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By Root 1321 0

Fraction and Decimal provide ways to get exact results, albeit at the cost of some speed. For instance, in the following example (repeated from the prior section), floating-point numbers do not accurately give the zero answer expected, but both of the other types do:

>>> 0.1 + 0.1 + 0.1 - 0.3 # This should be zero (close, but not exact)

5.5511151231257827e-17

>>> from fractions import Fraction

>>> Fraction(1, 10) + Fraction(1, 10) + Fraction(1, 10) - Fraction(3, 10)

Fraction(0, 1)

>>> from decimal import Decimal

>>> Decimal('0.1') + Decimal('0.1') + Decimal('0.1') - Decimal('0.3')

Decimal('0.0')

Moreover, fractions and decimals both allow more intuitive and accurate results than floating points sometimes can, in different ways (by using rational representation and by limiting precision):

>>> 1 / 3 # Use 3.0 in Python 2.6 for true "/"

0.33333333333333331

>>> Fraction(1, 3) # Numeric accuracy

Fraction(1, 3)

>>> import decimal

>>> decimal.getcontext().prec = 2

>>> decimal.Decimal(1) / decimal.Decimal(3)

Decimal('0.33')

In fact, fractions both retain accuracy and automatically simplify results. Continuing the preceding interaction:

>>> (1 / 3) + (6 / 12) # Use ".0" in Python 2.6 for true "/"

0.83333333333333326

>>> Fraction(6, 12) # Automatically simplified

Fraction(1, 2)

>>> Fraction(1, 3) + Fraction(6, 12)

Fraction(5, 6)

>>> decimal.Decimal(str(1/3)) + decimal.Decimal(str(6/12))

Decimal('0.83')

>>> 1000.0 / 1234567890

8.1000000737100011e-07

>>> Fraction(1000, 1234567890)

Fraction(100, 123456789)

Conversions and mixed types

To support fraction conversions, floating-point objects now have a method that yields their numerator and denominator ratio, fractions have a from_float method, and float accepts a Fraction as an argument. Trace through the following interaction to see how this pans out (the * in the second test is special syntax that expands a tuple into individual arguments; more on this when we study function argument passing in Chapter 18):

>>> (2.5).as_integer_ratio() # float object method

(5, 2)

>>> f = 2.5

>>> z = Fraction(*f.as_integer_ratio()) # Convert float -> fraction: two args

>>> z # Same as Fraction(5, 2)

Fraction(5, 2)

>>> x # x from prior interaction

Fraction(1, 3)

>>> x + z

Fraction(17, 6) # 5/2 + 1/3 = 15/6 + 2/6

>>> float(x) # Convert fraction -> float

0.33333333333333331

>>> float(z)

2.5

>>> float(x + z)

2.8333333333333335

>>> 17 / 6

2.8333333333333335

>>> Fraction.from_float(1.75) # Convert float -> fraction: other way

Fraction(7, 4)

>>> Fraction(*(1.75).as_integer_ratio())

Fraction(7, 4)

Finally, some type mixing is allowed in expressions, though Fraction must sometimes be manually propagated to retain accuracy. Study the following interaction to see how this works:

>>> x

Fraction(1, 3)

>>> x + 2 # Fraction + int -> Fraction

Fraction(7, 3)

>>> x + 2.0 # Fraction + float -> float

2.3333333333333335

>>> x + (1./3) # Fraction + float -> float

0.66666666666666663

>>> x + (4./3)

1.6666666666666665

>>> x + Fraction(4, 3) # Fraction + Fraction -> Fraction

Fraction(5, 3)

Caveat: although you can convert from floating-point to fraction, in some cases there is an unavoidable precision loss when you do so, because the number is inaccurate in its original floating-point form. When needed, you can simplify such results by limiting the maximum denominator value:

>>> 4.0 / 3

1.3333333333333333

>>> (4.0 / 3).as_integer_ratio() # Precision loss from float

(6004799503160661, 4503599627370496)

>>> x

Fraction(1, 3)

>>> a = x + Fraction(*(4.0 / 3).as_integer_ratio())

>>> a

Fraction(22517998136852479, 13510798882111488)

>>> 22517998136852479 / 13510798882111488. # 5 / 3 (or close to it!)

1.6666666666666667

>>> a.limit_denominator(10) # Simplify to closest fraction

Fraction(5, 3)

For more details on the Fraction type, experiment further on your own and consult the Python 2.6 and 3.0 library manuals and other documentation.

Sets

Python 2.4 also introduced a new collection type, the