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9.4. decimal — Decimal fixed point and floating point arithmetic
New in version 2.4.
The decimal
module provides support for decimal floating point arithmetic. It offers several advantages over the float
datatype:
Decimal “is based on a floatingpoint model which was designed with people in mind, and necessarily has a paramount guiding principle – computers must provide an arithmetic that works in the same way as the arithmetic that people learn at school.” – excerpt from the decimal arithmetic specification.
Decimal numbers can be represented exactly. In contrast, numbers like
1.1
and2.2
do not have exact representations in binary floating point. End users typically would not expect1.1 + 2.2
to display as3.3000000000000003
as it does with binary floating point.The exactness carries over into arithmetic. In decimal floating point,
0.1 + 0.1 + 0.1  0.3
is exactly equal to zero. In binary floating point, the result is5.5511151231257827e017
. While near to zero, the differences prevent reliable equality testing and differences can accumulate. For this reason, decimal is preferred in accounting applications which have strict equality invariants.The decimal module incorporates a notion of significant places so that
1.30 + 1.20
is2.50
. The trailing zero is kept to indicate significance. This is the customary presentation for monetary applications. For multiplication, the “schoolbook” approach uses all the figures in the multiplicands. For instance,1.3 * 1.2
gives1.56
while1.30 * 1.20
gives1.5600
.Unlike hardware based binary floating point, the decimal module has a user alterable precision (defaulting to 28 places) which can be as large as needed for a given problem:
>>> from decimal import * >>> getcontext().prec = 6 >>> Decimal(1) / Decimal(7) Decimal('0.142857') >>> getcontext().prec = 28 >>> Decimal(1) / Decimal(7) Decimal('0.1428571428571428571428571429')
Both binary and decimal floating point are implemented in terms of published standards. While the builtin float type exposes only a modest portion of its capabilities, the decimal module exposes all required parts of the standard. When needed, the programmer has full control over rounding and signal handling. This includes an option to enforce exact arithmetic by using exceptions to block any inexact operations.
The decimal module was designed to support “without prejudice, both exact unrounded decimal arithmetic (sometimes called fixedpoint arithmetic) and rounded floatingpoint arithmetic.” – excerpt from the decimal arithmetic specification.
The module design is centered around three concepts: the decimal number, the context for arithmetic, and signals.
A decimal number is immutable. It has a sign, coefficient digits, and an exponent. To preserve significance, the coefficient digits do not truncate trailing zeros. Decimals also include special values such as Infinity
, Infinity
, and NaN
. The standard also differentiates 0
from +0
.
The context for arithmetic is an environment specifying precision, rounding rules, limits on exponents, flags indicating the results of operations, and trap enablers which determine whether signals are treated as exceptions. Rounding options include ROUND_CEILING
, ROUND_DOWN
, ROUND_FLOOR
, ROUND_HALF_DOWN
, ROUND_HALF_EVEN
, ROUND_HALF_UP
, ROUND_UP
, and ROUND_05UP
.
Signals are groups of exceptional conditions arising during the course of computation. Depending on the needs of the application, signals may be ignored, considered as informational, or treated as exceptions. The signals in the decimal module are: Clamped
, InvalidOperation
, DivisionByZero
, Inexact
, Rounded
, Subnormal
, Overflow
, and Underflow
.
For each signal there is a flag and a trap enabler. When a signal is encountered, its flag is set to one, then, if the trap enabler is set to one, an exception is raised. Flags are sticky, so the user needs to reset them before monitoring a calculation.
See also
IBM’s General Decimal Arithmetic Specification, The General Decimal Arithmetic Specification .
9.4.1. Quickstart Tutorial
The usual start to using decimals is importing the module, viewing the current context with getcontext()
and, if necessary, setting new values for precision, rounding, or enabled traps:
>>> from decimal import *
>>> getcontext()
Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=999999999, Emax=999999999,
capitals=1, flags=[], traps=[Overflow, DivisionByZero,
InvalidOperation])
>>> getcontext().prec = 7 # Set a new precision
Decimal instances can be constructed from integers, strings, floats, or tuples. Construction from an integer or a float performs an exact conversion of the value of that integer or float. Decimal numbers include special values such as NaN
which stands for “Not a number”, positive and negative Infinity
, and 0
.
>>> getcontext().prec = 28
>>> Decimal(10)
Decimal('10')
>>> Decimal('3.14')
Decimal('3.14')
>>> Decimal(3.14)
Decimal('3.140000000000000124344978758017532527446746826171875')
>>> Decimal((0, (3, 1, 4), 2))
Decimal('3.14')
>>> Decimal(str(2.0 ** 0.5))
Decimal('1.41421356237')
>>> Decimal(2) ** Decimal('0.5')
Decimal('1.414213562373095048801688724')
>>> Decimal('NaN')
Decimal('NaN')
>>> Decimal('Infinity')
Decimal('Infinity')
The significance of a new Decimal is determined solely by the number of digits input. Context precision and rounding only come into play during arithmetic operations.
>>> getcontext().prec = 6
>>> Decimal('3.0')
Decimal('3.0')
>>> Decimal('3.1415926535')
Decimal('3.1415926535')
>>> Decimal('3.1415926535') + Decimal('2.7182818285')
Decimal('5.85987')
>>> getcontext().rounding = ROUND_UP
>>> Decimal('3.1415926535') + Decimal('2.7182818285')
Decimal('5.85988')
Decimals interact well with much of the rest of Python. Here is a small decimal floating point flying circus:
>>> data = map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())
>>> max(data)
Decimal('9.25')
>>> min(data)
Decimal('0.03')
>>> sorted(data)
[Decimal('0.03'), Decimal('1.00'), Decimal('1.34'), Decimal('1.87'),
Decimal('2.35'), Decimal('3.45'), Decimal('9.25')]
>>> sum(data)
Decimal('19.29')
>>> a,b,c = data[:3]
>>> str(a)
'1.34'
>>> float(a)
1.34
>>> round(a, 1) # round() first converts to binary floating point
1.3
>>> int(a)
1
>>> a * 5
Decimal('6.70')
>>> a * b
Decimal('2.5058')
>>> c % a
Decimal('0.77')
And some mathematical functions are also available to Decimal:
>>> getcontext().prec = 28
>>> Decimal(2).sqrt()
Decimal('1.414213562373095048801688724')
>>> Decimal(1).exp()
Decimal('2.718281828459045235360287471')
>>> Decimal('10').ln()
Decimal('2.302585092994045684017991455')
>>> Decimal('10').log10()
Decimal('1')
The quantize()
method rounds a number to a fixed exponent. This method is useful for monetary applications that often round results to a fixed number of places:
>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
Decimal('7.32')
>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
Decimal('8')
As shown above, the getcontext()
function accesses the current context and allows the settings to be changed. This approach meets the needs of most applications.
For more advanced work, it may be useful to create alternate contexts using the Context() constructor. To make an alternate active, use the setcontext()
function.
In accordance with the standard, the decimal
module provides two ready to use standard contexts, BasicContext
and ExtendedContext
. The former is especially useful for debugging because many of the traps are enabled:
>>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
>>> setcontext(myothercontext)
>>> Decimal(1) / Decimal(7)
Decimal('0.142857142857142857142857142857142857142857142857142857142857')
>>> ExtendedContext
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=999999999, Emax=999999999,
capitals=1, flags=[], traps=[])
>>> setcontext(ExtendedContext)
>>> Decimal(1) / Decimal(7)
Decimal('0.142857143')
>>> Decimal(42) / Decimal(0)
Decimal('Infinity')
>>> setcontext(BasicContext)
>>> Decimal(42) / Decimal(0)
Traceback (most recent call last):
File "<pyshell#143>", line 1, in toplevel
Decimal(42) / Decimal(0)
DivisionByZero: x / 0
Contexts also have signal flags for monitoring exceptional conditions encountered during computations. The flags remain set until explicitly cleared, so it is best to clear the flags before each set of monitored computations by using the clear_flags()
method.
>>> setcontext(ExtendedContext)
>>> getcontext().clear_flags()
>>> Decimal(355) / Decimal(113)
Decimal('3.14159292')
>>> getcontext()
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=999999999, Emax=999999999,
capitals=1, flags=[Rounded, Inexact], traps=[])
The flags entry shows that the rational approximation to Pi
was rounded (digits beyond the context precision were thrown away) and that the result is inexact (some of the discarded digits were nonzero).
Individual traps are set using the dictionary in the traps
field of a context:
>>> setcontext(ExtendedContext)
>>> Decimal(1) / Decimal(0)
Decimal('Infinity')
>>> getcontext().traps[DivisionByZero] = 1
>>> Decimal(1) / Decimal(0)
Traceback (most recent call last):
File "<pyshell#112>", line 1, in toplevel
Decimal(1) / Decimal(0)
DivisionByZero: x / 0
Most programs adjust the current context only once, at the beginning of the program. And, in many applications, data is converted to Decimal
with a single cast inside a loop. With context set and decimals created, the bulk of the program manipulates the data no differently than with other Python numeric types.
9.4.2. Decimal objects
 class
decimal.
Decimal
( [ value [, context ] ] ) 
Construct a new
Decimal
object based from value.value can be an integer, string, tuple,
float
, or anotherDecimal
object. If no value is given, returnsDecimal('0')
. If value is a string, it should conform to the decimal numeric string syntax after leading and trailing whitespace characters are removed:sign ::= '+'  '' digit ::= '0'  '1'  '2'  '3'  '4'  '5'  '6'  '7'  '8'  '9' indicator ::= 'e'  'E' digits ::= digit [digit]... decimalpart ::= digits '.' [digits]  ['.'] digits exponentpart ::= indicator [sign] digits infinity ::= 'Infinity'  'Inf' nan ::= 'NaN' [digits]  'sNaN' [digits] numericvalue ::= decimalpart [exponentpart]  infinity numericstring ::= [sign] numericvalue  [sign] nan
If value is a unicode string then other Unicode decimal digits are also permitted where
digit
appears above. These include decimal digits from various other alphabets (for example, ArabicIndic and Devanāgarī digits) along with the fullwidth digitsu'\uff10'
throughu'\uff19'
.If value is a
tuple
, it should have three components, a sign (0
for positive or1
for negative), atuple
of digits, and an integer exponent. For example,Decimal((0, (1, 4, 1, 4), 3))
returnsDecimal('1.414')
.If value is a
float
, the binary floating point value is losslessly converted to its exact decimal equivalent. This conversion can often require 53 or more digits of precision. For example,Decimal(float('1.1'))
converts toDecimal('1.100000000000000088817841970012523233890533447265625')
.The context precision does not affect how many digits are stored. That is determined exclusively by the number of digits in value. For example,
Decimal('3.00000')
records all five zeros even if the context precision is only three.The purpose of the context argument is determining what to do if value is a malformed string. If the context traps
InvalidOperation
, an exception is raised; otherwise, the constructor returns a new Decimal with the value ofNaN
.Once constructed,
Decimal
objects are immutable.Changed in version 2.6: leading and trailing whitespace characters are permitted when creating a Decimal instance from a string.
Changed in version 2.7: The argument to the constructor is now permitted to be a
float
instance.Decimal floating point objects share many properties with the other builtin numeric types such as
float
andint
. All of the usual math operations and special methods apply. Likewise, decimal objects can be copied, pickled, printed, used as dictionary keys, used as set elements, compared, sorted, and coerced to another type (such asfloat
orlong
).There are some small differences between arithmetic on Decimal objects and arithmetic on integers and floats. When the remainder operator
%
is applied to Decimal objects, the sign of the result is the sign of the dividend rather than the sign of the divisor:>>> (7) % 4 1 >>> Decimal(7) % Decimal(4) Decimal('3')
The integer division operator
//
behaves analogously, returning the integer part of the true quotient (truncating towards zero) rather than its floor, so as to preserve the usual identityx == (x // y) * y + x % y
:>>> 7 // 4 2 >>> Decimal(7) // Decimal(4) Decimal('1')
The
%
and//
operators implement theremainder
anddivideinteger
operations (respectively) as described in the specification.Decimal objects cannot generally be combined with floats in arithmetic operations: an attempt to add a
Decimal
to afloat
, for example, will raise aTypeError
. There’s one exception to this rule: it’s possible to use Python’s comparison operators to compare afloat
instancex
with aDecimal
instancey
. Without this exception, comparisons betweenDecimal
andfloat
instances would follow the general rules for comparing objects of different types described in the Expressions section of the reference manual, leading to confusing results.Changed in version 2.7: A comparison between a
float
instancex
and aDecimal
instancey
now returns a result based on the values ofx
andy
. In earlier versionsx < y
returned the same (arbitrary) result for anyDecimal
instancex
and anyfloat
instancey
.In addition to the standard numeric properties, decimal floating point objects also have a number of specialized methods:
adjusted
( )
Return the adjusted exponent after shifting out the coefficient’s rightmost digits until only the lead digit remains:
Decimal('321e+5').adjusted()
returns seven. Used for determining the position of the most significant digit with respect to the decimal point.
as_tuple
( )
Return a named tuple representation of the number:
DecimalTuple(sign, digits, exponent)
.Changed in version 2.6: Use a named tuple.
canonical
( )
Return the canonical encoding of the argument. Currently, the encoding of a
Decimal
instance is always canonical, so this operation returns its argument unchanged.New in version 2.6.
compare
( other [, context ] )
Compare the values of two Decimal instances. This operation behaves in the same way as the usual comparison method
__cmp__()
, except thatcompare()
returns a Decimal instance rather than an integer, and if either operand is a NaN then the result is a NaN:a or b is a NaN ==> Decimal('NaN') a < b ==> Decimal('1') a == b ==> Decimal('0') a > b ==> Decimal('1')
compare_signal
( other [, context ] )
This operation is identical to the
compare()
method, except that all NaNs signal. That is, if neither operand is a signaling NaN then any quiet NaN operand is treated as though it were a signaling NaN.New in version 2.6.
compare_total
( other )
Compare two operands using their abstract representation rather than their numerical value. Similar to the
compare()
method, but the result gives a total ordering onDecimal
instances. TwoDecimal
instances with the same numeric value but different representations compare unequal in this ordering:>>> Decimal('12.0').compare_total(Decimal('12')) Decimal('1')
Quiet and signaling NaNs are also included in the total ordering. The result of this function is
Decimal('0')
if both operands have the same representation,Decimal('1')
if the first operand is lower in the total order than the second, andDecimal('1')
if the first operand is higher in the total order than the second operand. See the specification for details of the total order.New in version 2.6.
compare_total_mag
( other )
Compare two operands using their abstract representation rather than their value as in
compare_total()
, but ignoring the sign of each operand.x.compare_total_mag(y)
is equivalent tox.copy_abs().compare_total(y.copy_abs())
.New in version 2.6.
conjugate
( )
Just returns self, this method is only to comply with the Decimal Specification.
New in version 2.6.
copy_abs
( )
Return the absolute value of the argument. This operation is unaffected by the context and is quiet: no flags are changed and no rounding is performed.
New in version 2.6.
copy_negate
( )
Return the negation of the argument. This operation is unaffected by the context and is quiet: no flags are changed and no rounding is performed.
New in version 2.6.
copy_sign
( other )
Return a copy of the first operand with the sign set to be the same as the sign of the second operand. For example:
>>> Decimal('2.3').copy_sign(Decimal('1.5')) Decimal('2.3')
This operation is unaffected by the context and is quiet: no flags are changed and no rounding is performed.
New in version 2.6.
exp
( [ context ] )
Return the value of the (natural) exponential function
e**x
at the given number. The result is correctly rounded using theROUND_HALF_EVEN
rounding mode.>>> Decimal(1).exp() Decimal('2.718281828459045235360287471') >>> Decimal(321).exp() Decimal('2.561702493119680037517373933E+139')
New in version 2.6.
from_float
( f )
Classmethod that converts a float to a decimal number, exactly.
Note Decimal.from_float(0.1) is not the same as Decimal(‘0.1’). Since 0.1 is not exactly representable in binary floating point, the value is stored as the nearest representable value which is 0x1.999999999999ap4. That equivalent value in decimal is 0.1000000000000000055511151231257827021181583404541015625.
>>> Decimal.from_float(0.1) Decimal('0.1000000000000000055511151231257827021181583404541015625') >>> Decimal.from_float(float('nan')) Decimal('NaN') >>> Decimal.from_float(float('inf')) Decimal('Infinity') >>> Decimal.from_float(float('inf')) Decimal('Infinity')
New in version 2.7.
fma
( other, third [, context ] )
Fused multiplyadd. Return self*other+third with no rounding of the intermediate product self*other.
>>> Decimal(2).fma(3, 5) Decimal('11')
New in version 2.6.
is_canonical
( )
Return
True
if the argument is canonical andFalse
otherwise. Currently, aDecimal
instance is always canonical, so this operation always returnsTrue
.New in version 2.6.
is_finite
( )
Return
True
if the argument is a finite number, andFalse
if the argument is an infinity or a NaN.New in version 2.6.
is_infinite
( )
Return
True
if the argument is either positive or negative infinity andFalse
otherwise.New in version 2.6.
is_nan
( )
Return
True
if the argument is a (quiet or signaling) NaN andFalse
otherwise.New in version 2.6.
is_normal
( )
Return
True
if the argument is a normal finite nonzero number with an adjusted exponent greater than or equal to Emin. ReturnFalse
if the argument is zero, subnormal, infinite or a NaN. Note, the term normal is used here in a different sense with thenormalize()
method which is used to create canonical values.New in version 2.6.
is_signed
( )
Return
True
if the argument has a negative sign andFalse
otherwise. Note that zeros and NaNs can both carry signs.New in version 2.6.
is_subnormal
( )
Return
True
if the argument is subnormal, andFalse
otherwise. A number is subnormal is if it is nonzero, finite, and has an adjusted exponent less than Emin.New in version 2.6.
is_zero
( )
Return
True
if the argument is a (positive or negative) zero andFalse
otherwise.New in version 2.6.
ln
( [ context ] )
Return the natural (base e) logarithm of the operand. The result is correctly rounded using the
ROUND_HALF_EVEN
rounding mode.New in version 2.6.
log10
( [ context ] )
Return the base ten logarithm of the operand. The result is correctly rounded using the
ROUND_HALF_EVEN
rounding mode.New in version 2.6.
logb
( [ context ] )
For a nonzero number, return the adjusted exponent of its operand as a
Decimal
instance. If the operand is a zero thenDecimal('Infinity')
is returned and theDivisionByZero
flag is raised. If the operand is an infinity thenDecimal('Infinity')
is returned.New in version 2.6.
logical_and
( other [, context ] )
logical_and()
is a logical operation which takes two logical operands (see Logical operands). The result is the digitwiseand
of the two operands.New in version 2.6.
logical_invert
( [ context ] )
logical_invert()
is a logical operation. The result is the digitwise inversion of the operand.New in version 2.6.
logical_or
( other [, context ] )
logical_or()
is a logical operation which takes two logical operands (see Logical operands). The result is the digitwiseor
of the two operands.New in version 2.6.
logical_xor
( other [, context ] )
logical_xor()
is a logical operation which takes two logical operands (see Logical operands). The result is the digitwise exclusive or of the two operands.New in version 2.6.
max
( other [, context ] )
Like
max(self, other)
except that the context rounding rule is applied before returning and thatNaN
values are either signaled or ignored (depending on the context and whether they are signaling or quiet).
max_mag
( other [, context ] )
Similar to the
max()
method, but the comparison is done using the absolute values of the operands.New in version 2.6.
min
( other [, context ] )
Like
min(self, other)
except that the context rounding rule is applied before returning and thatNaN
values are either signaled or ignored (depending on the context and whether they are signaling or quiet).
min_mag
( other [, context ] )
Similar to the
min()
method, but the comparison is done using the absolute values of the operands.New in version 2.6.
next_minus
( [ context ] )
Return the largest number representable in the given context (or in the current thread’s context if no context is given) that is smaller than the given operand.
New in version 2.6.
next_plus
( [ context ] )
Return the smallest number representable in the given context (or in the current thread’s context if no context is given) that is larger than the given operand.
New in version 2.6.
next_toward
( other [, context ] )
If the two operands are unequal, return the number closest to the first operand in the direction of the second operand. If both operands are numerically equal, return a copy of the first operand with the sign set to be the same as the sign of the second operand.
New in version 2.6.
normalize
( [ context ] )
Normalize the number by stripping the rightmost trailing zeros and converting any result equal to
Decimal('0')
toDecimal('0e0')
. Used for producing canonical values for attributes of an equivalence class. For example,Decimal('32.100')
andDecimal('0.321000e+2')
both normalize to the equivalent valueDecimal('32.1')
.
number_class
( [ context ] )
Return a string describing the class of the operand. The returned value is one of the following ten strings.
"Infinity"
, indicating that the operand is negative infinity."Normal"
, indicating that the operand is a negative normal number."Subnormal"
, indicating that the operand is negative and subnormal."Zero"
, indicating that the operand is a negative zero."+Zero"
, indicating that the operand is a positive zero."+Subnormal"
, indicating that the operand is positive and subnormal."+Normal"
, indicating that the operand is a positive normal number."+Infinity"
, indicating that the operand is positive infinity."NaN"
, indicating that the operand is a quiet NaN (Not a Number)."sNaN"
, indicating that the operand is a signaling NaN.
New in version 2.6.
quantize
( exp [, rounding [, context [, watchexp ] ] ] )
Return a value equal to the first operand after rounding and having the exponent of the second operand.
>>> Decimal('1.41421356').quantize(Decimal('1.000')) Decimal('1.414')
Unlike other operations, if the length of the coefficient after the quantize operation would be greater than precision, then an
InvalidOperation
is signaled. This guarantees that, unless there is an error condition, the quantized exponent is always equal to that of the righthand operand.Also unlike other operations, quantize never signals Underflow, even if the result is subnormal and inexact.
If the exponent of the second operand is larger than that of the first then rounding may be necessary. In this case, the rounding mode is determined by the
rounding
argument if given, else by the givencontext
argument; if neither argument is given the rounding mode of the current thread’s context is used.If watchexp is set (default), then an error is returned whenever the resulting exponent is greater than
Emax
or less thanEtiny
.
radix
( )
Return
Decimal(10)
, the radix (base) in which theDecimal
class does all its arithmetic. Included for compatibility with the specification.New in version 2.6.
remainder_near
( other [, context ] )
Return the remainder from dividing self by other. This differs from
self % other
in that the sign of the remainder is chosen so as to minimize its absolute value. More precisely, the return value isself  n * other
wheren
is the integer nearest to the exact value ofself / other
, and if two integers are equally near then the even one is chosen.If the result is zero then its sign will be the sign of self.
>>> Decimal(18).remainder_near(Decimal(10)) Decimal('2') >>> Decimal(25).remainder_near(Decimal(10)) Decimal('5') >>> Decimal(35).remainder_near(Decimal(10)) Decimal('5')
rotate
( other [, context ] )
Return the result of rotating the digits of the first operand by an amount specified by the second operand. The second operand must be an integer in the range precision through precision. The absolute value of the second operand gives the number of places to rotate. If the second operand is positive then rotation is to the left; otherwise rotation is to the right. The coefficient of the first operand is padded on the left with zeros to length precision if necessary. The sign and exponent of the first operand are unchanged.
New in version 2.6.
same_quantum
( other [, context ] )
Test whether self and other have the same exponent or whether both are
NaN
.
scaleb
( other [, context ] )
Return the first operand with exponent adjusted by the second. Equivalently, return the first operand multiplied by
10**other
. The second operand must be an integer.New in version 2.6.
shift
( other [, context ] )
Return the result of shifting the digits of the first operand by an amount specified by the second operand. The second operand must be an integer in the range precision through precision. The absolute value of the second operand gives the number of places to shift. If the second operand is positive then the shift is to the left; otherwise the shift is to the right. Digits shifted into the coefficient are zeros. The sign and exponent of the first operand are unchanged.
New in version 2.6.
to_eng_string
( [ context ] )
Convert to a string, using engineering notation if an exponent is needed.
Engineering notation has an exponent which is a multiple of 3. This can leave up to 3 digits to the left of the decimal place and may require the addition of either one or two trailing zeros.
For example, this converts
Decimal('123E+1')
toDecimal('1.23E+3')
.
to_integral
( [ rounding [, context ] ] )
Identical to the
to_integral_value()
method. Theto_integral
name has been kept for compatibility with older versions.
to_integral_exact
( [ rounding [, context ] ] )
Round to the nearest integer, signaling
Inexact
orRounded
as appropriate if rounding occurs. The rounding mode is determined by therounding
parameter if given, else by the givencontext
. If neither parameter is given then the rounding mode of the current context is used.New in version 2.6.
to_integral_value
( [ rounding [, context ] ] )
Round to the nearest integer without signaling
Inexact
orRounded
. If given, applies rounding; otherwise, uses the rounding method in either the supplied context or the current context.Changed in version 2.6: renamed from
to_integral
toto_integral_value
. The old name remains valid for compatibility.
9.4.2.1. Logical operands
The logical_and()
, logical_invert()
, logical_or()
, and logical_xor()
methods expect their arguments to be logical operands. A logical operand is a Decimal
instance whose exponent and sign are both zero, and whose digits are all either 0
or 1
.
9.4.3. Context objects
Contexts are environments for arithmetic operations. They govern precision, set rules for rounding, determine which signals are treated as exceptions, and limit the range for exponents.
Each thread has its own current context which is accessed or changed using the getcontext()
and setcontext()
functions:
Beginning with Python 2.5, you can also use the with
statement and the localcontext()
function to temporarily change the active context.
decimal.
localcontext
( [ c ] )
Return a context manager that will set the current context for the active thread to a copy of c on entry to the withstatement and restore the previous context when exiting the withstatement. If no context is specified, a copy of the current context is used.
New in version 2.5.
For example, the following code sets the current decimal precision to 42 places, performs a calculation, and then automatically restores the previous context:
from decimal import localcontext with localcontext() as ctx: ctx.prec = 42 # Perform a high precision calculation s = calculate_something() s = +s # Round the final result back to the default precision with localcontext(BasicContext): # temporarily use the BasicContext print Decimal(1) / Decimal(7) print Decimal(355) / Decimal(113)
New contexts can also be created using the Context
constructor described below. In addition, the module provides three premade contexts:
 class
decimal.
BasicContext

This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to
ROUND_HALF_UP
. All flags are cleared. All traps are enabled (treated as exceptions) exceptInexact
,Rounded
, andSubnormal
.Because many of the traps are enabled, this context is useful for debugging.
 class
decimal.
ExtendedContext

This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to
ROUND_HALF_EVEN
. All flags are cleared. No traps are enabled (so that exceptions are not raised during computations).Because the traps are disabled, this context is useful for applications that prefer to have result value of
NaN
orInfinity
instead of raising exceptions. This allows an application to complete a run in the presence of conditions that would otherwise halt the program.
 class
decimal.
DefaultContext

This context is used by the
Context
constructor as a prototype for new contexts. Changing a field (such a precision) has the effect of changing the default for new contexts created by theContext
constructor.This context is most useful in multithreaded environments. Changing one of the fields before threads are started has the effect of setting systemwide defaults. Changing the fields after threads have started is not recommended as it would require thread synchronization to prevent race conditions.
In single threaded environments, it is preferable to not use this context at all. Instead, simply create contexts explicitly as described below.
The default values are precision=28, rounding=ROUND_HALF_EVEN, and enabled traps for Overflow, InvalidOperation, and DivisionByZero.
In addition to the three supplied contexts, new contexts can be created with the Context
constructor.
 class
decimal.
Context
( prec=None, rounding=None, traps=None, flags=None, Emin=None, Emax=None, capitals=1 ) 
Creates a new context. If a field is not specified or is
None
, the default values are copied from theDefaultContext
. If the flags field is not specified or isNone
, all flags are cleared.The prec field is a positive integer that sets the precision for arithmetic operations in the context.
The rounding option is one of:
ROUND_CEILING
(towardsInfinity
),ROUND_DOWN
(towards zero),ROUND_FLOOR
(towardsInfinity
),ROUND_HALF_DOWN
(to nearest with ties going towards zero),ROUND_HALF_EVEN
(to nearest with ties going to nearest even integer),ROUND_HALF_UP
(to nearest with ties going away from zero), orROUND_UP
(away from zero).ROUND_05UP
(away from zero if last digit after rounding towards zero would have been 0 or 5; otherwise towards zero)
The traps and flags fields list any signals to be set. Generally, new contexts should only set traps and leave the flags clear.
The Emin and Emax fields are integers specifying the outer limits allowable for exponents.
The capitals field is either
0
or1
(the default). If set to1
, exponents are printed with a capitalE
; otherwise, a lowercasee
is used:Decimal('6.02e+23')
.Changed in version 2.6: The
ROUND_05UP
rounding mode was added.The
Context
class defines several general purpose methods as well as a large number of methods for doing arithmetic directly in a given context. In addition, for each of theDecimal
methods described above (with the exception of theadjusted()
andas_tuple()
methods) there is a correspondingContext
method. For example, for aContext
instanceC
andDecimal
instancex
,C.exp(x)
is equivalent tox.exp(context=C)
. EachContext
method accepts a Python integer (an instance ofint
orlong
) anywhere that a Decimal instance is accepted.create_decimal
( num )
Creates a new Decimal instance from num but using self as context. Unlike the
Decimal
constructor, the context precision, rounding method, flags, and traps are applied to the conversion.This is useful because constants are often given to a greater precision than is needed by the application. Another benefit is that rounding immediately eliminates unintended effects from digits beyond the current precision. In the following example, using unrounded inputs means that adding zero to a sum can change the result:
>>> getcontext().prec = 3 >>> Decimal('3.4445') + Decimal('1.0023') Decimal('4.45') >>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023') Decimal('4.44')
This method implements the tonumber operation of the IBM specification. If the argument is a string, no leading or trailing whitespace is permitted.
create_decimal_from_float
( f )
Creates a new Decimal instance from a float f but rounding using self as the context. Unlike the
Decimal.from_float()
class method, the context precision, rounding method, flags, and traps are applied to the conversion.>>> context = Context(prec=5, rounding=ROUND_DOWN) >>> context.create_decimal_from_float(math.pi) Decimal('3.1415') >>> context = Context(prec=5, traps=[Inexact]) >>> context.create_decimal_from_float(math.pi) Traceback (most recent call last): ... Inexact: None
New in version 2.7.
Etiny
( )
Returns a value equal to
Emin  prec + 1
which is the minimum exponent value for subnormal results. When underflow occurs, the exponent is set toEtiny
.
The usual approach to working with decimals is to create
Decimal
instances and then apply arithmetic operations which take place within the current context for the active thread. An alternative approach is to use context methods for calculating within a specific context. The methods are similar to those for theDecimal
class and are only briefly recounted here.compare_total_mag
( x, y )
Compares two operands using their abstract representation, ignoring sign.
plus
( x )
Plus corresponds to the unary prefix plus operator in Python. This operation applies the context precision and rounding, so it is not an identity operation.
power
( x, y [, modulo ] )
Return
x
to the power ofy
, reduced modulomodulo
if given.With two arguments, compute
x**y
. Ifx
is negative theny
must be integral. The result will be inexact unlessy
is integral and the result is finite and can be expressed exactly in ‘precision’ digits. The result should always be correctly rounded, using the rounding mode of the current thread’s context.With three arguments, compute
(x**y) % modulo
. For the three argument form, the following restrictions on the arguments hold:all three arguments must be integral
y
must be nonnegativeat least one of
x
ory
must be nonzeromodulo
must be nonzero and have at most ‘precision’ digits
The value resulting from
Context.power(x, y, modulo)
is equal to the value that would be obtained by computing(x**y) % modulo
with unbounded precision, but is computed more efficiently. The exponent of the result is zero, regardless of the exponents ofx
,y
andmodulo
. The result is always exact.Changed in version 2.6:
y
may now be nonintegral inx**y
. Stricter requirements for the threeargument version.
remainder
( x, y )
Returns the remainder from integer division.
The sign of the result, if nonzero, is the same as that of the original dividend.
remainder_near
( x, y )
Returns
x  y * n
, where n is the integer nearest the exact value ofx / y
(if the result is 0 then its sign will be the sign of x).
9.4.4. Signals
Signals represent conditions that arise during computation. Each corresponds to one context flag and one context trap enabler.
The context flag is set whenever the condition is encountered. After the computation, flags may be checked for informational purposes (for instance, to determine whether a computation was exact). After checking the flags, be sure to clear all flags before starting the next computation.
If the context’s trap enabler is set for the signal, then the condition causes a Python exception to be raised. For example, if the DivisionByZero
trap is set, then a DivisionByZero
exception is raised upon encountering the condition.
 class
decimal.
Clamped

Altered an exponent to fit representation constraints.
Typically, clamping occurs when an exponent falls outside the context’s
Emin
andEmax
limits. If possible, the exponent is reduced to fit by adding zeros to the coefficient.
 class
decimal.
DecimalException

Base class for other signals and a subclass of
ArithmeticError
.
 class
decimal.
DivisionByZero

Signals the division of a noninfinite number by zero.
Can occur with division, modulo division, or when raising a number to a negative power. If this signal is not trapped, returns
Infinity
orInfinity
with the sign determined by the inputs to the calculation.
 class
decimal.
Inexact

Indicates that rounding occurred and the result is not exact.
Signals when nonzero digits were discarded during rounding. The rounded result is returned. The signal flag or trap is used to detect when results are inexact.
 class
decimal.
InvalidOperation

An invalid operation was performed.
Indicates that an operation was requested that does not make sense. If not trapped, returns
NaN
. Possible causes include:Infinity  Infinity 0 * Infinity Infinity / Infinity x % 0 Infinity % x x._rescale( noninteger ) sqrt(x) and x > 0 0 ** 0 x ** (noninteger) x ** Infinity
 class
decimal.
Overflow

Numerical overflow.
Indicates the exponent is larger than
Emax
after rounding has occurred. If not trapped, the result depends on the rounding mode, either pulling inward to the largest representable finite number or rounding outward toInfinity
. In either case,Inexact
andRounded
are also signaled.
 class
decimal.
Rounded

Rounding occurred though possibly no information was lost.
Signaled whenever rounding discards digits; even if those digits are zero (such as rounding
5.00
to5.0
). If not trapped, returns the result unchanged. This signal is used to detect loss of significant digits.
 class
decimal.
Subnormal

Exponent was lower than
Emin
prior to rounding.Occurs when an operation result is subnormal (the exponent is too small). If not trapped, returns the result unchanged.
 class
decimal.
Underflow

Numerical underflow with result rounded to zero.
Occurs when a subnormal result is pushed to zero by rounding.
Inexact
andSubnormal
are also signaled.
The following table summarizes the hierarchy of signals:
exceptions.ArithmeticError(exceptions.StandardError)
DecimalException
Clamped
DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
Inexact
Overflow(Inexact, Rounded)
Underflow(Inexact, Rounded, Subnormal)
InvalidOperation
Rounded
Subnormal
9.4.5. Floating Point Notes
9.4.5.1. Mitigating roundoff error with increased precision
The use of decimal floating point eliminates decimal representation error (making it possible to represent 0.1
exactly); however, some operations can still incur roundoff error when nonzero digits exceed the fixed precision.
The effects of roundoff error can be amplified by the addition or subtraction of nearly offsetting quantities resulting in loss of significance. Knuth provides two instructive examples where rounded floating point arithmetic with insufficient precision causes the breakdown of the associative and distributive properties of addition:
# Examples from Seminumerical Algorithms, Section 4.2.2.
>>> from decimal import Decimal, getcontext
>>> getcontext().prec = 8
>>> u, v, w = Decimal(11111113), Decimal(11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal('9.5111111')
>>> u + (v + w)
Decimal('10')
>>> u, v, w = Decimal(20000), Decimal(6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal('0.01')
>>> u * (v+w)
Decimal('0.0060000')
The decimal
module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance:
>>> getcontext().prec = 20
>>> u, v, w = Decimal(11111113), Decimal(11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal('9.51111111')
>>> u + (v + w)
Decimal('9.51111111')
>>>
>>> u, v, w = Decimal(20000), Decimal(6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal('0.0060000')
>>> u * (v+w)
Decimal('0.0060000')
9.4.5.2. Special values
The number system for the decimal
module provides special values including NaN
, sNaN
, Infinity
, Infinity
, and two zeros, +0
and 0
.
Infinities can be constructed directly with: Decimal('Infinity')
. Also, they can arise from dividing by zero when the DivisionByZero
signal is not trapped. Likewise, when the Overflow
signal is not trapped, infinity can result from rounding beyond the limits of the largest representable number.
The infinities are signed (affine) and can be used in arithmetic operations where they get treated as very large, indeterminate numbers. For instance, adding a constant to infinity gives another infinite result.
Some operations are indeterminate and return NaN
, or if the InvalidOpera