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How many bits are required to represent an arbitrary number X?

(assume unsigned number). I get that to represent 5 states, you need 3 bits (b/c you can represent 8 total with 3 bits). But when you represent a certain number like 5,435, how do you find out the number of bits needed? How many bits are then required to represent an arbitrary number X?

Public Comments

1. Take the base 2 log and round up. If you don't have access to a base 2 log button, remember you can do log2(x)=log(x)/log(2). So log2(5435)~~12.4, so you need 13 bits.
2. The number of states in a binary number of digits is 2^n where n is the number of bits. Three bits can represent 8 distinct states. This can be found by calculating 2^3. Each increase of a bit doubles the number of unique states possible. Eight bits gives you 256 states (2^8). Ten bits is 1024 states (2^10). Three steps up from that, which would allow you to represent your 5,435 value, would be 13 bits with its 8192 states (2^13).