number representation and binary arithmetic

Number RepresentationandBinary Arithmetic

ECEn 224 Winter 2002

Unsigned Addition


Binary Addition0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 0 and carry 1 to the next columnExamples: 0101 (510)+ 0010 (210) 0111 (710)Carries 1 1 1 0101 (510)+ 0011 (310) 1000 (810)


Binary Addition w/OverflowAdd 4510 and 4410 in binaryCarries 1 1 1 101101 (4510)+ 101100 (4410) 1011001 (8910)If the operands are unsigned, you can use the final carry-out as the MSB of the result.

Adding 2 k bit numbers k+1 bit result


More Binary Addition w/Overflow 1 1 1 1 1111 (1510)+ 0001 (110) 0000 (010)If you dont want a 5-bit result, just keep the lower 4 bits.

This is insufficient to hold the result (16).

It rolls over back to 0.


Signed Numbers


Negative Binary NumbersSeveral ways of representing negative numbersMost obvious is to add a sign (+ or -) to the binary integer


Sign-Magnitude0 is +1 is -Easy to interpret number value


Sheet1

[( A B' ) + C ) D' + E ]= [ ( A B' + C ) D' ] E'(2-1)

= [ ( A B' + C )' + D ] E'(2-2)

= [ ( A B' ) C )' + D ] E'(2-3)

= [ ( A' + B' ) C' + D ] E'(2-4)

F = A ( A' + B )

Lect 1

valuesymbolvaluesymbol

0088

1199

2210A

3311B

4412C

5513D

6614E

7715F

20480012048

10240000

5120000

256001256

1281128011282430

641640164

320013200

161160000

8181800

4001400

2120000

1111100

2194524960

8-4-2-1

DecimalCodeDecimal2-out-of-5

Digit(BCD)DigitCodeNumberSignMagnitudeFull Number

00 0 0 000 0 0 1 1+10000100001110110

10 0 0 110 0 1 0 1-11000110001x101101

20 0 1 020 0 1 1 0+50010100101_________________

30 0 1 130 1 0 0 1-51010110101

40 1 0 040 1 0 1 0+00000000000110111

50 1 0 150 1 1 0 0-01000010000110110

60 1 1 061 0 0 0 160 1 0 1\000000

70 1 1 171 0 0 1 070 1 0 0110110

81 0 0 081 0 1 0 081 1 0 0110110

91 0 0 191 1 0 0 091 1 0 1000000

101 1 1 1110110

111 1 1 0__________________________________

121 0 1 0100101111110

131 0 1 1

141 0 0 1

151 0 0 0

161 1 0 0 0

Sheet2

564.4810

Sheet3

Sign-Magnitude Examples 0 0 0 0 0011 (+310)+ 0 0100 (+410) 0 0111 (+710)Signs are the same Just add the magnitudes 0 0 0 1 0011 (-310)+ 1 0100 (-410) 1 0111 (-710)


Another Sign-Magnitude Example 0 0101 (+510)+ 1 0011 (-310) Signs are different -determine which is larger magnitudePut larger magnitude number on top

Subtract

Result has sign of larger magnitude number 0 0101 (+510)- 1 0011 (-310) 0 0010 (+210)


Yet Another Sign-Magnitude Example 0 0010 (+210)+ 1 0101 (-510) Signs are different -determine which is larger magnitudePut larger magnitude number on top

Subtract

Result has sign of larger magnitude number 1 0101 (-510)- 0 0010 (+210) 1 0011 (-310)


Sign-MagnitudeAddition requires two separate operationsadditionsubtractionSeveral decisions:Signs same or different?Which operand is larger?What is sign of final result?Two zeroes (+0, -0)


Sign-MagnitudeAdvantagesEasy to understand

DisadvantagesTwo different 0sHard to implement in logic


Ones ComplementPositive numbers are the same as sign-magnitude-N is represented as the complement of N+ -N = N


Sheet1

[( A B' ) + C ) D' + E ]= [ ( A B' + C ) D' ] E'(2-1)

= [ ( A B' + C )' + D ] E'(2-2)

= [ ( A B' ) C )' + D ] E'(2-3)

= [ ( A' + B' ) C' + D ] E'(2-4)

F = A ( A' + B )

Lect 1


0088

1199

2210A

3311B

4412C

5513D

6614E

7715F

20480012048

10240000

5120000

256001256

1281128011282430

641640164

320013200

161160000

8181800

4001400

2120000

1111100

2194524960

8-4-2-1


Digit(BCD)DigitCodeNumber1's ComplementMagnitudeFull Number

00 0 0 000 0 0 1 1+10001000100001110110

10 0 0 110 0 1 0 1-11110000110001x101101

20 0 1 020 0 1 1 0+50101010100101_________________

30 0 1 130 1 0 0 1-51010010110101

40 1 0 040 1 0 1 0+00000000000000110111

50 1 0 150 1 1 0 0-01111000010000110110

60 1 1 061 0 0 0 160 1 0 1\000000

70 1 1 171 0 0 1 070 1 0 0110110

81 0 0 081 0 1 0 081 1 0 0110110

91 0 0 191 1 0 0 091 1 0 1000000

101 1 1 1110110

111 1 1 0__________________________________

121 0 1 0100101111110

131 0 1 1

141 0 0 1

151 0 0 0

161 1 0 0 0

Sheet2

564.4810

Sheet3

Ones Complement Examples 000101 (510)+ 010100 (2010) 011001 (2510) 1 1 0 0 0101 (510)+ 1100 (-310) 0001 0010 (+210) +If there is a carry out on the left, it must be wrapped around and added back in on the right


Ones ComplementAddition complicated by end around carryNo decisions (like SM)Still two zeroes (+0, -0)


Ones ComplementAdvantagesEasy to generate NOnly one addition process DisadvantagesEnd around carryTwo different 0s


Twos ComplementTreat positional digits differently01112C = -0 x 23 + 1 x 22 + 1 x 21 + 1 x 20 = 71011112C = -1 x 23 + 1 x 22 + 1 x 21 + 1 x 20 = -110Most significant bit (MSB) given negative weightOther bits same as in unsigned


Twos Complement


Sheet1

[( A B' ) + C ) D' + E ]= [ ( A B' + C ) D' ] E'(2-1)

= [ ( A B' + C )' + D ] E'(2-2)

= [ ( A B' ) C )' + D ] E'(2-3)

= [ ( A' + B' ) C' + D ] E'(2-4)

F = A ( A' + B )

Lect 1


0088

1199

2210A

3311B

4412C

5513D

6614E

7715F

20480012048

10240000

5120000

256001256

1281128011282430

641640164

320013200

161160000

8181800

4001400

2120000

1111100

2194524960

8-4-2-1


Digit(BCD)DigitCodeNumber2's ComplementMagnitudeFull Number

00 0 0 000 0 0 1 1+10001000100001110110

10 0 0 110 0 1 0 1-11111000110001x101101

20 0 1 020 0 1 1 0+50101010100101_________________

30 0 1 130 1 0 0 1-51011010110101

40 1 0 040 1 0 1 0+00000000000000110111

50 1 0 150 1 1 0 0-0none000010000110110

60 1 1 061 0 0 0 160 1 0 1\000000

70 1 1 171 0 0 1 070 1 0 0110110

81 0 0 081 0 1 0 081 1 0 0110110

91 0 0 191 1 0 0 091 1 0 1000000

101 1 1 1110110

111 1 1 0__________________________________

121 0 1 0100101111110

131 0 1 1

141 0 0 1

151 0 0 0

161 1 0 0 0

Sheet2

564.4810

Sheet3

Sign-Extension in 2s ComplementTo make a k-bit number widerreplicate sign bit1102C = -1 x 22 + 1 x 21 = -21011102C = -1 x 23 + 1 x 22 + 1 x 21 = -210111102C = -1 x 24 + 1 x 23 + 1 x 22 + 1 x 21 = -210


More Sign-Extension0102C = 1 x 21 = 21000102C = 1 x 21 = 210

000102C = 1 x 21 = 210

Works for both positive and negative numbers


Negating a 2s Complement NumberInvert all the bitsAdd 1+210 = 00102C 1101 + 1 = 11102C = -210invert-210 = 11102C 0001 + 1 = 00102C = +210invert


Twos Complement Addition 0 0 1 0101 (+510)+ 0001 (+110) 0110 (+610) 1 1 1 1011 (-510)+ 1111 (-110) 1010 (-610) 1 1 1 0110 (+610)+ 1111 (-110) 0101 (+510)Operation is same as for unsignedSame rules, same procedure

Interpretation of operands and results are different


Twos ComplementAddition always the sameOnly 1 zero

Representation of choice


Twos Complement OverflowAll representations can overflowFocus on 2s complement here 1 0 0 0 1011 (-510)+ 1100 (-410) 0111 (+710) ??The correct answer (-9) cannot be represented by 4 bitsThe correct answer is 10111


OverflowCan you use leftmost carry-out as new MSB?

The answer is no, in general 1 0 0 0 1011 (-510)+ 1100 (-410) 10111 (-910)Works here 0 0 0 0 1000 (-810)+ 0111 (+710) 01111 (+1510) ??Does NOT work here


Handling OverflowSign-extend the operands Do the addition 0 0 0 0 11000 (-810)+00111 (+710) 11111 (-110)


When Can Overflow Occur?Adding two positive numbers - yes

Adding two negative numbers yes

Adding a positive to a negative no

Adding a negative to a positive - no


General Handling of OverflowAdding two k bit numbers gives k+1 bit result Two options include:

Sign extend and keep entire k+1 bit resultThrow away new bit generated (keep only k bits)Detect overflow and signal an errorIgnore the overflow (this is what most computers do)

The choice is up to the designer


Detecting OverflowOverflow occurs if:Adding two positive numbers gives what looks like a negative resultAdding two negative numbers gives what looks like a positive result

Overflow will never occur when adding positive to negative

There are other ways of detecting overflow vs. what is suggested here


Arithmetic Circuits


Binary AdderAn example of an iterative network

FullAdderA0S0C0S1S2C3...B0 C1A1B1FullAdder C2FullAdderA2B2Sn-1Cn Cn-1FullAdderAn-1Bn-1 This type of adder is also called a ripple-carry adder because the carry ripples through from cell to cell


Full Adder DerivationSiCi+1AiBiCiAiBiBiCiAiCi


Sheet1

[( A B' ) + C ) D' + E ]= [ ( A B' + C ) D' ] E'(2-1)

= [ ( A B' + C )' + D ] E'(2-2)

= [ ( A B' ) C )' + D ] E'(2-3)

= [ ( A' + B' ) C' + D ] E'(2-4)

F = A ( A' + B )

Lect 2


0088

1199

2210A

3311B

4412C

5513D

6614E

7715F

20480012048

10240000

5120000

256001256

1281128011282430

641640164

320013200

161160000

8181800

4001400

2120000

1111100

2194524960

8-4-2-1

DecimalCodeDecimal2-out-of-5GrayCharacterA6

Digit(BCD)DigitCodeNumberCode

00 0 0 000 0 0 1 100 0 0 0110110

10 0 0 110 0 1 0 110 0 0 1x101101c1100011

20 0 1 020 0 1 1 020 0 1 1_________________d1100100

30 0 1 130 1 0 0 130 0 1 0e1100101

40 1 0 040 1 0 1 040 1 1 0110111f1100110

50 1 0 150 1 1 0 050 1 1 1110110g1100111

60 1 1 061 0 0 0 160 1 0 1000000h1101000

70 1 1 171 0 0 1 070 1 0 0110110I1101001

81 0 0 081 0 1 0 081 1 0 0110110j1101010

91 0 0 191 1 0 0 091 1 0 1000000k1101011

101 1 1 1110110l1101100

111 1 1 0__________________________________m1101101

121 0 1 0100101111110n1101110

131 0 1 1o1101111

141 0 0 1p1111000

151 0 0 0q1110001

161 1 0 0 0

Lect 3

AiBiCiCi+1Si

00000A'B'

00101A'B

01001

01110AB

10001

10110

11010

11111

Binary SubtracterFullSub.X1D1B1D2...Y1B2X2Y2FullSub. B3FullSub.Dn-1Bn Bn-1FullSub.Xn-1Yn-1D(ifference) = X - YB is the borrow signalEquations generated similarly to full adderB0D0X0Y0


Another Way to Make a Subtracter(2s complement)FullAdderFullAdderFullAdderFullAdderC0 = 1A0B0B1A1B2A2B3A3B0B1B2B3S3C4S2C3S1C2S0S = A + (-B)-B is generated by inverting and adding 1adding 1C1inverting


Binary Arithmetic Comparison


Sign Magnitude

One's Complement

Two's Complement

Negative Number

Easiest to Understand Simple to Compute

Easy to Compute

Hardest to Compute

Zeroes

2 Zeroes

2 Zeroes

1 Zero

Largest Number

Same number of

+ and - Numbers

Same number of

+ and - Numbers

One Extra Negative Number

Logic Required

Requires Adder and Subtracter

Only Adder Required

Only Adder Required

Extra Logic to Identify Larger Operand, Compute Sign, etc.

Carry Wraps Around

-

Overflow Detection

Overflow: Carry from High Order Adder Bits

Overflow: Sign of Both Operands is the Same and Sign of Sum is Different

Overflow: Sign of Both Operands is the Same and Sign of Sum is Different

SummaryUnderstand the three main representation of binary integersSign-MagnitudeOnes ComplementTwos ComplementDesign binary adders and subtracters for each representation


number representation and binary arithmetic

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