Figure 1.1 Block diagram of a digital computer.

Functional Units

REPRESENTATION OF NUMBERS - POSITIONAL NUMBERS

Decimal Binary Octal Hexadecimal 00 0000 00 0 01 0001 01 1 02 0010 02 2 03 0011 03 3 04 0100 04 4 05 0101 05 5 06 0110 06 6 07 0111 07 7 08 1000 10 8 09 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F

Binary, octal, and hexadecimal conversion

1 0 1 0 1 1 1 1 0 1 1 0 0 0 1 11 2 7 5 4 3

A F 6 3

OctalBinaryHexa

CONVERSION OF BASES

Decimal to Base R number

Base R to Decimal Conversion

V(A) = akRk

A = an-1 an-2 an-3 … a0 . a-1 … a-m

(736.4)8 = 7 x 82 + 3 x 81 + 6 x 80 + 4 x 8-1

= 7 x 64 + 3 x 8 + 6 x 1 + 4/8 = (478.5)10(110110)2 = ... = (54)10

(110.111)2 = ... = (6.785)10

(F3)16 = ... = (243)10

(0.325)6 = ... = (0.578703703 .................)10

- Separate the number into its integer and fraction parts and convert each part separately.

- Convert integer part into the base R number

→ successive divisions by R and accumulation of the remainders.

- Convert fraction part into the base R number

→ successive multiplications by R and accumulation of integer digits

EXAMPLE

Convert 41.687510 to base 2.

Integer = 414120 110 0 5 0 2 1 1 0 0 1

Fraction = 0.68750.6875x 21.3750x 20.7500x 21.5000 x 21.0000

(41)10 = (101001)2 (0.6875)10 = (0.1011)2

(41.6875)10 = (101001.1011)2

Convert (63)10 to base 5: (223)5

Convert (1863)10 to base 8: (3507)8

Convert (0.63671875)10 to hexadecimal: (0.A3)16

Exercise

COMPLEMENT OF NUMBERS

Two types of complements for base R number system: - R's complement and (R-1)'s complement

The (R-1)'s Complement

Subtract each digit of a number from (R-1)

Example

- 9's complement of 83510 is 16410

- 1's complement of 10102 is 01012(bit by bit complement operation)

The R's Complement

Add 1 to the low-order digit of its (R-1)'s complement

Example

- 10's complement of 83510 is 16410 + 1 = 16510

- 2's complement of 10102 is 01012 + 1 = 01102

FIXED POINT NUMBERS

Binary Fixed-Point Representation

X = xnxn-1xn-2 ... x1x0. x-1x-2 ... x-m

Sign Bit(xn): 0 for positive - 1 for negative

Remaining Bits(xn-1xn-2 ... x1x0. x-1x-2 ... x-m)

Numbers: Fixed Point Numbers and Floating Point Numbers

SIGNED NUMBERS

Signed magnitude representation Signed 1's complement representation Signed 2's complement representation

Example: Represent +9 and -9 in 7 bit-binary number

Only one way to represent +9 ==> 0 001001 Three different ways to represent -9: In signed-magnitude: 1 001001 In signed-1's complement: 1 110110 In signed-2's complement: 1 110111

In general, in computers, fixed point numbers are represented either integer part only or fractional part only.

Need to be able to represent both positive and negative numbers

- Following 3 representations

ARITHMETIC ADDITION: SIGNED MAGNITUDE[1] Compare their signs[2] If two signs are the same , ADD the two magnitudes - Look out for an overflow[3] If not the same , compare the relative magnitudes of the numbers and then SUBTRACT the smaller from the larger --> need a subtractor to add[4] Determine the sign of the result

6 0110+) 9 1001 15 1111 -> 01111

9 1001- ) 6 0110 3 0011 -> 00011

9 1001 -) 6 0110 - 3 0011 -> 10011

6 0110+) 9 1001 -15 1111 -> 11111

6 + 9 -6 + 9

6 + (- 9) -6 + (-9)

Overflow 9 + 9 or (-9) + (-9) 9 1001+) 9 1001 (1)0010overflow

Fixed Point Representations

ARITHMETIC ADDITION: SIGNED 2’s COMPLEMENT

Example 6 0 0110 9 0 1001 15 0 1111

-6 1 1010 9 0 1001 3 0 0011

6 0 0110 -9 1 0111 -3 1 1101

-9 1 0111 -9 1 0111 -18 (1)0 1110

Add the two numbers, including their sign bit, and discard any carry out of leftmost (sign) bit - Look out for an overflow

overflow9 0 10019 0 1001+)

+) +)

+) +)

18 1 00102 operands have the same signand the result sign changes

xn-1yn-1s’n-1 + x’n-1y’n-1sn-1 = cn-1 cn

x’n-1y’n-1sn-1

(cn-1 cn)

xn-1yn s’n-1

(cn-1 cn)

Fixed Point Representations

ARITHMETIC ADDITION: SIGNED 1’s COMPLEMENT

Add the two numbers, including their sign bits. - If there is a carry out of the most significant (sign) bit, the result is incremented by 1 and the carry is discarded.

6 0 0110 -9 1 0110 -3 1 1100

-6 1 1001 9 0 1001 (1) 0(1)0010 1 3 0 0011

+) +)

+)

end-around carry

-9 1 0110-9 1 0110 (1)0 1100 1 0 1101

+)

+)

9 0 10019 0 1001 1 (1)0010

+)

overflow

Example

not overflow (cn-1 cn) = 0

(cn-1 cn)

Fixed Point Representations

FLOATING POINT NUMBER REPRESENTATION* The location of the fractional point is not fixed to a certain location* The range of the representable numbers is wide

F = EM

mn ekek-1 ... e0 mn-1mn-2 … m0 . m-1 … m-m

sign exponent mantissa

- Mantissa Signed fixed point number, either an integer or a fractional number

- Exponent Designates the position of the radix point

Decimal Value

V(F) = V(M) * RV(E) M: MantissaE: ExponentR: Radix

Floating Point Representation

FLOATING POINT NUMBERS

0 .1234567 0 04sign sign

mantissa exponent==> +.1234567 x 10+04

Example A binary number +1001.11 in 16-bit floating point number representation (6-bit exponent and 10-bit fractional mantissa)

0 0 00100 100111000

0 0 00101 010011100

Example

Note: In Floating Point Number representation, only Mantissa(M) and Exponent(E) are explicitly represented. The Radix(R) and the position of the Radix Point are implied.

Exponent MantissaSignor

Floating Point Representation

IEEE Standard for FLOATING POINT NUMBERS

• To represent IEEE standard for floating-point numbers is called IEEE floating-point. This standard specifies how single precision (32 bit) and double precision (64 bit) floating point numbers are to be represented.

• It is divided in two formats as follows:

1. Single Precision: The IEEE single precision floating point standard representation

requires a 32 bit word, which may be represented as numbered from 0 to 31, left to right. The first bit is the sign bit, S, the next eight bits are the exponent bits, 'E', and the final 23 bits are the fraction 'F':

• S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF

0 1 8 9 31

• The value V represented by the word may be determined as follows:

• If E=255 and F is nonzero, then V=NaN ("Not a number")

• If E=255 and F is zero and S is 1, then V=-Infinity

• If E=255 and F is zero and S is 0, then V=Infinity

• If 0<E<255 then V=(-1)**S * 2 ** (E-127) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point.

• If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-126) * (0.F) These are "unnormalized" values.

• If E=0 and F is zero and S is 1, then V=-0

• If E=0 and F is zero and S is 0, then V=0

• In particular,

• 0 00000000 00000000000000000000000 = 0

• 1 00000000 00000000000000000000000 = -0

• 0 11111111 00000000000000000000000 = Infinity

• 1 11111111 00000000000000000000000 = -Infinity

• 0 11111111 00000100000000000000000 = NaN

• 1 11111111 00100010001001010101010 = NaN

• 0 10000000 00000000000000000000000 = +1 * 2**(128-127) * 1.0 = 2

• 0 10000001 10100000000000000000000 = +1 * 2**(129-127) * 1.101 = 6.5

• 1 10000001 10100000000000000000000 = -1 * 2**(129-127) * 1.101 = -6.5

• 0 00000001 00000000000000000000000 = +1 * 2**(1-127) * 1.0 = 2**(-126)

• 0 00000000 10000000000000000000000 = +1 * 2**(-126) * 0.1 = 2**(-127)

• 0 00000000 00000000000000000000001 = +1 * 2**(-126) * 0.00000000000000000000001 = 2**(- 149) (Smallest positive value)

• Double Precision• The IEEE double precision floating point standard representation requires a 64 bit

word, which may be represented as numbered from 0 to 63, left to right. The first bit is the sign bit, S, the next eleven bits are the exponent bits, 'E', and the final 52 bits are the fraction 'F':

• S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF • 0 1 11 12 63• The value V represented by the word may be determined as follows:• If E=2047 and F is nonzero, then V=NaN ("Not a number") • If E=2047 and F is zero and S is 1, then V=-Infinity • If E=2047 and F is zero and S is 0, then V=Infinity • If 0<E<2047 then V=(-1)**S * 2 ** (E-1023) * (1.F) where "1.F" is intended to

represent the binary number created by prefixing F with an implicit leading 1 and a binary point.

• If E=0 and F is nonzero, then V=(-1)**S * 2 ** (-1022) * (0.F) These are "unnormalized" values.

• If E=0 and F is zero and S is 1, then V=-0 • If E=0 and F is zero and S is 0, then V=0

Example of Single Precision Format• Ex-1

Represent a binary positive number 1101011 in the IEEE aingle-precision format.

The binary positive number 1101011= + 1.101011 X 2^6

The 23-bit mantissa M= 0.101011 000000 000000 00000

The biased exponent E’= E+ 127 = 6+127=133 = 1000 0101

The sign bit S=0, since the number is positive. So 32-bit representation is

Ex-2

Represent the decimal number -0.75 in the IEEE single-precision format.

The decimal number -0.75= - 0.11 in binary= - 1.1X2^(-1)

The 23-bit mantissa M= 0.100000 000000 000000 00000

The biased exponent E’= E+127= -1+127=126=0111 1110

Since the number is negative, the sign bit S=1

0 1000 0101 101011 000000 000000 00000

0 0111 1110 100000 000000 000000 00000

17

Algorithms for Multiplication (Manual Method)

1101 Multiplicand M

X 1011 Multiplier Q

1101

1101

0000

1101____

10001111 Product P

Partial product

18

4) Algorithms for Multiplication

Multiplicand

Multiplier

Product

Sequential Multiplication Method for Unsigned Number

START

A=0; F=0, Q= MultiplierM=Multiplicand, size=n

Q[0]=1? A=A+M

Right Shift(FAQ)

Size=size-1

Size=0?

Product=AQ

End

Yes

No

F=flip-flop holds the end carry generated in the addition

Illustration of Sequential Multiplication Method Multiplier Q=14=1110 and multiplicand M=6=0110

Initial Config M F A Q Size

Step 1,Q[0] =0 0110 0 0000 0111 3

R.S(FAQ)

Step 2,Q[0] =1 0110 0 0110 0111

A=A+M, R.S(FAQ) 0110 0 0011 0011 2

Step3,Q[0] =1

A=A+M, R.S(FAQ) 0110 0 1001 0011 1

0110 0 0100 1001

Step4,Q[0] =1 0110 0 1010 1001 0

A=A+M, R.S(FAQ) 0110 0 0101 0100

Product= AQ= 0101 0100 =84

Booth’s Multiplication Method (for signed numbers) START

A=0; Q[-1]=0, Q= MultiplierM=Multiplicand, size=n

Q[0]=0? and Q[- 1]=1

A=A-M

Right Shift(AQ)

Size=size-1

Size=0?

Product=AQ

End

No

Q[0]=1 and Q[- 1]=0

A=A+M

Yes

Yes

Yes

No

No

Illustration of Booth’s Algorithm Multiplier Q=7=0111 and multiplicand M= --6 = 1010(as negetive results are automatically in 2’s

complement form). [A flip-flop (a fictitious bit position)is used to the right of lsb of the multiplier and it is initialized to 0]

Initial Config M A Q Size

Step 1,Q[0] =0 1010 0000 0111 0 4

R.S(FAQ)

Step 2,Q[0] =1,Q[-1] =0 1010 0110 0111 0

A=A-M, R.S(AQ) 1010 0011 0011 1 3

Step3,Q[0] =1

A=A+M, R.S(AQ) 1010 0001 1001 1 2

Step4,Q[0] =1 1010 0000 1100 1 1

A=A+M, R.S(AQ)

Step5,Q[0] =1 1010 1010 1100 1 0

A=A+M, R.S(AQ) 1010 1101 0110 0

Product= AQ= 1101 0110, it shows that the product is negative number so to get the number in familiar form we take 2’s complement of magnitude. The result is -42

Division Algorithm

START

A=0; F=0, Q= DividendM=Divisor, size=n

Sign of A=0?

A=A+M Q[0] =1

Size=size-1

Size=0?

quotient=Q , Remainder=A

End

No

Left shift(AQ)

A=A-M

Q[0]=0

No

Yes

Illustration of Restoring division Algorithmdividend Q=7=0111 and divisor M= 3 =0011

Initial Config M A Q Size

00011 00000 0111 4

Step 1

LS(AQ) 00011 00000 111- __

A=A-M 00011 11101 111- __

Sign of A=-ve

Set Q[0]=0 &

Restore A by (A+M) 00011 00000 1110 3

Step 2

LS(AQ) 00011 00001 110- __

A=A-M, 00011 11110 110- __

As sign of A=-ve

Set Q[0]=0

Restore A 00011 00001 1100 2

Step3

LS(AQ) 00011 00011 100- __ A=A-M, 00011 00000 100- __

As sign of A=+ve

Set Q[0]=0 00011 00000 1001 1

Restoring Division Algorithm

M A Q Size

Step4LS(AQ) 00011 00001 001- __ A=A-M, 00011 11110 001- __As sign of A=+veSet Q[0]=0 Restore A 00011 00001 0010 0

RESULT :

Quotient = Q = 0010

Remainder = A = 00001 = 1

Illustration of Non-Restoring division Algorithmdividend Q=7=0111 and divisor M= 3 =0011

Initial Config M A Q Size

00011 00000 0111 4

Step 1

As sign of A=+ve

LS(AQ) 00011 00000 111- __

A=A-M 00011 11101 111- __

As sign of A=-ve

Set Q[0]=0 00011 11101 1110 3

Step 2

As sign of A=-ve

LS(AQ) 00011 11011 110- __

A=A+M, 00011 11110 110- __

As sign of A=-ve

Set Q[0]=0

00011 00001 1100 2

Non-restoring Division Algorithm

M A Q Size

Step3

As sign of A=-ve

LS(AQ) 00011 11101 100- __ A=A+M, 00011 00000 100- __

As sign of A=+ve

Set Q[0]=1 00011 00000 1001 1

Step4

As sign of A=+veLS(AQ) 00011 00001 001- __ A=A-M, 00011 11110 001- __As sign of A=-veSet Q[0]=0 00011 00001 0010 0

RESULT : Quotient = Q = 0010Remainder = A = 00001 = 1