AComputer Arithmatic

8/13/2019 AComputer Arithmatic

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Computer Arithmetic Pada

Computer Organization

Computer ArithmeticRepresentasi bilangan padacomputer adalah :

1. Sign and magnitude2. Ones Complement 3. Twos Complement


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ALU Inputs and Outputs


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Integer Representation

Operasi menggunakan sirkuit logika dengandua nilai sinyal listrik yang di representasikan

dengan 0 & 1 .Positive numbers stored in binarye.g. 41=00101001

No minus signNo periodSign-MagnitudeTwos compliment


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Bilangan Binary Digit

Tuliskan bilangan binary sebagai berikut : 1 = 0001 7 = 14 =

2 = 0010 8 = 15 = 3 = 0011 9 = 16 =10000 4 = 0100 10 = 17 =

5 = 11 = 18 = 6 = 13 = 19 = 10011


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Penambahan Bilangan Binary

5 = 0101 4 = 6 =3 = 0011 5 = 2 = ---------+ ----------------+ ----------------+

15247 bil Desimal di konversi Hexa :15247 : 16 = 952 sisanya 15 = F 952 : 16 = 59 sisanya 8 = 8 59 : 16 = 3 sisanya 11 = B 3 : 16 = 0 sisanya 3 = 3Jadi (15247 ) Des = (3B8F) Hex


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Konversi Desimal ke Octal ke biner

Desimal Octal Biner 0 0 000 1 1 001

2 2 010 3 3 4 4 5 5 6 6 7 7


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Konversi Desimal ke Hexa ke Biner

Desimal Hexa Biner 0 0 0000


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Konversi Biner ke Octal

1 101 000 101 = 1 5 0 5 oct


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Konversi Biner ke Hexa

1 1101 1100 0101 1001 = 1 D C 5 9


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Representasi Binary Code Decimal (BCD)

Desimal Biner BCD 0 0000 0000 1 0001 0001


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Konversi Desimal ke BCD

(3729)Dec =


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Sign-Magnitude

Bit paling kiri sign bit0 means positive

1 means negative+18 = 00010010 -18 = 10010010

ProblemsNeed to consider both sign and magnitude inarithmeticTwo representations of zero (+0 and -0)


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Pada Sistem Sign Magnitude denganBilangan 4 Bit

Nilai negatif dinyatakan dengan mengubah Most Significant bit dari0 ke 1. Misalnya : + 5 adalah 0101 dan -5 adalah 1101 danseterusnya .BilaDalan Representasi ones coplement , nilai negatif diperolehdengan menggantikan tiap bit pada ngan positif yang sesuai.Bilangan ones complement : +7 = 0111 +0 = 0000 -1 = 1110+6 = 0110 -7 = 1000 -0 = 1111+5 = 0101 -6 = 1001+4 = 0100 -5 = 1010+3 = 0011 -4 = 1011+2 = 0010 -3 = 1100+1 = 0001 -2 = 1101


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Twos Compliment

+3 = 00000011 +4 = 0100 -7 = 1001+2 = 00000010 +5 = 0101 -8 = 1000

+1 = 00000001 +6 = 0110+0 = 00000000 +7 = 0111 -1 = 11111111 -4 = 1100

-2 = 11111110 -5 = 1011 -3 = 11111101 -6 = 1010


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Benefits

One representation of zero Arithmetic works easily (see later)

Negating is fairly easy3 = 00000011Boolean complement gives 11111100

Add 1 to LSB 11111101


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Geometric Depiction of TwosComplement Integers


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Negation Special Case 2

-128 = 10000000bitwise not 01111111

Add 1 to LSB +1Result 10000000So:

-(-128) = -128 XMonitor MSB (sign bit)It should change during negation


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Range of Numbers

8 bit 2s compliment+127 = 01111111 = 2 7 -1 -128 = 10000000 = -2 7

16 bit 2s compliment+32767 = 011111111 11111111 = 2 15 - 1 -32768 = 100000000 00000000 = -2 15


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Conversion Between Lengths

Positive number pack with leading zeros+18 = 00010010

+18 = 00000000 00010010Negative numbers pack with leading ones-18 = 10010010

-18 = 11111111 10010010i.e. pack with MSB (sign bit)


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Addition and Subtraction

Normal binary additionMonitor sign bit for overflow

Take twos compliment of substahend and addto minuend

i.e. a - b = a + (-b)

So we only need addition and complementcircuits


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Multiplication

ComplexWork out partial product for each digit

Take care with place value (column) Add partial products


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Multiplication Example

1011 Multiplicand (11 dec) x 1101 Multiplier (13 dec)

1011 Partial products 0000 Note: if multiplier bit is 1 copy 1011 multiplicand (place value)

1011 otherwise zero 10001111 Product (143 dec) Note: need double length result


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Unsigned Binary Multiplication


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Execution of Example


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Flowchart for Unsigned BinaryMultiplication


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Multiplying Negative Numbers

This does not work!Solution 1

Convert to positive if requiredMultiply as aboveIf signs were different, negate answer

Solution 2Booths algorithm


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Booths Algorithm


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Example of Booths Algorithm


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Real Numbers

Numbers with fractionsCould be done in pure binary

1001.1010 = 24

+ 20

+2-1

+ 2-3

=9.625Where is the binary point?Fixed?

Very limited

Moving?How do you show where it is?


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Floating Point

+/- .significand x 2 exponent MisnomerPoint is actually fixed between sign bit and body

of mantissaExponent indicates place value (point position)

S i g

n b i t

BiasedExponent

Significand or Mantissa


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Floating Point Examples


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Signs for Floating Point

Mantissa is stored in 2s complimentExponent is in excess or biased notation

e.g. Excess (bias) 128 means8 bit exponent fieldPure value range 0-255Subtract 128 to get correct value

Range -128 to +127


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Normalization

FP numbers are usually normalizedi.e. exponent is adjusted so that leading bit

(MSB) of mantissa is 1Since it is always 1 there is no need to store it(c.f. Scientific notation where numbers arenormalized to give a single digit before thedecimal pointe.g. 3.123 x 10 3)


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FP Ranges

For a 32 bit number8 bit exponent+/- 2 256 1.5 x 10 77

AccuracyThe effect of changing lsb of mantissa23 bit mantissa 2 -23 1.2 x 10 -7

About 6 decimal places


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Expressible Numbers


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IEEE 754

Standard for floating point storage32 and 64 bit standards

8 and 11 bit exponent respectivelyExtended formats (both mantissa and exponent)for intermediate results


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FP Arithmetic +/-

Check for zeros Align significands (adjusting exponents)

Add or subtract significandsNormalize result


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FP Arithmetic x/

Check for zero Add/subtract exponents

Multiply/divide significands (watch sign)NormalizeRound

All intermediate results should be in doublelength storage

Floating


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FloatingPointMultiplication

Floating


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FloatingPointDivision


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Required Reading

IEEE 754 on IEEE Web site

Documents

AComputer Arithmatic