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

Question

There have been several questions posted to SO about floating-point representation. For example, the decimal number 0.1 doesn\'t have an exact binary representation, so it\'

Answer 1

This is a good question.

All your question is based on "how do we represent a number?"

ALL the numbers can be represented with decimal representation or with binary (2's complement) representation. All of them !!

BUT some (most of them) require infinite number of elements ("0" or "1" for the binary position, or "0", "1" to "9" for the decimal representation).

Like 1/3 in decimal representation (1/3 = 0.3333333... <- with an infinite number of "3")

Like 0.1 in binary ( 0.1 = 0.00011001100110011.... <- with an infinite number of "0011")

Everything is in that concept. Since your computer can only consider finite set of digits (decimal or binary), only some numbers can be exactly represented in your computer...

And as said Jon, 3 is a prime number which isn't a factor of 10, so 1/3 cannot be represented with a finite number of elements in base 10.

Even with arithmetic with arbitrary precision, the numbering position system in base 2 is not able to fully describe 6.1, although it can represent 61.

For 6.1, we must use another representation (like decimal representation, or IEEE 854 that allows base 2 or base 10 for the representation of floating-point values)