# Arbitrary-precision numbers

Rug provides integers and floating-point numbers with arbitrary precision and correct rounding:

`Integer`

is a bignum integer with arbitrary precision,`Rational`

is a bignum rational number with arbitrary precision,`Float`

is a multi-precision floating-point number with correct rounding, and`Complex`

is a multi-precision complex number with correct rounding.

Rug is a high-level interface to the following GNU libraries:

- GMP for integers and rational numbers,
- MPFR for floating-point numbers, and
- MPC for complex numbers.

Rug is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. See the full text of the GNU LGPL and GNU GPL for details.

## Whatâ€™s new

### Version 1.9.0 news (2020-06-01)

- Displaying
`Float`

and`Complex`

numbers, and converting them to strings, now avoids outputting an exponent term if the radix point can be moved to the correct place without inserting any extra digits. For example`"42.0"`

will be produced rather than`"4.20e1"`

(issue 18). This change does not affect output when`LowerExp`

(`"{:e}"`

format) or`UpperExp`

(`"{:E}"`

format) is used. - New methods Float::to_sign_string_exp and Float::to_sign_string_exp_round were added.
- A new function float::allowed_exp_range was added.
- A new method Float::clamp_exp was added.
- The following methods are now const functions:
- The
*az*crate is now a public dependency, and wrapping and checked casts to/from primitives and big numbers are provided through the traits of the crate.

#### Compatibility note

The output of `Float`

and `Complex`

numbers was
changed as specified above.

### Other releases

Details on other releases can be found in *RELEASES.md*.

## Quick example

```
use rug::{Assign, Integer};
let mut int = Integer::new();
assert_eq!(int, 0);
int.assign(14);
assert_eq!(int, 14);
let decimal = "98_765_432_109_876_543_210";
int.assign(Integer::parse(decimal).unwrap());
assert!(int > 100_000_000);
let hex_160 = "ffff0000ffff0000ffff0000ffff0000ffff0000";
int.assign(Integer::parse_radix(hex_160, 16).unwrap());
assert_eq!(int.significant_bits(), 160);
int = (int >> 128) - 1;
assert_eq!(int, 0xfffe_ffff_u32);
```

- Integer::new creates a new
`Integer`

intialized to zero. - To assign values to Rug types, we use the
`Assign`

trait and its method`Assign::assign`

. We do not use the assignment operator`=`

as that would drop the left-hand-side operand and replace it with a right-hand-side operand of the same type, which is not what we want here. - Arbitrary precision numbers can hold numbers that are too large to fit in a primitive type. To assign such a number to the large types, we use strings rather than primitives; in the example this is done using Integer::parse and Integer::parse_radix.
- We can compare Rug types to primitive types or to other Rug types
using the normal comparison operators, for example
`int > 100_000_000`

. - Most arithmetic operations are supported with Rug types and
primitive types on either side of the operator, for example
`int >> 128`

.

## Using with primitive types

With Rust primitive types, arithmetic operators usually operate on two
values of the same type, for example `12i32 + 5i32`

. Unlike primitive
types, conversion to and from Rug types can be expensive, so the
arithmetic operators are overloaded to work on many combinations of
Rug types and primitives. More details are available in the
documentation.

## Operators

Operators are overloaded to work on Rug types alone or on a combination of Rug types and Rust primitives. When at least one operand is an owned value of a Rug type, the operation will consume that value and return a value of the Rug type. For example

```
use rug::Integer;
let a = Integer::from(10);
let b = 5 - a;
assert_eq!(b, 5 - 10);
```

Here `a`

is consumed by the subtraction, and `b`

is an owned
`Integer`

.

If on the other hand there are no owned Rug types and there are references instead, the returned value is not the final value, but an incomplete-computation value. For example

```
use rug::Integer;
let (a, b) = (Integer::from(10), Integer::from(20));
let incomplete = &a - &b;
// This would fail to compile: assert_eq!(incomplete, -10);
let sub = Integer::from(incomplete);
assert_eq!(sub, -10);
```

Here `a`

and `b`

are not consumed, and `incomplete`

is not the final
value. It still needs to be converted or assigned into an `Integer`

.
This is covered in more detail in the documentationâ€™s
*Incomplete-computation values* section.

More details on operators are available in the documentation.

## Using Rug

Rug is available on crates.io. To use Rug in your crate,
add it as a dependency inside *Cargo.toml*:

```
[dependencies]
rug = "1.9"
```

Rug requires rustc version 1.37.0 or later.

Rug also depends on the GMP, MPFR and MPC libraries through the low-level FFI bindings in the gmp-mpfr-sys crate, which needs some setup to build; the gmp-mpfr-sys documentation has some details on usage under GNU/Linux, macOS and Windows.

## Optional features

The Rug crate has six optional features:

`integer`

, enabled by default. Required for the`Integer`

type and its supporting features.`rational`

, enabled by default. Required for the`Rational`

number type and its supporting features. This feature requires the`integer`

feature.`float`

, enabled by default. Required for the`Float`

type and its supporting features.`complex`

, enabled by default. Required for the`Complex`

number type and its supporting features. This feature requires the`float`

feature.`rand`

, enabled by default. Required for the`RandState`

type and its supporting features. This feature requires the`integer`

feature.`serde`

, disabled by default. This provides serialization support for the`Integer`

,`Rational`

,`Float`

and`Complex`

number types, providing that they are enabled. This feature requires the serde crate.

The first five optional features are enabled by default; to use
features selectively, you can add the dependency like this to
*Cargo.toml*:

```
[dependencies.rug]
version = "1.9"
default-features = false
features = ["integer", "float", "rand"]
```

Here only the `integer`

, `float`

and `rand`

features are enabled. If
none of the features are selected, the gmp-mpfr-sys crate
is not required and thus not enabled. In that case, only the
`Assign`

trait and the traits that are in the `ops`

module are
provided by the crate.