Why can't decimal numbers be represented exactly in binary?

Decimal numbers can be represented exactly, if you have enough space - just not by floating binary point numbers. If you use a floating decimal point type (e.g. System. Decimal in .

NET) then plenty of values which can't be represented exactly in binary floating point can be exactly represented.

The reason for the imprecision is the nature of number bases. In base 10, you can't exactly represent 1/3. It becomes 0.333... However, in base 3, 1/3 is exactly represented by 0.1 and 1/2 is an infinitely repeating decimal (tresimal?).

The values that can be finitely represented depend on the number of unique prime factors of the base, so base 30 2 * 3 * 5 can represent more fractions than base 2 or base 10. Even more for base 210 2 * 3 * 5 * 7.

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