Decimal to 2's complement binary conversion (least bits required to represent decimal number)

1
votes

I was asked in a class:

What is the least number of bits are required to represent the following decimal numbers as 2’s complement binary numbers? -64 31 128

  1. So, first of all, what is the difference between 1 and 2's complement binary? How can I convert a decimal number into a 2's complement binary representation?

  2. I also am wondering, is it possible to have multiple representations of that decimal number? How do I find out the least number of required bits?

1
binary
There are broadly two types of number system Non-positional e.g. ROMAN-NUMBERS and Positional. Decimal number system, Binary number system, Hexadecimal all are positional number system. so many representation are possible. its long theory you have to explore...Actually you are not asking question but a Chapter – Grijesh Chauhan
@melpomene Thanks I edited that – jordan
1's complement in binary = change all bits (0 -> 1 , 1 -> 0) ...2's complement = 1's + 1 ... – Grijesh Chauhan
A number can have multiple representation in different number system. The word TEN actually English word that can be represented as 10 in Decimal-number-system, A in Hexadecimal-number-system, 1010 in Binary Number-System, 11 in Octal-number-system and every time it would be called TEN. – Grijesh Chauhan

1 Answers

3
votes

  1. The difference between 1's and 2's complement:

    • In 1's complement negative numbers are represented as the positive number with all of its bits flipped. So for example -6 would be ..111001 because 6 is ..000110.
    • In 2's complement negative numbers are represented as the positive number with all its bits flipped, plus one. So for example -6 would be ..111001 + 1 = ..111010

  2. Multiple representations of a decimal number in binary: The representation of a number in binary is unique, but for one exception: in 1's complement the number 0 can be written both as 0 and as ..111.

Note that in both cases negative numbers have a representation with an infinite string of 1's on the left. In practice the number of bits available is finite, so we use only some of the rightmost bits. The convention is that if the leftmost bit is 1 we assume that the number has an infinite string on ones to the left before this first bit. This means that the leftmost bit gives the sign of the number: if it's 1 the number is negative, if it's 0 the number is positive. The rest of the bits give the magnitude. (Of course, there are other ways to represent signed numbers where this doesn't hold.)

The least number of bits required to represent a number may differ between 1's and 2's complement. For example, 1's complement needs at least 5 bits to represent -8, while 2's complement can make do with 4:

        +8 in binary:  1000
-8 in 1's complement: 10111
-8 in 2's complement:  1000