IEEE 754 Floating-Point Standard#

Stage 1 · 📋 Foundations · Lesson 06 of 56 · beginner

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Important

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Storing real numbers#

Computers store real numbers in a finite number of bits, and the near-universal scheme for doing so is the IEEE 754 standard. Understanding it explains a whole class of surprises — why sums do not quite add up, why you should never test two floats for exact equality — that otherwise look like bugs.

Sign, exponent, mantissa#

A floating-point number is stored in three parts, like scientific notation in binary: a sign bit, an exponent (which scales the value), and a mantissa (the significant digits). The two common sizes are single precision (32 bits: 1 sign, 8 exponent, 23 mantissa) and double precision (64 bits: 1 sign, 11 exponent, 52 mantissa) — the float64 that numpy and pandas use by default. More mantissa bits mean more precision.

Why 0.1 + 0.2 ≠ 0.3#

With finite mantissa bits, most decimal fractions cannot be represented exactly — \(0.1\) in binary is a repeating fraction, rounded to fit. The rounding errors accumulate, so the famous result is

\[0.1 + 0.2 = 0.30000000000000004 \neq 0.3.\]

It is not a language bug; it is the unavoidable cost of squeezing infinite decimals into 64 bits.

What it means for data work#

Three habits follow. Never test floats for exact equality — compare within a tolerance (numpy.isclose) instead. Beware accumulated error when summing many values, and prefer stable formulations. And know the special values the standard defines — positive and negative infinity, and NaN (not-a-number) — because NaN in particular is how missing or undefined numeric results surface throughout pandas.

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