# fractions – Rational Numbers¶

Purpose: Implements a class for working with rational numbers. 2.6 and later

The Fraction class implements numerical operations for rational numbers based on the API defined by Rational in numbers.

## Creating Fraction Instances¶

As with decimal, new values can be created in several ways. One easy way is to create them from separate numerator and denominator values:

```import fractions

for n, d in [ (1, 2), (2, 4), (3, 6) ]:
f = fractions.Fraction(n, d)
print '%s/%s = %s' % (n, d, f)
```

The lowest common denominator is maintained as new values are computed.

```\$ python fractions_create_integers.py

1/2 = 1/2
2/4 = 1/2
3/6 = 1/2
```

Another way to create a Fraction is using a string representation of <numerator> / <denominator>:

```import fractions

for s in [ '1/2', '2/4', '3/6' ]:
f = fractions.Fraction(s)
print '%s = %s' % (s, f)
```
```\$ python fractions_create_strings.py

1/2 = 1/2
2/4 = 1/2
3/6 = 1/2
```

Strings can also use the more usual decimal or floating point notation of [<digits>].[<digits>].

```import fractions

for s in [ '0.5', '1.5', '2.0' ]:
f = fractions.Fraction(s)
print '%s = %s' % (s, f)
```
```\$ python fractions_create_strings_floats.py

0.5 = 1/2
1.5 = 3/2
2.0 = 2
```

There are class methods for creating Fraction instances directly from other representations of rational values such as float or decimal.

```import fractions

for v in [ 0.1, 0.5, 1.5, 2.0 ]:
print '%s = %s' % (v, fractions.Fraction.from_float(v))

```

Notice that for floating point values that cannot be expressed exactly the rational representation may yield unexpected results.

```\$ python fractions_from_float.py

0.1 = 3602879701896397/36028797018963968
0.5 = 1/2
1.5 = 3/2
2.0 = 2
```

Using decimal representations of the values gives the expected results.

```import decimal
import fractions

for v in [ decimal.Decimal('0.1'),
decimal.Decimal('0.5'),
decimal.Decimal('1.5'),
decimal.Decimal('2.0'),
]:
print '%s = %s' % (v, fractions.Fraction.from_decimal(v))
```
```\$ python fractions_from_decimal.py

0.1 = 1/10
0.5 = 1/2
1.5 = 3/2
2.0 = 2
```

## Arithmetic¶

Once the fractions are instantiated, they can be used in mathematical expressions as you would expect.

```import fractions

f1 = fractions.Fraction(1, 2)
f2 = fractions.Fraction(3, 4)

print '%s + %s = %s' % (f1, f2, f1 + f2)
print '%s - %s = %s' % (f1, f2, f1 - f2)
print '%s * %s = %s' % (f1, f2, f1 * f2)
print '%s / %s = %s' % (f1, f2, f1 / f2)
```
```\$ python fractions_arithmetic.py

1/2 + 3/4 = 5/4
1/2 - 3/4 = -1/4
1/2 * 3/4 = 3/8
1/2 / 3/4 = 2/3
```

## Approximating Values¶

A useful feature of Fraction is the ability to convert a floating point number to an approximate rational value by limiting the size of the denominator.

```import fractions
import math

print 'PI       =', math.pi

f_pi = fractions.Fraction(str(math.pi))
print 'No limit =', f_pi

for i in range(1, 100, 5):
limited = f_pi.limit_denominator(i)
print '{0:8} = {1}'.format(i, limited)
```
```\$ python fractions_limit_denominator.py

PI       = 3.14159265359
No limit = 314159265359/100000000000
1 = 3
6 = 19/6
11 = 22/7
16 = 22/7
21 = 22/7
26 = 22/7
31 = 22/7
36 = 22/7
41 = 22/7
46 = 22/7
51 = 22/7
56 = 22/7
61 = 179/57
66 = 201/64
71 = 223/71
76 = 223/71
81 = 245/78
86 = 267/85
91 = 267/85
96 = 289/92
```