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 do not have an exact representation in binary floating point. End
users typically would not expect 1.1 to display as
1.1000000000000001 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
is 5.5511151231257827e017. 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 is 2.50. 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 gives 1.56 while 1.30 *
1.20 gives 1.5600.
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:
>>> 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.
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, or tuples. To
create a Decimal from a float, first convert it to a string. This
serves as an explicit reminder of the details of the conversion (including
representation error). 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((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)
1.3400000000000001
>>> 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.
Decimal objects

class decimal.Decimal([value[, context]])
Construct a new Decimal object based from value.
value can be an integer, string, tuple, or another Decimal
object. If no value is given, returns Decimal('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 tuple, it should have three components, a sign
(0 for positive or 1 for negative), a tuple of
digits, and an integer exponent. For example, Decimal((0, (1, 4, 1, 4), 3))
returns Decimal('1.414').
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 of
NaN.
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.
Decimal floating point objects share many properties with the other builtin
numeric types such as float and int. 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 as float or
long).
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 that
compare() 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 on Decimal instances. Two
Decimal 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, and Decimal('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 to
x.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 the
ROUND_HALF_EVEN rounding mode.
>>> Decimal(1).exp()
Decimal('2.718281828459045235360287471')
>>> Decimal(321).exp()
Decimal('2.561702493119680037517373933E+139')
New in version 2.6.

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 and False
otherwise. Currently, a Decimal instance is always canonical, so
this operation always returns True.
New in version 2.6.

is_finite()
Return True if the argument is a finite number, and
False 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 and False otherwise.
New in version 2.6.

is_nan()
Return True if the argument is a (quiet or signaling) NaN and
False otherwise.
New in version 2.6.

is_normal()
Return True if the argument is a normal finite number. Return
False if the argument is zero, subnormal, infinite or a NaN.
New in version 2.6.

is_qnan()
Return True if the argument is a quiet NaN, and
False otherwise.
New in version 2.6.

is_signed()
Return True if the argument has a negative sign and
False otherwise. Note that zeros and NaNs can both carry signs.
New in version 2.6.

is_snan()
Return True if the argument is a signaling NaN and False
otherwise.
New in version 2.6.

is_subnormal()
Return True if the argument is subnormal, and False
otherwise.
New in version 2.6.

is_zero()
Return True if the argument is a (positive or negative) zero and
False 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 then
Decimal('Infinity') is returned and the DivisionByZero flag
is raised. If the operand is an infinity then Decimal('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
digitwise and of the two operands.
New in version 2.6.

logical_invert(other[, context])
logical_invert() is a logical operation. The argument must
be a logical operand (see Logical operands). 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
digitwise or 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 that NaN 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 that NaN 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') to
Decimal('0e0'). Used for producing canonical values for members
of an equivalence class. For example, Decimal('32.100') and
Decimal('0.321000e+2') both normalize to the equivalent value
Decimal('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 given
context 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 than
Etiny.

radix()
Return Decimal(10), the radix (base) in which the Decimal
class does all its arithmetic. Included for compatibility with the
specification.
New in version 2.6.

remainder_near(other[, context])
Compute the modulo as either a positive or negative value depending on
which is closest to zero. For instance, Decimal(10).remainder_near(6)
returns Decimal('2') which is closer to zero than Decimal('4').
If both are equally close, the one chosen will have the same sign as
self.

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.

sqrt([context])
 Return the square root of the argument to full precision.

to_eng_string([context])
Convert to an engineeringtype string.
Engineering notation has an exponent which is a multiple of 3, so there
are up to 3 digits left of the decimal place. For example, converts
Decimal('123E+1') to Decimal('1.23E+3')

to_integral([rounding[, context]])
 Identical to the to_integral_value() method. The to_integral
name has been kept for compatibility with older versions.

to_integral_exact([rounding[, context]])
Round to the nearest integer, signaling Inexact or
Rounded as appropriate if rounding occurs. The rounding mode is
determined by the rounding parameter if given, else by the given
context. 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 or
Rounded. 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 to to_integral_value. The old name
remains valid for compatibility.
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.
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:

decimal.getcontext()
 Return the current context for the active thread.

decimal.setcontext(c)
 Set the current context for the active thread to c.
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
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) except Inexact, Rounded, and
Subnormal.
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 or Infinity 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 creating by the Context 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 the DefaultContext. If the flags
field is not specified or is None, 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 (towards Infinity),
 ROUND_DOWN (towards zero),
 ROUND_FLOOR (towards Infinity),
 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), or
 ROUND_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 or 1 (the default). If set to
1, exponents are printed with a capital E; otherwise, a
lowercase e 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 the Decimal methods described above (with
the exception of the adjusted() and as_tuple() methods) there is
a corresponding Context method. For example, C.exp(x) is
equivalent to x.exp(context=C).

clear_flags()
 Resets all of the flags to 0.

copy()
 Return a duplicate of the context.

copy_decimal(num)
 Return a copy of the Decimal instance num.

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.

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
to Etiny.

Etop()
 Returns a value equal to Emax  prec + 1.
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 the Decimal class and are only briefly
recounted here.

abs(x)
 Returns the absolute value of x.

add(x, y)
 Return the sum of x and y.

canonical(x)
 Returns the same Decimal object x.

compare(x, y)
 Compares x and y numerically.

compare_signal(x, y)
 Compares the values of the two operands numerically.

compare_total(x, y)
 Compares two operands using their abstract representation.

compare_total_mag(x, y)
 Compares two operands using their abstract representation, ignoring sign.

copy_abs(x)
 Returns a copy of x with the sign set to 0.

copy_negate(x)
 Returns a copy of x with the sign inverted.

copy_sign(x, y)
 Copies the sign from y to x.

divide(x, y)
 Return x divided by y.

divide_int(x, y)
 Return x divided by y, truncated to an integer.

divmod(x, y)
 Divides two numbers and returns the integer part of the result.

exp(x)
 Returns e ** x.

fma(x, y, z)
 Returns x multiplied by y, plus z.

is_canonical(x)
 Returns True if x is canonical; otherwise returns False.

is_finite(x)
 Returns True if x is finite; otherwise returns False.

is_infinite(x)
 Returns True if x is infinite; otherwise returns False.

is_nan(x)
 Returns True if x is a qNaN or sNaN; otherwise returns False.

is_normal(x)
 Returns True if x is a normal number; otherwise returns False.

is_qnan(x)
 Returns True if x is a quiet NaN; otherwise returns False.

is_signed(x)
 Returns True if x is negative; otherwise returns False.

is_snan(x)
 Returns True if x is a signaling NaN; otherwise returns False.

is_subnormal(x)
 Returns True if x is subnormal; otherwise returns False.

is_zero(x)
 Returns True if x is a zero; otherwise returns False.

ln(x)
 Returns the natural (base e) logarithm of x.

log10(x)
 Returns the base 10 logarithm of x.

logb(x)
 Returns the exponent of the magnitude of the operand’s MSD.

logical_and(x, y)
 Applies the logical operation and between each operand’s digits.

logical_invert(x)
 Invert all the digits in x.

logical_or(x, y)
 Applies the logical operation or between each operand’s digits.

logical_xor(x, y)
 Applies the logical operation xor between each operand’s digits.

max(x, y)
 Compares two values numerically and returns the maximum.

max_mag(x, y)
 Compares the values numerically with their sign ignored.

min(x, y)
 Compares two values numerically and returns the minimum.

min_mag(x, y)
 Compares the values numerically with their sign ignored.

minus(x)
 Minus corresponds to the unary prefix minus operator in Python.

multiply(x, y)
 Return the product of x and y.

next_minus(x)
 Returns the largest representable number smaller than x.

next_plus(x)
 Returns the smallest representable number larger than x.

next_toward(x, y)
 Returns the number closest to x, in direction towards y.

normalize(x)
 Reduces x to its simplest form.

number_class(x)
 Returns an indication of the class of x.

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 of y, reduced modulo modulo if given.
With two arguments, compute x**y. If x is negative then y
must be integral. The result will be inexact unless y 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 nonnegative
 at least one of x or y must be nonzero
 modulo must be nonzero and have at most ‘precision’ digits
The result of Context.power(x, y, modulo) is identical to the result
that would be obtained by computing (x**y) % modulo with unbounded
precision, but is computed more efficiently. It is always exact.
Changed in version 2.6: y may now be nonintegral in x**y.
Stricter requirements for the threeargument version.

quantize(x, y)
 Returns a value equal to x (rounded), having the exponent of y.

radix()
 Just returns 10, as this is Decimal, :)

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
of x / y (if the result is 0 then its sign will be the sign of x).

rotate(x, y)
 Returns a rotated copy of x, y times.

same_quantum(x, y)
 Returns True if the two operands have the same exponent.

scaleb(x, y)
 Returns the first operand after adding the second value its exp.

shift(x, y)
 Returns a shifted copy of x, y times.

sqrt(x)
 Square root of a nonnegative number to context precision.

subtract(x, y)
 Return the difference between x and y.

to_eng_string(x)
 Converts a number to a string, using scientific notation.

to_integral_exact(x)
 Rounds to an integer.

to_sci_string(x)
 Converts a number to a string using scientific notation.
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 and Emax 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 or
Infinity 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
to Infinity. In either case, Inexact and Rounded
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 to 5.0). 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
and Subnormal 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
Floating Point Notes
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')
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
InvalidOperation signal is trapped, raise an exception. For example,
0/0 returns NaN which means “not a number”. This variety of
NaN is quiet and, once created, will flow through other computations
always resulting in another NaN. This behavior can be useful for a
series of computations that occasionally have missing inputs — it allows the
calculation to proceed while flagging specific results as invalid.
A variant is sNaN which signals rather than remaining quiet after every
operation. This is a useful return value when an invalid result needs to
interrupt a calculation for special handling.
The behavior of Python’s comparison operators can be a little surprising where a
NaN is involved. A test for equality where one of the operands is a
quiet or signaling NaN always returns False (even when doing
Decimal('NaN')==Decimal('NaN')), while a test for inequality always returns
True. An attempt to compare two Decimals using any of the <,
<=, > or >= operators will raise the InvalidOperation signal
if either operand is a NaN, and return False if this signal is
not trapped. Note that the General Decimal Arithmetic specification does not
specify the behavior of direct comparisons; these rules for comparisons
involving a NaN were taken from the IEEE 854 standard (see Table 3 in
section 5.7). To ensure strict standardscompliance, use the compare()
and comparesignal() methods instead.
The signed zeros can result from calculations that underflow. They keep the sign
that would have resulted if the calculation had been carried out to greater
precision. Since their magnitude is zero, both positive and negative zeros are
treated as equal and their sign is informational.
In addition to the two signed zeros which are distinct yet equal, there are
various representations of zero with differing precisions yet equivalent in
value. This takes a bit of getting used to. For an eye accustomed to
normalized floating point representations, it is not immediately obvious that
the following calculation returns a value equal to zero:
>>> 1 / Decimal('Infinity')
Decimal('0E1000000026')
Working with threads
The getcontext() function accesses a different Context object for
each thread. Having separate thread contexts means that threads may make
changes (such as getcontext.prec=10) without interfering with other threads.
Likewise, the setcontext() function automatically assigns its target to
the current thread.
If setcontext() has not been called before getcontext(), then
getcontext() will automatically create a new context for use in the
current thread.
The new context is copied from a prototype context called DefaultContext. To
control the defaults so that each thread will use the same values throughout the
application, directly modify the DefaultContext object. This should be done
before any threads are started so that there won’t be a race condition between
threads calling getcontext(). For example:
# Set applicationwide defaults for all threads about to be launched
DefaultContext.prec = 12
DefaultContext.rounding = ROUND_DOWN
DefaultContext.traps = ExtendedContext.traps.copy()
DefaultContext.traps[InvalidOperation] = 1
setcontext(DefaultContext)
# Afterwards, the threads can be started
t1.start()
t2.start()
t3.start()
. . .
Recipes
Here are a few recipes that serve as utility functions and that demonstrate ways
to work with the Decimal class:
def moneyfmt(value, places=2, curr='', sep=',', dp='.',
pos='', neg='', trailneg=''):
"""Convert Decimal to a money formatted string.
places: required number of places after the decimal point
curr: optional currency symbol before the sign (may be blank)
sep: optional grouping separator (comma, period, space, or blank)
dp: decimal point indicator (comma or period)
only specify as blank when places is zero
pos: optional sign for positive numbers: '+', space or blank
neg: optional sign for negative numbers: '', '(', space or blank
trailneg:optional trailing minus indicator: '', ')', space or blank
>>> d = Decimal('1234567.8901')
>>> moneyfmt(d, curr='$')
'$1,234,567.89'
>>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='')
'1.234.568'
>>> moneyfmt(d, curr='$', neg='(', trailneg=')')
'($1,234,567.89)'
>>> moneyfmt(Decimal(123456789), sep=' ')
'123 456 789.00'
>>> moneyfmt(Decimal('0.02'), neg='<', trailneg='>')
'<0.02>'
"""
q = Decimal(10) ** places # 2 places > '0.01'
sign, digits, exp = value.quantize(q).as_tuple()
result = []
digits = map(str, digits)
build, next = result.append, digits.pop
if sign:
build(trailneg)
for i in range(places):
build(next() if digits else '0')
build(dp)
if not digits:
build('0')
i = 0
while digits:
build(next())
i += 1
if i == 3 and digits:
i = 0
build(sep)
build(curr)
build(neg if sign else pos)
return ''.join(reversed(result))
def pi():
"""Compute Pi to the current precision.
>>> print pi()
3.141592653589793238462643383
"""
getcontext().prec += 2 # extra digits for intermediate steps
three = Decimal(3) # substitute "three=3.0" for regular floats
lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
while s != lasts:
lasts = s
n, na = n+na, na+8
d, da = d+da, da+32
t = (t * n) / d
s += t
getcontext().prec = 2
return +s # unary plus applies the new precision
def exp(x):
"""Return e raised to the power of x. Result type matches input type.
>>> print exp(Decimal(1))
2.718281828459045235360287471
>>> print exp(Decimal(2))
7.389056098930650227230427461
>>> print exp(2.0)
7.38905609893
>>> print exp(2+0j)
(7.38905609893+0j)
"""
getcontext().prec += 2
i, lasts, s, fact, num = 0, 0, 1, 1, 1
while s != lasts:
lasts = s
i += 1
fact *= i
num *= x
s += num / fact
getcontext().prec = 2
return +s
def cos(x):
"""Return the cosine of x as measured in radians.
>>> print cos(Decimal('0.5'))
0.8775825618903727161162815826
>>> print cos(0.5)
0.87758256189
>>> print cos(0.5+0j)
(0.87758256189+0j)
"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i1)
num *= x * x
sign *= 1
s += num / fact * sign
getcontext().prec = 2
return +s
def sin(x):
"""Return the sine of x as measured in radians.
>>> print sin(Decimal('0.5'))
0.4794255386042030002732879352
>>> print sin(0.5)
0.479425538604
>>> print sin(0.5+0j)
(0.479425538604+0j)
"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i1)
num *= x * x
sign *= 1
s += num / fact * sign
getcontext().prec = 2
return +s
Decimal FAQ
Q. It is cumbersome to type decimal.Decimal('1234.5'). Is there a way to
minimize typing when using the interactive interpreter?
A. Some users abbreviate the constructor to just a single letter:
>>> D = decimal.Decimal
>>> D('1.23') + D('3.45')
Decimal('4.68')
Q. In a fixedpoint application with two decimal places, some inputs have many
places and need to be rounded. Others are not supposed to have excess digits
and need to be validated. What methods should be used?
A. The quantize() method rounds to a fixed number of decimal places. If
the Inexact trap is set, it is also useful for validation:
>>> TWOPLACES = Decimal(10) ** 2 # same as Decimal('0.01')
>>> # Round to two places
>>> Decimal('3.214').quantize(TWOPLACES)
Decimal('3.21')
>>> # Validate that a number does not exceed two places
>>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact]))
Decimal('3.21')
>>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact]))
Traceback (most recent call last):
...
Inexact
Q. Once I have valid two place inputs, how do I maintain that invariant
throughout an application?
A. Some operations like addition, subtraction, and multiplication by an integer
will automatically preserve fixed point. Others operations, like division and
noninteger multiplication, will change the number of decimal places and need to
be followedup with a quantize() step:
>>> a = Decimal('102.72') # Initial fixedpoint values
>>> b = Decimal('3.17')
>>> a + b # Addition preserves fixedpoint
Decimal('105.89')
>>> a  b
Decimal('99.55')
>>> a * 42 # So does integer multiplication
Decimal('4314.24')
>>> (a * b).quantize(TWOPLACES) # Must quantize noninteger multiplication
Decimal('325.62')
>>> (b / a).quantize(TWOPLACES) # And quantize division
Decimal('0.03')
In developing fixedpoint applications, it is convenient to define functions
to handle the quantize() step:
>>> def mul(x, y, fp=TWOPLACES):
... return (x * y).quantize(fp)
>>> def div(x, y, fp=TWOPLACES):
... return (x / y).quantize(fp)
>>> mul(a, b) # Automatically preserve fixedpoint
Decimal('325.62')
>>> div(b, a)
Decimal('0.03')
Q. There are many ways to express the same value. The numbers 200,
200.000, 2E2, and 02E+4 all have the same value at
various precisions. Is there a way to transform them to a single recognizable
canonical value?
A. The normalize() method maps all equivalent values to a single
representative:
>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
>>> [v.normalize() for v in values]
[Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]
Q. Some decimal values always print with exponential notation. Is there a way
to get a nonexponential representation?
A. For some values, exponential notation is the only way to express the number
of significant places in the coefficient. For example, expressing
5.0E+3 as 5000 keeps the value constant but cannot show the
original’s twoplace significance.
If an application does not care about tracking significance, it is easy to
remove the exponent and trailing zeroes, losing significance, but keeping the
value unchanged:
>>> def remove_exponent(d):
... return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()
>>> remove_exponent(Decimal('5E+3'))
Decimal('5000')
Q. Is there a way to convert a regular float to a Decimal?
A. Yes, all binary floating point numbers can be exactly expressed as a
Decimal. An exact conversion may take more precision than intuition would
suggest, so we trap Inexact to signal a need for more precision:
def float_to_decimal(f):
"Convert a floating point number to a Decimal with no loss of information"
n, d = f.as_integer_ratio()
with localcontext() as ctx:
ctx.traps[Inexact] = True
while True:
try:
return Decimal(n) / Decimal(d)
except Inexact:
ctx.prec += 1
>>> float_to_decimal(math.pi)
Decimal('3.141592653589793115997963468544185161590576171875')
Q. Why isn’t the float_to_decimal() routine included in the module?
A. There is some question about whether it is advisable to mix binary and
decimal floating point. Also, its use requires some care to avoid the
representation issues associated with binary floating point:
>>> float_to_decimal(1.1)
Decimal('1.100000000000000088817841970012523233890533447265625')
Q. Within a complex calculation, how can I make sure that I haven’t gotten a
spurious result because of insufficient precision or rounding anomalies.
A. The decimal module makes it easy to test results. A best practice is to
rerun calculations using greater precision and with various rounding modes.
Widely differing results indicate insufficient precision, rounding mode issues,
illconditioned inputs, or a numerically unstable algorithm.
Q. I noticed that context precision is applied to the results of operations but
not to the inputs. Is there anything to watch out for when mixing values of
different precisions?
A. Yes. The principle is that all values are considered to be exact and so is
the arithmetic on those values. Only the results are rounded. The advantage
for inputs is that “what you type is what you get”. A disadvantage is that the
results can look odd if you forget that the inputs haven’t been rounded:
>>> getcontext().prec = 3
>>> Decimal('3.104') + Decimal('2.104')
Decimal('5.21')
>>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104')
Decimal('5.20')
The solution is either to increase precision or to force rounding of inputs
using the unary plus operation:
>>> getcontext().prec = 3
>>> +Decimal('1.23456789') # unary plus triggers rounding
Decimal('1.23')
Alternatively, inputs can be rounded upon creation using the
Context.create_decimal() method:
>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
Decimal('1.2345')
