Python fractions Module — Core Concepts

What the fractions module provides

The fractions module gives you a Fraction class that represents rational numbers as exact numerator/denominator pairs. Arithmetic on fractions produces exact results — no floating-point rounding.

from fractions import Fraction

a = Fraction(1, 3)
b = Fraction(1, 6)
print(a + b)  # 1/2 (exact)
print(a * 3)  # 1 (exact)

Creating fractions

Method	Example	Result
Two integers	Fraction(3, 4)	3/4
Single integer	Fraction(5)	5
String	Fraction("3.14")	157/50
Float	Fraction(0.1)	3602879701896397/36028797018963968
Decimal	Fraction(Decimal("0.1"))	1/10

Notice the float vs string difference. Fraction(0.1) captures the actual float value (which isn’t exactly 0.1 in binary). Fraction("0.1") gives you exactly 1/10. Always use strings or integers for precise input.

Automatic simplification

Fractions are always stored in lowest terms:

Fraction(4, 8)   # → Fraction(1, 2)
Fraction(6, -9)  # → Fraction(-2, 3)

The sign is always on the numerator. The GCD is computed automatically using math.gcd.

Arithmetic

All standard operations return exact Fraction results:

from fractions import Fraction

a = Fraction(2, 3)
b = Fraction(3, 4)

a + b   # 17/12
a - b   # -1/12
a * b   # 1/2
a / b   # 8/9
a ** 2  # 4/9
abs(Fraction(-5, 3))  # 5/3

Mixing Fraction with int preserves exactness. Mixing with float converts to float (losing precision).

Fraction(1, 3) + 1     # Fraction(4, 3) — exact
Fraction(1, 3) + 0.5   # 0.8333333333333333 — float, not exact

limit_denominator — practical approximation

Sometimes you want a simpler fraction that’s close enough:

Fraction(355, 113)  # ≈ π

pi_approx = Fraction(355, 113)
pi_approx.limit_denominator(10)   # Fraction(22, 7)
pi_approx.limit_denominator(100)  # Fraction(311, 99)
pi_approx.limit_denominator(1000) # Fraction(355, 113) — already simple enough

limit_denominator(max) returns the closest fraction with denominator at most max. This is useful for converting messy floating-point numbers back to clean fractions:

Fraction(0.1).limit_denominator(1000)  # Fraction(1, 10)

Float vs Decimal vs Fraction

Type	Precision	Speed	Use case
float	~15 decimal digits	Fastest	General-purpose math, science
Decimal	Configurable (28+ digits)	Medium	Finance, currency, fixed-point
Fraction	Exact (arbitrary precision)	Slowest	Exact ratios, symbolic math, algorithms

Choose float unless you need more. Choose Decimal for money. Choose Fraction when rounding errors are unacceptable.

Common misconception

People assume Fraction is always better than float because it’s exact. But Fraction arithmetic creates progressively larger numerators and denominators. After many operations, a Fraction might have numerators with hundreds of digits. Arithmetic on such large integers is slower than float arithmetic. Fractions trade speed for exactness — make sure you need that exactness.

Practical use cases

Probability calculations:

from fractions import Fraction

# Probability of drawing 2 aces from a 52-card deck
p = Fraction(4, 52) * Fraction(3, 51)
print(p)  # 1/221 (exact)

Recipe scaling:

# Scale a recipe from 4 servings to 6
scale = Fraction(6, 4)  # 3/2
flour = Fraction(2, 3) * scale  # 1 cup (exact)

Continued fraction approximations:

# Find rational approximation of √2
import math
approx = Fraction(math.sqrt(2)).limit_denominator(100)
print(approx)  # 99/70

The one thing to remember: Use Fraction when you need exact rational arithmetic — create from strings or integers (not floats) to keep the input exact, and use limit_denominator when you need a cleaner approximation.