# Decimal (Decimal v2.0.0) View Source

Decimal arithmetic on arbitrary precision floating-point numbers.

A number is represented by a signed coefficient and exponent such that: `sign * coefficient * 10 ^ exponent`

. All numbers are represented and calculated
exactly, but the result of an operation may be rounded depending on the
context the operation is performed with, see: `Decimal.Context`

. Trailing
zeros in the coefficient are never truncated to preserve the number of
significant digits unless explicitly done so.

There are also special values such as NaN (not a number) and ±Infinity. -0 and +0 are two distinct values. Some operation results are not defined and will return NaN. This kind of NaN is quiet, any operation returning a number will return NaN when given a quiet NaN (the NaN value will flow through all operations).

Exceptional conditions are grouped into signals, each signal has a flag and a
trap enabler in the context. Whenever a signal is triggered it's flag is set
in the context and will be set until explicitly cleared. If the signal is trap
enabled `Decimal.Error`

will be raised.

## Specifications

This library follows the above specifications for reference of arithmetic operation implementations, but the public APIs may differ to provide a more idiomatic Elixir interface.

The specification models the sign of the number as 1, for a negative number,
and 0 for a positive number. Internally this implementation models the sign as
1 or -1 such that the complete number will be `sign * coefficient * 10 ^ exponent`

and will refer to the sign in documentation as either *positive*
or *negative*.

There is currently no maximum or minimum values for the exponent. Because of that all numbers are "normal". This means that when an operation should, according to the specification, return a number that "underflows" 0 is returned instead of Etiny. This may happen when dividing a number with infinity. Additionally, overflow, underflow and clamped may never be signalled.

# Link to this section Summary

## Types

The coefficient of the power of `10`

. Non-negative because the sign is stored separately in `sign`

.

The exponent to which `10`

is raised.

Rounding algorithm.

`1`

for positive`-1`

for negative

This implementation models the `sign`

as `1`

or `-1`

such that the complete number will be: `sign * coef * 10 ^ exp`

.

## Functions

The absolute value of given number. Sets the number's sign to positive.

Adds two numbers together.

Applies the context to the given number rounding it to specified precision.

Creates a new decimal number from an integer, string, float, or existing decimal number.

Compares two numbers numerically. If the first number is greater than the second
`:gt`

is returned, if less than `:lt`

is returned, if both numbers are equal
`:eq`

is returned.

Divides two numbers.

Divides two numbers and returns the integer part.

Compares two numbers numerically and returns `true`

if they are equal,
otherwise `false`

. If one of the operands is a quiet NaN this operation
will always return `false`

.

Compares two numbers numerically and returns `true`

if they are equal,
otherwise `false`

. If one of the operands is a quiet NaN this operation
will always return `false`

.

Creates a new decimal number from a floating point number.

Compares two numbers numerically and returns `true`

if the the first argument
is greater than the second, otherwise `false`

. If one the operands is a
quiet NaN this operation will always return `false`

.

Returns `true`

if number is ±Infinity, otherwise `false`

.

Returns `true`

when the given `decimal`

has no significant digits after the decimal point.

Returns `true`

if argument is a decimal number, otherwise `false`

.

Compares two numbers numerically and returns `true`

if the the first number is
less than the second number, otherwise `false`

. If one of the operands is a
quiet NaN this operation will always return `false`

.

Compares two values numerically and returns the maximum. Unlike most other
functions in `Decimal`

if a number is NaN the result will be the other number.
Only if both numbers are NaN will NaN be returned.

Compares two values numerically and returns the minimum. Unlike most other
functions in `Decimal`

if a number is NaN the result will be the other number.
Only if both numbers are NaN will NaN be returned.

Multiplies two numbers.

Returns `true`

if number is NaN, otherwise `false`

.

Negates the given number.

Check if given number is negative

Creates a new decimal number from an integer or a string representation.

Creates a new decimal number from the sign, coefficient and exponent such that
the number will be: `sign * coefficient * 10 ^ exponent`

.

Normalizes the given decimal: removes trailing zeros from coefficient while keeping the number numerically equivalent by increasing the exponent.

Parses a binary into a decimal.

Check if given number is positive

Remainder of integer division of two numbers. The result will have the sign of the first number.

Rounds the given number to specified decimal places with the given strategy (default is to round to nearest one). If places is negative, at least that many digits to the left of the decimal point will be zero.

Finds the square root.

Subtracts second number from the first. Equivalent to `Decimal.add/2`

when the
second number's sign is negated.

Returns the decimal converted to a float.

Returns the decimal represented as an integer.

Converts given number to its string representation.

# Link to this section Types

## Specs

coefficient() :: non_neg_integer() | :NaN | :inf

The coefficient of the power of `10`

. Non-negative because the sign is stored separately in `sign`

.

`non_neg_integer`

- when the`t`

represents a number, instead of one of the special values below.`:NaN`

- Not a Number.`:inf`

- Infinity.

## Specs

## Specs

exponent() :: integer()

The exponent to which `10`

is raised.

## Specs

rounding() :: :down | :half_up | :half_even | :ceiling | :floor | :half_down | :up

Rounding algorithm.

See `Decimal.Context`

for more information.

## Specs

sign() :: 1 | -1

`1`

for positive`-1`

for negative

## Specs

signal() :: :invalid_operation | :division_by_zero | :rounded | :inexact

## Specs

t() :: %Decimal{coef: coefficient(), exp: exponent(), sign: sign()}

This implementation models the `sign`

as `1`

or `-1`

such that the complete number will be: `sign * coef * 10 ^ exp`

.

`coef`

- the coefficient of the power of`10`

.`exp`

- the exponent of the power of`10`

.`sign`

-`1`

for positive,`-1`

for negative.

# Link to this section Functions

## Specs

The absolute value of given number. Sets the number's sign to positive.

## Specs

Adds two numbers together.

## Exceptional conditions

- If one number is -Infinity and the other +Infinity,
`:invalid_operation`

will be signalled.

## Examples

```
iex> Decimal.add(1, "1.1")
#Decimal<2.1>
iex> Decimal.add(1, "Inf")
#Decimal<Infinity>
```

## Specs

Applies the context to the given number rounding it to specified precision.

## Specs

Creates a new decimal number from an integer, string, float, or existing decimal number.

Because conversion from a floating point number is not exact, it's recommended
to instead use `new/1`

or `from_float/1`

when the argument's type is certain.
See `from_float/1`

.

## Examples

```
iex> {:ok, decimal} = Decimal.cast(3)
iex> decimal
#Decimal<3>
iex> Decimal.cast("bad")
:error
```

## Specs

## Specs

Compares two numbers numerically. If the first number is greater than the second
`:gt`

is returned, if less than `:lt`

is returned, if both numbers are equal
`:eq`

is returned.

Neither number can be a NaN.

## Examples

```
iex> Decimal.compare("1.0", 1)
:eq
iex> Decimal.compare("Inf", -1)
:gt
```

## Specs

Divides two numbers.

## Exceptional conditions

- If both numbers are ±Infinity
`:invalid_operation`

is signalled. - If both numbers are ±0
`:invalid_operation`

is signalled. - If second number (denominator) is ±0
`:division_by_zero`

is signalled.

## Examples

```
iex> Decimal.div(3, 4)
#Decimal<0.75>
iex> Decimal.div("Inf", -1)
#Decimal<-Infinity>
```

## Specs

Divides two numbers and returns the integer part.

## Exceptional conditions

- If both numbers are ±Infinity
`:invalid_operation`

is signalled. - If both numbers are ±0
`:invalid_operation`

is signalled. - If second number (denominator) is ±0
`:division_by_zero`

is signalled.

## Examples

```
iex> Decimal.div_int(5, 2)
#Decimal<2>
iex> Decimal.div_int("Inf", -1)
#Decimal<-Infinity>
```

## Specs

Integer division of two numbers and the remainder. Should be used when both
`div_int/2`

and `rem/2`

is needed. Equivalent to: `{Decimal.div_int(x, y), Decimal.rem(x, y)}`

.

## Exceptional conditions

- If both numbers are ±Infinity
`:invalid_operation`

is signalled. - If both numbers are ±0
`:invalid_operation`

is signalled. - If second number (denominator) is ±0
`:division_by_zero`

is signalled.

## Examples

```
iex> Decimal.div_rem(5, 2)
{Decimal.new(2), Decimal.new(1)}
```

## Specs

Compares two numbers numerically and returns `true`

if they are equal,
otherwise `false`

. If one of the operands is a quiet NaN this operation
will always return `false`

.

## Examples

```
iex> Decimal.eq?("1.0", 1)
true
iex> Decimal.eq?(1, -1)
false
```

## Specs

`true`

if they are equal,
otherwise `false`

. If one of the operands is a quiet NaN this operation
will always return `false`

.

## Examples

```
iex> Decimal.equal?("1.0", 1)
true
iex> Decimal.equal?(1, -1)
false
```

## Specs

Creates a new decimal number from a floating point number.

Floating point numbers use a fixed number of binary digits to represent a decimal number which has inherent inaccuracy as some decimal numbers cannot be represented exactly in limited precision binary.

Floating point numbers will be converted to decimal numbers with
`:io_lib_format.fwrite_g/1`

. Since this conversion is not exact and
because of inherent inaccuracy mentioned above, we may run into counter-intuitive results:

```
iex> Enum.reduce([0.1, 0.1, 0.1], &+/2)
0.30000000000000004
iex> Enum.reduce([Decimal.new("0.1"), Decimal.new("0.1"), Decimal.new("0.1")], &Decimal.add/2)
#Decimal<0.3>
```

For this reason, it's recommended to build decimals with `new/1`

, which is always precise, instead.

## Examples

```
iex> Decimal.from_float(3.14)
#Decimal<3.14>
```

## Specs

Compares two numbers numerically and returns `true`

if the the first argument
is greater than the second, otherwise `false`

. If one the operands is a
quiet NaN this operation will always return `false`

.

## Examples

```
iex> Decimal.gt?("1.3", "1.2")
true
iex> Decimal.gt?("1.2", "1.3")
false
```

## Specs

Returns `true`

if number is ±Infinity, otherwise `false`

.

## Specs

Returns `true`

when the given `decimal`

has no significant digits after the decimal point.

## Examples

```
iex> Decimal.integer?("1.00")
true
iex> Decimal.integer?("1.10")
false
```

Returns `true`

if argument is a decimal number, otherwise `false`

.

## Examples

```
iex> Decimal.is_decimal(Decimal.new(42))
true
iex> Decimal.is_decimal(42)
false
```

Allowed in guard tests on OTP 21+.

## Specs

Compares two numbers numerically and returns `true`

if the the first number is
less than the second number, otherwise `false`

. If one of the operands is a
quiet NaN this operation will always return `false`

.

## Examples

```
iex> Decimal.lt?("1.1", "1.2")
true
iex> Decimal.lt?("1.4", "1.2")
false
```

## Specs

Compares two values numerically and returns the maximum. Unlike most other
functions in `Decimal`

if a number is NaN the result will be the other number.
Only if both numbers are NaN will NaN be returned.

## Examples

```
iex> Decimal.max(1, "2.0")
#Decimal<2.0>
iex> Decimal.max(1, "NaN")
#Decimal<1>
iex> Decimal.max("NaN", "NaN")
#Decimal<NaN>
```

## Specs

Compares two values numerically and returns the minimum. Unlike most other
functions in `Decimal`

if a number is NaN the result will be the other number.
Only if both numbers are NaN will NaN be returned.

## Examples

```
iex> Decimal.min(1, "2.0")
#Decimal<1>
iex> Decimal.min(1, "NaN")
#Decimal<1>
iex> Decimal.min("NaN", "NaN")
#Decimal<NaN>
```

## Specs

Multiplies two numbers.

## Exceptional conditions

- If one number is ±0 and the other is ±Infinity
`:invalid_operation`

is signalled.

## Examples

```
iex> Decimal.mult("0.5", 3)
#Decimal<1.5>
iex> Decimal.mult("Inf", -1)
#Decimal<-Infinity>
```

## Specs

Returns `true`

if number is NaN, otherwise `false`

.

## Specs

Negates the given number.

## Examples

```
iex> Decimal.negate(1)
#Decimal<-1>
iex> Decimal.negate("-Inf")
#Decimal<Infinity>
```

## Specs

Check if given number is negative

## Specs

Creates a new decimal number from an integer or a string representation.

A decimal number will always be created exactly as specified with all digits kept - it will not be rounded with the context.

## Backus–Naur form

```
sign ::= "+" | "-"
digit ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
indicator ::= "e" | "E"
digits ::= digit [digit]...
decimal-part ::= digits "." [digits] | ["."] digits
exponent-part ::= indicator [sign] digits
infinity ::= "Infinity" | "Inf"
nan ::= "NaN" [digits]
numeric-value ::= decimal-part [exponent-part] | infinity
numeric-string ::= [sign] numeric-value | [sign] nan
```

## Floats

See also `from_float/1`

.

## Examples

```
iex> Decimal.new(1)
#Decimal<1>
iex> Decimal.new("3.14")
#Decimal<3.14>
```

## Specs

new(1 | -1, non_neg_integer() | :NaN | :inf, integer()) :: t()

Creates a new decimal number from the sign, coefficient and exponent such that
the number will be: `sign * coefficient * 10 ^ exponent`

.

A decimal number will always be created exactly as specified with all digits kept - it will not be rounded with the context.

## Specs

Normalizes the given decimal: removes trailing zeros from coefficient while keeping the number numerically equivalent by increasing the exponent.

## Examples

```
iex> Decimal.normalize(Decimal.new("1.00"))
#Decimal<1>
iex> Decimal.normalize(Decimal.new("1.01"))
#Decimal<1.01>
```

## Specs

Parses a binary into a decimal.

If successful, returns a tuple in the form of `{decimal, remainder_of_binary}`

,
otherwise `:error`

.

## Examples

```
iex> Decimal.parse("3.14")
{%Decimal{coef: 314, exp: -2, sign: 1}, ""}
iex> Decimal.parse("3.14.15")
{%Decimal{coef: 314, exp: -2, sign: 1}, ".15"}
iex> Decimal.parse("-1.1e3")
{%Decimal{coef: 11, exp: 2, sign: -1}, ""}
iex> Decimal.parse("bad")
:error
```

## Specs

Check if given number is positive

## Specs

Remainder of integer division of two numbers. The result will have the sign of the first number.

## Exceptional conditions

- If both numbers are ±Infinity
`:invalid_operation`

is signalled. - If both numbers are ±0
`:invalid_operation`

is signalled. - If second number (denominator) is ±0
`:division_by_zero`

is signalled.

## Examples

```
iex> Decimal.rem(5, 2)
#Decimal<1>
```

## Specs

Rounds the given number to specified decimal places with the given strategy (default is to round to nearest one). If places is negative, at least that many digits to the left of the decimal point will be zero.

See `Decimal.Context`

for more information about rounding algorithms.

## Examples

```
iex> Decimal.round("1.234")
#Decimal<1>
iex> Decimal.round("1.234", 1)
#Decimal<1.2>
```

## Specs

Finds the square root.

## Examples

```
iex> Decimal.sqrt("100")
#Decimal<10>
```

## Specs

Subtracts second number from the first. Equivalent to `Decimal.add/2`

when the
second number's sign is negated.

## Exceptional conditions

- If one number is -Infinity and the other +Infinity
`:invalid_operation`

will be signalled.

## Examples

```
iex> Decimal.sub(1, "0.1")
#Decimal<0.9>
iex> Decimal.sub(1, "Inf")
#Decimal<-Infinity>
```

## Specs

Returns the decimal converted to a float.

The returned float may have lower precision than the decimal. Fails if the decimal cannot be converted to a float.

## Specs

Returns the decimal represented as an integer.

Fails when loss of precision will occur.

## Specs

Converts given number to its string representation.

## Options

`:scientific`

- number converted to scientific notation.`:normal`

- number converted without a exponent.`:xsd`

- number converted to the canonical XSD representation.`:raw`

- number converted to its raw, internal format.