Floating-point numbers: why 0.1 + 0.2 is not 0.3

What it is

IEEE 754 double precision stores a sign, a 53-bit significand and an exponent in base 2. Decimal fractions such as 0.1 have no finite binary expansion, so they are stored as the nearest representable value; the Python tutorial walks through why 0.1 + 0.2 == 0.3 is false and how repr chooses the shortest string that round-trips.

Why it matters

Tests that compare computed floats for equality fail intermittently; totals drift by cents; large and small values added together lose the small ones; JSON numbers pass through languages with different precision.

How to apply

• Compare with a tolerance appropriate to the scale (math.isclose with relative and absolute tolerances), never with == after arithmetic.
• Use integers in the smallest unit (cents, milliseconds) or a decimal type for money, quantities and anything that is summed and reported.
• Sum many values with a compensated algorithm (math.fsum) when accuracy matters.
• Serialise with full precision (17 significant digits or the shortest round-trip form) and avoid formatting intermediates.
• Be aware of special values: NaN is not equal to itself; infinities propagate; -0.0 exists.

Pitfalls

round(2.675, 2) gives 2.67 because the stored value is slightly below 2.675. Integer-valued floats above 2⁵³ cannot represent all integers. Language or database column types (real vs double precision) differ in precision.