All integers are real numbers, but most computer representations of integers do not equal computer representations of real numbers.

To make the statement above precise, we have to be more specific about what we mean by computer integers and floating point numbers. I’ll use int32 and int64 to refer to 32-bit and 64-bit signed integers. I’ll use float32 and float64 to refer to IEEE 754 single precision and double precision numbers, what C calls float and double.

Most int32 numbers cannot be represented exactly by a float32. All int32 numbers can be represented approximately as float32 numbers, but usually not exactly. The previous statements remain true if you replace “32” everywhere by “64.”

32 bit details

The int32 data type represents integers −2³¹ through 2³¹ − 1. The float32 data type represents numbers of the form

± 1.f × 2^e

where 1 bit represents the sign, 23 bits represent f, and 8 bits represent e.

The number n = 2²⁴ can be represented by setting the fractional part f  to 0 and setting the exponent e to 24. But the number n + 1 cannot be represented as a float32 because its last bit would fall off the end of f:

2²⁴ + 1 = (1 + 2⁻²⁴) 2²⁴ = 1.000000000000000000000001_two × 2²⁴

The bits in f fill up representing the 23 zeros after the decimal place. There’s no 24th bit to store the final 1.

For each value of e, there are 2²³ possible values of f. So for each of e = 24, 25, 26, …, 31 there are 2²³ representable integers, for a total of 8 × 2²³.

So of the 2³¹ non-negative integers that can be represented by an int32 data type, only 9 × 2²³ can also be represented exactly as a float32 data type. This means about 3.5% of 32-bit integers can be represented exactly by a 32-bit float.

64 bit details

The calculations for 64-bit integers and 64-bit floating point numbers are analogous. Numbers represented by float64 data types have the form

± 1.f × 2^e

where now f has 52 bits and e has 11.

Of the 2⁶³ non-negative integers that can be represented by an int64 data type, only 11 × 2⁵² can also be represented exactly as a float64 data type. This means about 0.5% of 64-bit integers can be represented exactly by a 64-bit float.

A note on Python

Python’s integers have unlimited range, while its floating point numbers correspond to int64. So there are two reasons an integer might not be representable as a float: it may be larger than the largest float, or it may require more than 53 bits of precision.

Related posts

• Anatomy of a floating point number
• NaN, 1.#IND, 1.#INF, and all that
• Anatomy of a posit number
• Eight-bit floating point

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