Number of bits it takes to represent a number

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SUMMARY

The discussion centers on the accurate calculation of the number of bits required to represent a number in binary. The formula presented, $$bits(x)= \frac{log(x)}{log(2)}$$, is incorrect; the correct formula for positive integers is $$\lceil log_2(x+1) \rceil$$. It is established that representing the number 4 requires 3 bits in binary (100), and that the leading 1 in binary representations can be omitted for positive integers to save space. The conversation also touches on floating-point representations and the complexities of encoding integers.

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  • #31
There are two subjects being discussed, and confused.
1. How many bits to represent a quantity of symbols
2. How to code numbers in binary.

Maybe my wording is not precise, but how many symbols one might use, and what people decide the symbols represent are, for the most part, orthogonal concepts.

For example, let's design a 3 bit binary coding system using 8 symbols to represent 0,1,2,3,4,5,6,7.
Let's design a 3 bit binary coding system using 8 symbols to represent -4, -3, -2, -1 0, 1, 2, 3.

What people decide to have each symbol represent defines the coding system. (For example, unsigned, sign-mag, and 2's comp are commonly accepted methodologies, but one can define anything)
 

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