2’s Complement

• Another system that lets us represent negative numbers

• MSB is STILL the sign bit, but there is no negative zero

• Negative numbers count backwards and wrap around

• Calculating 2’s complement (Pos Neg)

1. Flip the bits ( 01 and 10 )

2. Add 1

Example ( Pos Neg )

110 -110

00012 11112

1. Flip bits: 1110

2. Add 1: 1110+0001 1111

Another Example ( Pos Neg )

2510 -2510

000110012 11100111 2

1. Flip bits: 11100110

2. Add 1: 11100110+00000001 11100111

Your Turn

• Assuming an 8-bit restriction, what is -2110 in 2’s complement?

1. Flip bits

2. Add 1

Answer: 111010112

Your Turn

• Assuming an 8-bit restriction, what is -3010 in 2’s complement?

1. Flip bits

2. Add 1

Answer: 111000102

Example ( Neg Pos )

-410 410

11002 01002

1. Flip bits: 0011

2. Add 1: 0011+0001 0100

Another Example ( Neg Pos )

-2910 2910

111000112 000111012

1. Flip bits: 00011100

2. Add 1: 00011100+00000001 00011101

Your Turn

• Assuming 2’s complement, what is the decimal value of 111110012?

1. Flip bits

2. Add 1

Answer: -710

Your Turn

• Assuming 2’s complement, what is the decimal value of 111010102?

1. Flip bits

2. Add 1

Answer: 22

2’s Complement ChartBinary Decimal

00000111 7

00000110 6

00000101 5

00000100 4

00000011 3

00000010 2

00000001 1

00000000 0

11111111 -1

11111110 -2

11111101 -3

11111100 -4

11111011 -5

11111010 -6

11111001 -7

11111000 -8

Binary Decimal

0111 7

0110 6

0101 5

0100 4

0011 3

0010 2

0001 1

0000 0

1111 -1

1110 -2

1101 -3

1100 -4

1011 -5

1010 -6

1001 -7

1000 -8

SHORTCUT!

1. Find the 1 on the farthest right

2. Flip all the bits to the left of the 1 (DO NOT FLIP THE 1)

Example:

4210 -4210

001010102 110101102

Awesomeness of 2’s Complement

• No more negative zero

• Lower minimum value: -(2N-1)

• So what’s the big deal?– Everything is addition– No need for special hardware to do

subtraction

2’s Complement Addition

• Just like normal positive binary addition

• You MUST restrict the number of bits

• IGNORE any overflow bits– maintain bit-size restriction

Positive Addition Example

1210 + 410 = 1610

Assuming 2’s complement

000010102 1210

+000000102 + 410

000011002 1610

Negative Addition Example

-1210 + -410 = -1610

111101002 -1210

+111111002 + -410

111100002 -1610

NOTE: We ignored the last overflow bit on the left!

Your Turn

• Show the binary addition of -14 + -3 = -17

Subtraction Example

1610 – 410 = 1610 + -410 = 1210

000100002

+111111002

000011002

NOTE: We ignored the last overflow bit on the left!

Your Turn

• Show the binary subtraction of 23 – 10 = 13

Overflow / Underflow Problem

• Addition and bit-size restriction allow for possible overflow / underflow

• Overflow – when the addition of two binary numbers yields a result that is greater than the maximum possible value

• Underflow – when the addition/subtraction of two binary numbers yields a result that is less than the minimum possible value

Overflow Example

• Assume 4-bit restriction and 2’s complement• Maximum possible value: 24-1 – 1 = 7

610 + 310 = 910

01102 610

+00112 +310

10012 -710 not good!

Underflow Example

• Assume 4-bit restriction and 2’s complement• Minimum possible value: -(24-1) = -8

-510 + -510 = -1010

10112 -510

+10112 +-510

01102 610 not good!