotient and the remainder, LIGO provides the `ediv` operation. `ediv x y` returns `Some (quotient, remainder)`, unless `y` is zero, in which case it returns `None`

const a : int = 37
const b : int = 5
const ediv1 : option (int * nat) = ediv(a, b) // Some (7, 2)
const c : nat = 37n
const ediv2 : option (int * nat) = ediv(c, b) // Some (7, 2)
const d : nat = 5n
const ediv3 : option (nat * nat) = ediv(c, d) // Some (7, 2)
const ediv4 : option (int * nat) = ediv(a, d) // Some (7, 2)

## 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)

## 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)