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# Binary Addition Binary Multiplication PowerPoint PPT Presentation

Binary Addition Binary Multiplication. Section 4.5 and 4.7 . Topics. Calculations Examples Signed Binary Number Unsigned Binary Number Hardware Implementation Overflow Condition Multiplication. Unsigned Number. (2-bit example). Unsigned Addition. 1+2=. Unsigned Addition. 1+3=.

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#### Presentation Transcript

Section 4.5 and 4.7

### Topics

• Calculations Examples

• Signed Binary Number

• Unsigned Binary Number

• Hardware Implementation

• Overflow Condition

• Multiplication

### Unsigned Number

(2-bit example)

• 1+2=

• 1+3=

(Indicates Overflow)

(Carry Out)

• 1-2=

(1’s complement)

(2’s complement)

### Unsigned Subtraction (2)

• 2-1=

• Overflow can be an issue in unsigned addition

• Unsigned Subtraction (M-N)

• If M≥N, and end carry will be produced. The end carry is discarded.

• If M<N,

• Take the 2’s complement of the sum

### Signed Binary Numbers

• 4-bit binary number

• 1 bit is used as a signed bit

• -8 to +7

• 2’s complement

(Indicates a negative number)

70=21+22+26=2+4+64

80=24+26=16+64

10010110→01101001 →01101010

21+23+25+26=2+8+32+64=106

10010110↔-106

010010110

010010110↔ 21+22+24+27=2+4+16+128=150

Conclusion: There is a problem of overflow

Fix: Use the end carry as the sign bit, and let b7 be

the extra bit.

### Signed Subtraction (70-80)

(Indicates a negative number)

70=21+22+26=2+4+64

80=24+26=16+64

11110110→00001001 →00001010

21+23=10

11110110↔-10

(No Problem)

### Signed Subtraction (-70-80)

(Indicates a positive number! A negative number expected.)

70=21+22+26=2+4+64

80=24+26=16+64

101101010 →010010101 → 010010110

010010110 ↔21+22+24+27=2+4+16+128=150

101101010 ↔-150

Conclusion: There is a problem of overflow

Fix: Use the end carry as the sign bit, and let b7 be

the extra bit.

### Observations

• Given the similarity between addition and subtraction, same hardware can be used.

• Overflow is an issue that needs to be addressed in the hardware implementation

• A signed number is not processed any different from an unsigned number. The programmer must interpret the results of addition and subtraction appropriately.

If M=0, =

If M=1, =

B0

If M=0,

If M=1,

B3

B2

B0

B1

0

### M=1

1

2’s complement is generated of B is generated!

When two unsigned numbers are added,

an overflow is detected from the end carry.

### Detect Overflow in Signed Addition

Observe

The cary into the sign bit

The carry out of the sign bit

If they are not equal,

they indicate an overflow.

### Two-Bit Binary Multiplier

(multiplicand)

(multiplier)

Use an AND gate to multiply A0 and B0

### Four-bit by three-bit Binary Multiplier

S10=A0B1+A1B0

S11=A0B2+A1B1+C1

S12=A0B3+A1B2+C2

S13=0+A1B3+C3

(S1X, where 1 is the first 4-bit adder)