How integers are stored in memory using two’s complement ?
What is an Integer ?
Integers (or also known as int) are a whole numbers that can have either zero, positive or negative values but it can’t have decimal values .
The size of int is actually 4 bytes (or 32 bites). Starting from -2147483648 (-2³¹)to 2147483647 (2³¹ — 1) .
Unsigned Numbers :
Unsigned numbers don’t have any sign, these can contain only magnitude of the number. So, representation of unsigned binary numbers are all positive numbers only. For example, representation of positive decimal numbers are positive by default. We always assume that there is a positive sign symbol in front of every number.
Representation of Unsigned Binary Numbers:
Since there is no sign bit in this unsigned binary number, so N bit binary number represent its magnitude only. Zero (0) is also unsigned number. This representation has only one zero (0), which is always positive. Every number in unsigned number representation has only one unique binary equivalent form, so this is unambiguous representation technique. The range of unsigned binary number is from 0 to (2n-1).
Example:
Represent decimal number 92 in unsigned binary number.
Simply convert it into Binary number, it contains only magnitude of the given number.
= (92)10= (1x26+0x25+1x24+1x23+1x22+0x21+0x20)10
= (1011100)2
It’s 7 bit binary magnitude of the decimal number 92.
Signed Numbers :
Signed numbers contain sign flag, this representation distinguish positive and negative numbers. This technique contains both sign bit and magnitude of a number. For example, in representation of negative decimal numbers, we need to put negative symbol in front of given decimal number.
Representation of Signed Binary Numbers:
There are three types of representations for signed binary numbers. Because of extra signed bit, binary number zero has two representation, either positive (0) or negative (1), so ambiguous representation. But 2’s complementation representation is unambiguous representation because of the absence of double representation of number 0. These are: Sign-Magnitude form, 1’s complement form, and 2’s complement form which are explained as following below :
a)-Sign-Magnitude form:
For n bit binary number, 1 bit is reserved for sign symbol. If the value of sign bit is 0, then the given number will be positive, else if the value of sign bit is 1, then the given number will be negative. Remaining (n-1) bits represent magnitude of the number. Since magnitude of number zero (0) is always 0, so there can be two representation of number zero (0), positive (+0) and negative (-0), which depends on value of sign bit. Hence these representations are ambiguous generally because of two representation of number zero (0). Generally sign bit is a most significant bit (MSB) of representation. The range of Sign-Magnitude form is from (2(n-1)-1) to (2(n-1)-1).
For example, range of 6 bit Sign-Magnitude form binary number is from (25–1) to (25–1) which is equal from minimum value -31 (i.e., 1 11111) to maximum value +31 (i.e., 0 11111). And zero (0) has two representation, -0 (i.e., 1 00000) and +0 (i.e., 0 00000).
b)- 1’s complement form:
Since, 1’s complement of a number is obtained by inverting each bit of given number. So, we represent positive numbers in binary form and negative numbers in 1’s complement form. There is extra bit for sign representation. If value of sign bit is 0, then number is positive and you can directly represent it in simple binary form, but if value of sign bit 1, then number is negative and you have to take 1’s complement of given binary number. You can get negative number by 1’s complement of a positive number and positive number by using 1’s complement of a negative number. Therefore, in this representation, zero (0) can have two representation, that’s why 1’s complement form is also ambiguous form. The range of 1’s complement form is from (2(n-1)-1) to (2(n-1)-1) .
For example: range of 6 bit 1’s complement form binary number is from (25–1) to (25–1) which is equal from minimum value -31 (i.e., 1 00000) to maximum value +31 (i.e., 0 11111). And zero (0) has two representation, -0 (i.e., 1 11111) and +0 (i.e., 0 00000).
c)- 2’s complement form:
Since, 2’s complement of a number is obtained by inverting each bit of given number plus 1 to least significant bit (LSB). So, we represent positive numbers in binary form and negative numbers in 2’s complement form. There is extra bit for sign representation. If value of sign bit is 0, then number is positive and you can directly represent it in simple binary form, but if value of sign bit 1, then number is negative and you have to take 2’s complement of given binary number. You can get negative number by 2’s complement of a positive number and positive number by directly using simple binary representation. If value of most significant bit (MSB) is 1, then take 2’s complement from, else not. Therefore, in this representation, zero (0) has only one (unique) representation which is always positive. The range of 2’s complement form is from (2(n-1)) to (2(n-1)-1).
For example: range of 6 bit 2’s complement form binary number is from (25) to (25–1) which is equal from minimum value -32 (i.e., 1 00000) to maximum value +31 (i.e., 0 11111). And zero (0) has two representation, -0 (i.e., 1 11111) and +0 (i.e., 0 00000).
You may also like to read :
Tutorialspoint : Unsigned and Signed Binary Numbers — Ankith Reddy
IBM: Signed and Unsigned Integers
Dev: Binary for Beginners: The ABCs of 0s and 1s — Aleksandr Hovhannisyan
