Two’s Complement (Signed: Positive and Negative Numbers)

Two’s Complement (Positive and Negative Numbers)

What is Two’s Complement?

Two’s complement is a way to represent signed numbers (both positive and negative) in binary. It helps computers easily perform addition and subtraction with positive and negative numbers.

Range of values for n bits

For n bits, the range of numbers is:

- 2^{n - 1} to + 2^{n - 1} - 1

For 8 bits, the range is:
−128 to +127

How to find the Two’s Complement (negative of a number)

To represent a negative number in two’s complement:

Write the positive number in binary using n bits.

Invert the bits (change 0 to 1, and 1 to 0).

Add 1 to the inverted bits.

Example: Find −5 in 8-bit two’s complement

Write +5 in 8 bits:
0000 0101

Invert bits:
1111 1010

Add 1:
1111 1010 + 1 = 1111 1011

So, 1111 1011 represents −5 in two’s complement.

Practice Questions & Answers

1. Convert +13 to 8-bit binary, then find −13 in two’s complement

+13 decimal → binary:

Divide by 2	Quotient	Remainder
13 ÷ 2	6	1
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Reading remainders bottom to top: 1101

Add leading zeros for 8 bits: 0000 1101

Find −13 two’s complement:

Invert bits: 1111 0010

Add 1: 1111 0010 + 1 = 1111 0011

Answer:
+13 = 0000 1101
−13 = 1111 0011

2. What decimal number does `1000 0000` represent in 8-bit two’s complement?

1000 0000 starts with 1 → negative number.

To find the value:

Invert bits: 0111 1111

Add 1: 0111 1111 + 1 = 1000 0000 (decimal 128)

So, 1000 0000 = −128

Summary Table: Two’s Complement Range for 8 bits

Binary	Decimal Value
`0000 0000`	0
`0000 0001`	1
...	...
`0111 1111`	+127
`1000 0000`	−128
`1000 0001`	−127
...	...
`1111 1111`	−1

Two’s Complement (Signed: Positive and Negative Numbers)

Two’s Complement (Positive and Negative Numbers)#

What is Two’s Complement?#

Range of values for n bits#

How to find the Two’s Complement (negative of a number)#

Example: Find −5 in 8-bit two’s complement#

Practice Questions & Answers#

1. Convert +13 to 8-bit binary, then find −13 in two’s complement#

2. What decimal number does 1000 0000 represent in 8-bit two’s complement?#

Summary Table: Two’s Complement Range for 8 bits#