Python has always supported floating-point (FP) numbers, based on the
underlying C double type, as a data type. However, while most
programming languages provide a floating-point type, many people (even
programmers) are unaware that floating-point numbers don't represent
certain decimal fractions accurately. The new Decimal type
can represent these fractions accurately, up to a user-specified
precision limit.
The limitations arise from the representation used for floating-point numbers.
FP numbers are made up of three components:
- The sign, which is positive or negative.
- The mantissa, which is a single-digit binary number
followed by a fractional part. For example,
1.01 in base-2 notation
is 1 + 0/2 + 1/4, or 1.25 in decimal notation.
- The exponent, which tells where the decimal point is located in the number represented.
For example, the number 1.25 has positive sign, a mantissa value of
1.01 (in binary), and an exponent of 0 (the decimal point doesn't need
to be shifted). The number 5 has the same sign and mantissa, but the
exponent is 2 because the mantissa is multiplied by 4 (2 to the power
of the exponent 2); 1.25 * 4 equals 5.
Modern systems usually provide floating-point support that conforms to
a standard called IEEE 754. C's double type is usually
implemented as a 64-bit IEEE 754 number, which uses 52 bits of space
for the mantissa. This means that numbers can only be specified to 52
bits of precision. If you're trying to represent numbers whose
expansion repeats endlessly, the expansion is cut off after 52 bits.
Unfortunately, most software needs to produce output in base 10, and
common fractions in base 10 are often repeating decimals in binary.
For example, 1.1 decimal is binary 1.0001100110011 ...; .1 =
1/16 + 1/32 + 1/256 plus an infinite number of additional terms. IEEE
754 has to chop off that infinitely repeated decimal after 52 digits,
so the representation is slightly inaccurate.
Sometimes you can see this inaccuracy when the number is printed:
>>> 1.1
1.1000000000000001
The inaccuracy isn't always visible when you print the number because
the FP-to-decimal-string conversion is provided by the C library, and
most C libraries try to produce sensible output. Even if it's not
displayed, however, the inaccuracy is still there and subsequent
operations can magnify the error.
For many applications this doesn't matter. If I'm plotting points and
displaying them on my monitor, the difference between 1.1 and
1.1000000000000001 is too small to be visible. Reports often limit
output to a certain number of decimal places, and if you round the
number to two or three or even eight decimal places, the error is
never apparent. However, for applications where it does matter,
it's a lot of work to implement your own custom arithmetic routines.
Hence, the Decimal type was created.
A new module, decimal, was added to Python's standard
library. It contains two classes, Decimal and
Context. Decimal instances represent numbers, and
Context instances are used to wrap up various settings such as
the precision and default rounding mode.
Decimal instances are immutable, like regular Python integers
and FP numbers; once it's been created, you can't change the value an
instance represents. Decimal instances can be created from
integers or strings:
>>> import decimal
>>> decimal.Decimal(1972)
Decimal("1972")
>>> decimal.Decimal("1.1")
Decimal("1.1")
You can also provide tuples containing the sign, the mantissa represented
as a tuple of decimal digits, and the exponent:
>>> decimal.Decimal((1, (1, 4, 7, 5), -2))
Decimal("-14.75")
Cautionary note: the sign bit is a Boolean value, so 0 is positive and
1 is negative.
Converting from floating-point numbers poses a bit of a problem:
should the FP number representing 1.1 turn into the decimal number for
exactly 1.1, or for 1.1 plus whatever inaccuracies are introduced?
The decision was to dodge the issue and leave such a conversion out of
the API. Instead, you should convert the floating-point number into a
string using the desired precision and pass the string to the
Decimal constructor:
>>> f = 1.1
>>> decimal.Decimal(str(f))
Decimal("1.1")
>>> decimal.Decimal('%.12f' % f)
Decimal("1.100000000000")
Once you have Decimal instances, you can perform the usual
mathematical operations on them. One limitation: exponentiation
requires an integer exponent:
>>> a = decimal.Decimal('35.72')
>>> b = decimal.Decimal('1.73')
>>> a+b
Decimal("37.45")
>>> a-b
Decimal("33.99")
>>> a*b
Decimal("61.7956")
>>> a/b
Decimal("20.64739884393063583815028902")
>>> a ** 2
Decimal("1275.9184")
>>> a**b
Traceback (most recent call last):
...
decimal.InvalidOperation: x ** (non-integer)
You can combine Decimal instances with integers, but not with
floating-point numbers:
>>> a + 4
Decimal("39.72")
>>> a + 4.5
Traceback (most recent call last):
...
TypeError: You can interact Decimal only with int, long or Decimal data types.
>>>
Decimal numbers can be used with the math and
cmath modules, but note that they'll be immediately converted to
floating-point numbers before the operation is performed, resulting in
a possible loss of precision and accuracy. You'll also get back a
regular floating-point number and not a Decimal.
>>> import math, cmath
>>> d = decimal.Decimal('123456789012.345')
>>> math.sqrt(d)
351364.18288201344
>>> cmath.sqrt(-d)
351364.18288201344j
Decimal instances have a sqrt() method that
returns a Decimal, but if you need other things such as
trigonometric functions you'll have to implement them.
>>> d.sqrt()
Decimal("351364.1828820134592177245001")
Instances of the Context class encapsulate several settings for
decimal operations:
- prec is the precision, the number of decimal places.
- rounding specifies the rounding mode. The decimal
module has constants for the various possibilities:
ROUND_DOWN, ROUND_CEILING,
ROUND_HALF_EVEN, and various others.
- traps is a dictionary specifying what happens on
encountering certain error conditions: either an exception is raised or
a value is returned. Some examples of error conditions are
division by zero, loss of precision, and overflow.
There's a thread-local default context available by calling
getcontext(); you can change the properties of this context
to alter the default precision, rounding, or trap handling. The
following example shows the effect of changing the precision of the default
context:
>>> decimal.getcontext().prec
28
>>> decimal.Decimal(1) / decimal.Decimal(7)
Decimal("0.1428571428571428571428571429")
>>> decimal.getcontext().prec = 9
>>> decimal.Decimal(1) / decimal.Decimal(7)
Decimal("0.142857143")
The default action for error conditions is selectable; the module can
either return a special value such as infinity or not-a-number, or
exceptions can be raised:
>>> decimal.Decimal(1) / decimal.Decimal(0)
Traceback (most recent call last):
...
decimal.DivisionByZero: x / 0
>>> decimal.getcontext().traps[decimal.DivisionByZero] = False
>>> decimal.Decimal(1) / decimal.Decimal(0)
Decimal("Infinity")
>>>
The Context instance also has various methods for formatting
numbers such as to_eng_string() and to_sci_string().
For more information, see the documentation for the decimal
module, which includes a quick-start tutorial and a reference.
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