Answer to Question #119199 in Combinatorics | Number Theory for Evelyn Teye

Question

Compute the sum of all the numbers represented by a 4-bit unsigned binary number.

Answer 1

We know that 4-bit unsigned integers range from 0000 to 1111 in binary numeral system (see http://www.cs.uwm.edu/classes/cs315/Bacon/Lecture/HTML/ch04s10.html).

$0000_2 = 0_{10}, \;\; 1111_2= (2^3+2^2+2^1+2^0)_{10} = 15_{10}.$

Therefore, there are 16 numbers from 0 to 15.

We know that the sum of numbers from 1 to n can be calculated as $\dfrac{n(n+1)}{2}$ (see https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF). So the sum of integers from 0 to 15 will be

$0 + \dfrac{15\cdot16}{2} = 120$