The document explains the concepts of signed and unsigned numbers in binary representation, detailing their characteristics and ranges. It discusses 1's and 2's complement methods for representing negative numbers, including their conversion processes and implications for zero representation. Key differences between these methods are highlighted, emphasizing the unambiguous nature of 2's complement versus the ambiguous representation in signed magnitude and 1's complement.

1. Signed and Unsigned
Numbers and Complement
Dr Anita Choudhary
Assistant Professor
Department of Computer Science
anitach312@gmail.com

2. Table of Content
Unsigned Numbers
Signed Numbers
1’s complement
2’s complement

3. Unsigned Numbers
• Unsigned numbers don't have any sign for representing negative
numbers.
• So the unsigned numbers are always positive.
• We always assume a positive sign in front of each decimal digit.
• There is no sign bit in unsigned binary numbers so it can only represent
its magnitude. In zero and one, zero is an unsigned binary number.
There is only one zero (0) in this representation, which is always
positive.
• The range of the unsigned binary numbers starts from 0 to (2^(n)-1).

4. Signed Numbers
• The signed numbers have a sign bit so that it can differentiate positive
and negative integer numbers.
• The signed binary number technique has both the sign bit and the
magnitude of the number.
• For representing the negative decimal number, the corresponding
symbol in front of the binary number will be added.
• The signed bit makes two possible representations of zero (positive (0)
and negative (1)), which is an ambiguous representation.
• In 2's complement representation no double representation of zero is
possible, which makes it unambiguous representation.

5. Signed Magnitude Method
• If the number is positive then sign bit will be 0 and if number is negative then sign
bit will be 1.
• Magnitude is represented with the binary form of the number to be represented.
• Example: Let we are using 5 bits register. The representation of -5 to +5 will be as
follows:

6. Sign Magnitude Representation

7. Unsigned and Signed Binary Numbers

8. 1’s complement
• 1’s complement of a binary number is another binary number obtained by
toggling all bits in it, i.e., transforming the 0 bit to 1 and the 1 bit to 0.
• In the 1’s complement format , the positive numbers remain unchanged .
• The negative numbers are obtained by taking the 1’s complement of positive
counterparts.
• example +9 will be represented as 00001001 in eight-bit notation and -9 will be
represented as 11110110, which is the 1’s complement of 00001001.
+9  00001001
↓ 1’s complement
-9  11110110

9. 1’s complement
• Let numbers be stored using 4 bits
1's complement of 7 (0111)
is 8 (1000)
1's complement of 12 (1100)
is 3 (0011)
• For finding 1's complement of the binary number, we can implement the
logic circuit also by using NOT gate. We use NOT gate for each bit of
the binary number. So, if we want to implement the logic circuit for 5-bit
1's complement, five NOT gates will be used.

10. 2’s complement
• 2’s complement of a binary number is 1, added to the 1’s complement
of the binary number.
• In 2’s complement representation of binary numbers, the MSB
represents the sign with a ‘0’ used for plus sign and a ‘1’ used for a
minus sign.
• The remaining bits are used for representing magnitude.
• Positive magnitudes are represented in the same way as in the case of
sign-bit or 1’s complement representation.
• Negative magnitudes are represented by the 2’s complement of their
positive counterparts.

11. 1’s Complement Representation

12. 2’s Complement Representation of Positive Numbers
• In 2’s complement representation also, the representation of the
positive number is same as1’s complement and sign-magnitude form.

13. 2’s Complement Representation of Negative Numbers
• But the representation of the negative number is different. For example, if we want to
represent -34 in 2’s complement form then
1.Write the number corresponding to +34.
2.Starting from Least Significant Bit (LSB), just copy all the bits until the first 1 is
encountered in the number.
3.After the first ‘1’ is encountered, invert all the 1s in the number with 0s and 0s in the
number with 1s (including the sign bit)
4.The resultant number is 2’s complement representation of the number -34.

14. • The second way of representing -34 in 2’s complement form is
1.Write the number corresponding to +34.
2.Find 1’s complement of +34
3.Add ‘1’ to the 1’s complement number
4.The resultant is 2’s complement representation of -34
2’s Complement Representation of Negative Numbers

15. Range of 1’s and 2’s complement
• Range of 1’s complement for n bit number is from –(2^(n)-1)-1 to (2 ^(n)-1)-1
whereas the range of 2’s complement for n bit is from –(2^(n)-1) to (2^(n)-1)-1.
• There are (2 ^(n)-1) valid numbers in 1’s complement and 2n valid numbers in 2’s
complement.

18. Thank You