Numbers behind sound, pictures and video

Binary addition works like decimal addition, column by column, from the right. The rules are 0 + 0 = 0, 0 + 1 = 1, and 1 + 1 = 10 (write 0, carry 1). When a column has a carry as well, 1 + 1 + 1 = 11 (write 1, carry 1). Shifting all bits one place to the left doubles a number: 0000 0101 (5) becomes 0000 1010 (10). Shifting right halves it, dropping the last bit: 0000 0111 (7) becomes 0000 0011 (3). To write a negative number in 8 bits, find the one’s complement (flip every bit) and add 1; this is the two’s complement. For −5: 0000 0101 flips to 1111 1010, plus 1 gives 1111 1011. Check by adding the number and its negative: 0000 0101 + 1111 1011 = 1 0000 0000, and the ninth bit does not fit in 8 bits, so the result is 0000 0000. With 8 bits this way we can write −128 to +127. Overflow happens when a result needs more bits than there are: in 4 bits, 1011 + 0110 = 10001, but only 0001 fits. The same idea works in decimal: with one digit, the 10’s complement of 3 is 10 − 3 = 7, and 3 + 7 = 10 drops its carry.