# Decimal numbers with fixed-point arithmetic¶

The package decimalfp provides a `Decimal` number type which can represent decimal numbers of (nearly) arbitrary magnitude and very large precision, i.e. with a very large number of fractional digits.

## Usage¶

`decimalfp.Decimal` instances are created by giving a value (default: 0) and a precision (i.e the number of fractional digits, default: None).

```>>> Decimal()
Decimal(0)
```

If precision is given, it must be of type int and >= 0.

```>>> Decimal(5, 3)
Decimal(5, 3)
>>> Decimal(5555, -3)
ValueError: Precision must be >= 0.
```

If value is given, it must either be a string, an instance of numbers.Integral, number.Rational (for example fractions.Fraction), decimal.Decimal, a finite instance of numbers.Real (for example float) or be convertable to a float or an int.

The value is always adjusted to the given precision or the precision is calculated from the given value, if no precision is given.

```>>> Decimal(b'12.345')
Decimal('12.345')
>>> Decimal(u'12.345')
Decimal('12.345')
>>> Decimal(u'12.345', 5)
Decimal('12.345', 5)
>>> Decimal(15, 4)
Decimal(15, 4)
>>> Decimal(10 ** 23)
Decimal(100000000000000000000000)
>>> Decimal(10 ** 48)
Decimal(1000000000000000000000000000000000000000000000000)
>>> Decimal(2 ** 48)
Decimal(281474976710656)
>>> Decimal(2 ** 98)
Decimal(316912650057057350374175801344)
>>> Decimal(Fraction(7, 56))
Decimal('0.125')
>>> Decimal(Fraction(8106479329266893, 4503599627370496), 7)
Decimal('1.8', 7)
>>> Decimal(1.8, 7)
Decimal('1.8', 7)
>>> Decimal(decimal.Decimal('-19.26', 5), 3)
Decimal('-19.26', 3)
```

When the given precision is lower than the precision of the given value, the result is rounded, according to the current default rounding mode (which itself defaults to ROUND_HALF_EVEN).

```>>> Decimal(u'12.345', 2)
Decimal('12.34')
>>> Decimal(u'12.3456', 3)
Decimal('12.346')
>>> Decimal(0.2, 3)
Decimal('0.2', 3)
>>> Decimal(0.2, 17)
Decimal('0.20000000000000001')
>>> Decimal(0.2, 55)
Decimal('0.200000000000000011102230246251565404236316680908203125', 55)
```

When no precision is given and the given value is a float or a numbers.Rational (but no `Decimal`), the `Decimal` constructor tries to convert value exactly. But this is done only up a fixed limit of fractional digits (imposed by the implementation, currently 65535). If value can not be represented as a `Decimal` within this limit, an exception is raised.

```>>> Decimal(Fraction(1, 7))
ValueError: Can't convert Fraction(1, 7) exactly to Decimal.
```

`Decimal` does not deal with infinity, division by 0 always raises a ZeroDivisionError. Likewise, infinite instances of type float or decimal.Decimal can not be converted to `Decimal` instances. The same is true for the ‘not a number’ instances of these types.

## Computations¶

When importing decimalfp, its `Decimal` type is registered in Pythons numerical stack as number.Rational. It supports all operations defined for that base class and its instances can be mixed in computations with instances of all numeric types mentioned above.

All numerical operations give an exact result, i.e. they are not automatically constraint to the precision of the operands or to a number of significant digits (like the floating-point Decimal type from the standard module decimal). When the result can not exactly be represented by a `Decimal` instance within the limit of fractional digits, an instance of fractions.Fraction is returned.

Adding or subtracting `Decimal` instances results in a `Decimal` instance with a precision equal to the maximum of the precisions of the operands.

```>>> Decimal('7.3') + Decimal('8.275')
Decimal('15.575')
>>> Decimal('-7.3', 4) + Decimal('8.275')
Decimal('0.975', 4)
```

In operations with other numerical types the precision of the result is at least equal to the precision of the involved `Decimal` instance, but may be greater, if neccessary. If the needed precision exceeds the limit of fractional digits, an instance of fractions.Fraction is returned.

```>>> 0.25 + Decimal(3)
Decimal('3.25')
>>> 0.25 - Decimal(-3, 5)
Decimal('3.25', 5)
>>> 0.725 + Decimal('3')
Decimal('3.72499999999999997779553950749686919152736663818359375')
>>> Decimal('3') + Fraction(1, 7)
Fraction(22, 7)
```

### Multiplication and division¶

Multiplying `Decimal` instances results in a `Decimal` instance with precision equal to the sum of the precisions of the operands.

```>>> Decimal('5.000') * Decimal('2.5')
Decimal('12.5', 4)
```

Dividing `Decimal` instances results in a `Decimal` instance with precision at least equal to max(0, numerator.precision - denominator.precision), but may be greater, if needed.

```>>> Decimal('5.2000') / Decimal('2.5')
Decimal('2.08', 3)
>>> Decimal('5.2003') / Decimal('2.5')
Decimal('2.08012')
```

In operations with other numerical types the precision of the result is at least equal to the precision of the involved `Decimal` instance, but may be greater, if neccessary. If the needed precision exceeds the limit of fractional digits, an instance of fractions.Fraction is returned.

```>>> 3 * Decimal('7.5')
Decimal('22.5')
>>> Decimal(5) * 0.25
Decimal('1.25')
>>> Decimal('3') * Fraction(1, 7)
Fraction(3, 7)
```

### Rounding¶

`Decimal` supports rounding via the built-in function round.

Note

In Python 3.x the function round uses the rounding mode ROUND_HALF_EVEN and returns an int when called with one argument, otherwise the same type as the number to be rounded.

```>>> round(Decimal('12.345'))
12
>>> round(Decimal('12.345'), 2)
Decimal('12.34')
>>> round(Decimal('1234.5'), -2)
Decimal(1200)
```

In addition, via the method `adjusted()` a `Decimal` with a different precision can be derived, supporting all rounding modes defined by the standard library module decimal.

The rounding modes defined in decimal are wrapped into the Enum `ROUNDING`.

```>>> d = Decimal('12.345')
>>> d.adjusted(2)           # default rounding mode is ROUND_HALF_EVEN !
Decimal('12.34')
Decimal('12.35')
Decimal('12.4')
```

For the details of the different rounding modes see the documentation of the standard library module decimal.

round and adjusted only allow to round to a quantum that’s a power to 10. The method `quantize()` can be used to round to any quantum and it does also support all rounding modes mentioned above.

```>>> d = Decimal('12.345')
>>># equivalent to round(d, 2) or d.adjusted(2)
>>># (default rounding mode ROUNDING.ROUND_HALF_EVEN):
>>> d.quantize(Decimal('0.01'))
Decimal('12.34')
>>> d.quantize(Decimal('0.05'))
Decimal('12.35')
>>> d.quantize('0.6')
Decimal('12.6')
>>> d.quantize(4)
Decimal('12')
```

## Class Decimal¶

DecValueT = Union[SupportsInt, SupportsFloat, SupportsAsIntegerRatio, str]

class Decimal

Decimal number with a given number of fractional digits.

Parameters
• value – numerical value (default: None)

• precision – number of fractional digits (default: None)

If value is given, it must either be a string, an instance of int, number.Rational (for example fractions.Fraction), decimal.Decimal, a finite instance of numbers.Real (for example float) or be convertable to a float or an int.

If a string is given as value, it must be a string in one of two formats:

• [+|-]<int>[.<frac>][<e|E>[+|-]<exp>] or

• [+|-].<frac>[<e|E>[+|-]<exp>].

If given value is None, Decimal(0) is returned.

Returns

`Decimal` instance derived from value according

to precision

The value is always adjusted to the given precision or the precision is calculated from the given value, if no precision is given.

Raises
• TypeErrorprecision is given, but not of type int.

• TypeErrorvalue is not an instance of the types listed above and not convertable to float or int.

• ValueErrorprecision is given, but not >= 0.

• ValueErrorprecision is given, but not <= MAX_DEC_PRECISION.

• ValueErrorvalue can not be converted to a Decimal (with a number of fractional digits <= MAX_DEC_PRECISION).

`Decimal` instances are immutable.

__abs__()

abs(self)

self + other

__bytes__() bytes

bytes(self)

__ceil__() int

math.ceil(self)

__eq__(other: Any) bool

self == other

__floor__() int

math.floor(self)

__format__(fmt_spec: str) str

Return self converted to a string according to fmt_spec.

Parameters

fmt_spec – a standard format specifier for a number

Returns

strself converted to a string according to fmt_spec

__ge__(other: Any) bool

self >= other

__gt__(other: Any) bool

self > other

__hash__() int

hash(self)

__init__()
__int__() int

math.trunc(self)

__le__(other: Any) bool

self <= other

__lt__(other: Any) bool

self < other

__mul__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational

self * other

__neg__()

-self

static __new__(cls, value: DecValueT = None, precision: Optional[Integer] = None)

Create and return new Decimal instance.

__pos__()

+self

__pow__(other: SupportsIntOrFloat, mod: Optional[Any] = None) numbers.Complex

self ** other

If other is an integer (or a Rational with denominator = 1), the result will be a Decimal or a Fraction. Otherwise, the result will be a float or a complex.

mod must always be None (otherwise a TypeError is raised).

other + self

__repr__() str

repr(self)

__rmul__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational

other * self

__round__(ndigits: Optional[int] = None) Union[int, Decimal]

round(self [, n_digits])

Round self to a given precision in decimal digits (default 0). n_digits may be negative.

This method is called by the built-in round function. It returns an int when called with one argument, otherwise a `Decimal`.

__rsub__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational

other - self

__rtruediv__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational

other / self

__str__() str

str(self)

__sub__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational

self - other

__truediv__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational

self / other

__trunc__() int

math.trunc(self)

adjusted(precision: Optional[int] = None, rounding: Optional[ROUNDING] = None)

Parameters
• precision – number of fractional digits (default: None)

• rounding – rounding mode (default: None)

Returns

`Decimal` instance derived from self, adjusted

to the given precision, using the given rounding mode

If no precision is given, the result is adjusted to the minimum precision preserving x == x.adjusted().

If no rounding mode is given, the current default rounding mode is used.

If the given precision is less than the precision of self, the result is rounded and thus information may be lost.

as_fraction() fractions.Fraction

Return an instance of Fraction equal to self.

Returns the Fraction with the smallest positive denominator, whose ratio is equal to self.

as_integer_ratio() Tuple[int, int]

Return a pair of integers whose ratio is equal to self.

Returns the pair of numerator and denominator with the smallest positive denominator, whose ratio is equal to self.

as_tuple() Tuple[int, int, int]

Return a tuple (sign, coeff, exp) equivalent to self.

self == sign * coeff * 10 ** exp.

sign in (-1, 0, 1), for self < 0, = 0, > 0. coeff = 0 only if self = 0.

classmethod from_decimal(d: Union[Decimal, int, decimal.Decimal])

Convert a finite decimal number to a `Decimal`.

Parameters

d – decimal number to be converted to a `Decimal`

Returns

`Decimal` instance derived from d

Raises
• TypeErrord is not an instance of the types listed above.

• ValueErrord can not be converted to a `Decimal`.

classmethod from_float(f: Union[float, int])

Convert a finite float (or int) to a `Decimal`.

Parameters

f – number to be converted to a Decimal

Returns

`Decimal` instance derived from f

Raises
• TypeErrorf is neither a float nor an int.

• ValueErrorf can not be converted to a `Decimal` with a precision <= MAX_DEC_PRECISION.

Beware that Decimal.from_float(0.3) != Decimal(‘0.3’).

classmethod from_real(r: numbers.Real, exact: bool = True)

Convert a finite Real number to a `Decimal`.

Parameters
Returns

`Decimal` instance derived from r

Raises
• TypeErrorr is not an instance of numbers.Real.

• ValueErrorexact is True and r can not exactly be converted to a `Decimal` with a precision <= MAX_DEC_PRECISION.

If exact is False and r can not exactly be represented by a Decimal with a precision <= MAX_DEC_PRECISION, the result is rounded to a precision = MAX_DEC_PRECISION.

quantize(quant: decimalfp.SupportsAsIntegerRatio, rounding: Optional[ROUNDING] = None) numbers.Rational

Return integer multiple of quant closest to self.

Parameters
• quant – quantum to get a multiple from

• rounding – rounding mode (default: None)

If no rounding mode is given, the current default rounding mode is used.

Returns

`Decimal` instance that is the integer multiple of quant

closest to self (according to rounding mode); if result can not be represented as `Decimal`, an instance of Fraction is returned

Raises
• TypeErrorquant is not a number or does not support as_integer_ratio

• ValueErrorquant is not convertable to a Rational

property denominator: int

Return the normalized denominator of ‘self’.

I. e. the smallest positive denominator from the pairs of integers, whose ratio is equal to self.

property imag: int

Return imaginary part of self.

Returns 0 (Real numbers have no imaginary component).

property magnitude: int

Return magnitude of self in terms of power to 10.

I.e. the largest integer exp so that 10 ** exp <= self.

property numerator: int

Return the normalized numerator of self.

I. e. the numerator from the pair of integers with the smallest positive denominator, whose ratio is equal to self.

property precision: int

Return precision of self.

property real: Decimal

Return real part of self.

Returns self (Real numbers are their real component).

## Rounding modes¶

Decimal supports rounding modes equivalent to those defined by the standard library module decimal: ROUND_DOWN, ROUND_UP, ROUND_HALF_DOWN, ROUND_HALF_UP, ROUND_HALF_EVEN, ROUND_CEILING, ROUND_FLOOR and ROUND_05UP.

The rounding modes are wrapped into the Enum `ROUNDING`.

class ROUNDING

Enumeration of rounding modes.

ROUND_05UP = 1

Round away from zero if last digit after rounding towards zero would have been 0 or 5; otherwise round towards zero.

ROUND_CEILING = 2

Round towards Infinity.

ROUND_DOWN = 3

Round towards zero.

ROUND_FLOOR = 4

Round towards -Infinity.

ROUND_HALF_DOWN = 5

Round to nearest with ties going towards zero.

ROUND_HALF_EVEN = 6

Round to nearest with ties going to nearest even integer.

ROUND_HALF_UP = 7

Round to nearest with ties going away from zero.

ROUND_UP = 8

Round away from zero.

Unless a rounding mode is explicitely given, the rounding mode set as current default is used. To get or set the default rounding mode, the package decimalfp provides the following two functions:

get_dflt_rounding_mode()

Return default rounding mode.

set_dflt_rounding_mode(rounding: ROUNDING) None

Set default rounding mode.

Parameters

rounding (ROUNDING) – rounding mode to be set as default

Raises

TypeError – given ‘rounding’ is not a valid rounding mode