# Numbers

Three types of numbers are available in Archetype: integers, naturals and rationals

# Naturals

Naturals are positive integers. They are defined as follows:

effect {  var n1 : nat = 345;  var n2 : nat = 9999999999999999999999999999999999999999999999999999999;  var n3 : nat = 10_000_000;}

Note that it is possible to structure large numbers by packs of 3 digits using the underscore character.

3 arithmetics operations + * % and the 6 comparison operators = <> < > <= >= are available. Note however that the difference operator returns an integer value (see integers section below).

effect {   var a : nat = 5;   var b : nat = 7;   var c : int = a - b; /* -2 typed as intger */   var d : int = b - a; /* 2 typed as integer */}

Note that de div operator is the euclidean division and returns a nat value.

The abs function is used to convert a integer value to a natural value.

effect {    var a = 5i; /* a is typed 'int'*/    var b = abs(5); /* b is typed 'nat' */}

# Integers

Integers are defined as follows:

effect {  var n1 : int = 3_458i;  var n2 : int = -5i;  var n3 : int = 99999999999999999999999999999999999999999999999999999999i;  var n4 : int = 10_000_000i;}

Note that the literal syntax of positive integers uses the i suffix.

Integer values are big integers, meaning there is no real constraint on the value and it can be negative.

The 5 arithmetics operations + * - div % and the 6 comparison operators = <> < > <= >= are available.

# Tezis

Tez is the type to specify an amount in Tezos cryptocurrency.

effect {  var t1 := 1tz;   // one tezis  var t2 := 1mtz:  // one 0.001 tezis  var t3 := 1utz;  // one mutez}

It is more constrained than integers since you can only add and subtract tez values. All comparison operators are available.

# Rationals

A rational is the quotient or fraction of two integers. You can define a rational with :

• a quotient of two integers

• a floating point notation

• a percentage notation

The following presents an example for each:

effect { var r1 := 6/4;    var r2 := 1.5; var r3 := 150%;}

r1 r2 and r3 represent the same rational, $3/2$ . These values are transcoded to the pair of integers (3,2). A fractions is simplified during transcoding process.

## Operations

Archetype provides the 4 arithmetic operations + - * / and the minus sign on rationals. They all return a rational. For example:

effect {  var r1 := 8/6;  // transcoded to (4,3)  var r2 := 1.8;    // transcoded to (9,5)  var rpl := r1+r2; // will execute to (6,15)  var rmi := r1-r2; // will execute to (-7,15)  var rmu := r1*r2; // will execute to (4,5)  var rdi := r1/r2: // will execute to (20,27)  var rms := -r1    // will execute to (-4,3) }

It is also possible to mix integers and rationals:

effect {  var r := 5/3;  var n := 4;  var rtn := r*n; // will execute to (20/3)  var rpn := r+n; // will execute to (17/3)}

It is also possible to mix rationals and tezis values, in that order. However the result is a value in tezis.

effect {  var r := 80%;  var a := 56tez;  var res := r*a;}

In the example above, the res value is 44800000utz . The process is the following:

1. convert a to utez (smallest unit)

2. compute the rational (here 3*56000000/4 = 168000000/4)

3. execute the euclidean division (44800000)

Note that the term a*r is not accepted as res value (line 4 above): rationals come first.

## Comparison

Rationals are comparable with = <> < <= > >= operators.

effect {   var r1 := 3/12;   var r2 := 0.25;   var r3 := 0.5;   if r1 = r2 and r3 > r2 then transfer 1tz to coder;}

It is possible to compare rationals and integers. It is not possible to compare rationals and tez values.

## Conversion to integers

Rational are converted to integers with the floor and ceil operators with the expected behaviour.

## Conversion to other types

There is no explicit cast (conversion operator) from a rational to a tez value, a duration or a date. You may just multiply by 1tez for example.

effect {  var a := 2.5;  transfer (a*1tez) from source to dest;}

# Casts

It is often necessary to cast (convert) a value from one type to another: for example, say you want to mulitply the transferred amount by a duration value to get a new duration value. It is then necessary to convert the tez amount and the duration value into integer values in order to multiply then.

The general rule is that conversions to integer are implicit, and conversions back to specific types (tez, duration) are not. For example the following converts the transferred amount to an integer value:

var t : int = transferred;

Note that it is necessary to explicitely type the t variable in order to trigger the implicit conversion of transferred to integer. The following types may be implicitly converted to integer:

• duration

• date

• tez

The resulting integer value is the number of the smallest value of the source type: namely second for duration, and utez for tez.

The following example illustrates this mecanism:

effect {  var t : int = transferred; // t is the number of utez   var d : int = 1h15m;       // d is the number of seconds (4500)  var res = t * d * 1s;      // res is a duration}

Note that:

• the conversion form int to a duration (line 4) is done by multiplying t*d by 1s

• the conversion from rational to int is done with operators ceil and floor

• the conversion from int to nat is done with abs