Programming Fundamentals › Programming Basics

Floating-Point Number

A binary approximation of real numbers, and why 0.1 + 0.2 != 0.3.

Also known as: float, double, IEEE 754, floating point

Computers store most non-whole numbers as floating-point values: a binary approximation of the real number. Many everyday decimals, like 0.1, have no exact binary form, just as 1/3 has no exact decimal form. The stored value is the closest one that fits.

>>> 0.1 + 0.2
0.30000000000000004
>>> 0.1 + 0.2 == 0.3
False

JavaScript prints the same result, because it uses the same standard (IEEE 754 doubles) for every number. Many other languages do too, so this isn’t a bug in one of them.

What this means in practice

Never compare floats with ==. Compare with a tolerance:

import math
math.isclose(0.1 + 0.2, 0.3)   # True

Never use floats for money. Errors are tiny, but they pile up across many additions and make totals off by a cent. Store money as an integer number of the smallest unit (cents), or use an exact decimal type. See money precision.

from decimal import Decimal
Decimal("0.1") + Decimal("0.2") == Decimal("0.3")   # True

(Build decimals from strings. Decimal(0.1) copies the float’s error.)

Other things to know

• Whole numbers are exact in a double only up to 2^53. Beyond that, neighbouring integers collapse into the same value. In JavaScript this limit is Number.MAX_SAFE_INTEGER.
• Rounding for display is fine (f"{x:.2f}"). Rounding stored values is a decision to make on purpose.
• Special values exist: NaN (“not a number”) and infinity. NaN is not equal to anything, including itself.

Rule of thumb: floats are for measurements and approximations (distance, temperature, averages); integers or decimals are for things you count or charge for.