Chapter3-CompArithmetic.ppt

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Computer Architecture and

Organization

Chapter 3: Data Representation

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Decimal to Binary Conversion – Fixed Point Numbers

Converting the Fractional Part —The Multiplication Method

A binary fraction is represented in the general form:

b_–1 X 2 –1 + b

–2 X 2 –2 + b–3 X 2 –3 + . . .

If we multiply the fraction by 2, then we will obtain:

b_–1 + b_–2 X 2 –1 + b

–3 X 2 –2 + . . .

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Decimal to Binary Conversion – Fixed Point Numbers

Remainder Method:

Conversion of the

decimal number 1234 to binary by

successive halving, starting at the top and working downward. For example, 77

divided by 2 yields a quotient of 38 and a remainder of 1

Quotient Remainder 1234 617 308 77 38 19 154 2 4 9 1 0 1 0 1 1 0 0 0 1 0 1 0

1 0 0 1 1 0 1 0 0 1 0

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Decimal to Binary Conversion – Fixed Point Numbers

Converting the Fractional Part —The Multiplication Method

.375 X 2 = 0.75

.75 X 2 = 1.5

.5 X 2 = 1.0

(0.375)₁₀ = (.011)₂

Most significant bit

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Decimal to Binary Conversion –

Fixed Point Numbers

Converting the Fractional Part —

The Multiplication Method

Terminating Fraction

Mixed Fraction

Check by converting back to Decimal

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Binary to Decimal Conversion

Conversion of the binary number 101011011 to decimal by

101011011

1 X 20 = 1

1 X 23_{= 8}

1 X 24_{= 16}

0 X 25_{= 0}

1 X 26_{= 64}

0 X 27_{= 0}

1 X 28_{= 256}

1 X 21 = 2

0 X 22_{= 0}

347₁₀

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Binary, Octal, and Hexadecimal Radix Representations

Decimal (Base 10)

Decimal (Base 10) Binary (Base 2)Binary (Base 2) Octal (Base 8)Octal (Base 8) Hexadecimal (Base 16)Hexadecimal (Base 16) 0

0 00 00 00

1

1 11 11 11

2

2 1010 22 22

3

3 1111 33 33

4

4 100100 44 44

5

5 101101 55 55

6

6 110110 66 66

7

7 111111 77 77

8

8 10001000 1010 88

9

9 10011001 1111 99

10

10 10101010 1212 AA

11

11 10111011 1313 BB

12

12 11001100 1414 CC

13

13 11011101 1515 DD

14

14 11101110 1616 EE

15

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Representing Negative Numbers

Signed magnitude

One’s complement

Note that: Both Signed magnitude and one’s

complement have two representation for

zero: a plus zero and a minus zero

Two’s complement

Excess 2m – 1 – Examples: for an eight bit

number, 0 becomes 128 + 0 = 128, 1

becomes 1 + 128 = 129, -5 becomes -5 +

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Negative Binary Numbers – 8 bits

N N Decimal Decimal N N Binary Binary -N Signed -N Signed Magnitude

Magnitude -N 1’s -N 1’s

Complement

Complement -N 2’s

-N 2’s

Complement

Complement -N excess

-N excess

128

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Binary Arithmetic

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Binary Arithmetic

• Addition

– 1’s Complement – 2’s Complement • Subtraction

– Use addition – add the complement of the subtrahend to the minuend

– Both are complement form, just add them together. • Overflow

– If the addend and augend are of opposite signs, overflow error cannot occur. If they are of the same sign and the

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Floating Point Numbers

n = f X 10

e

Radix

Exponent

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Floating-Point Numbers

3.14 = 0.314 × 101 = 3.14 × 100 0.000001 = 0.1 × 10-5 = 1.0 × 10-6 1941 = 0.1941 × 104 = 1.941 × 103

Consider a number representation:

• A signed three-digit fraction range in magnitude from 0.1 ≤ |f| < 1 or zero and a signed two-digit exponent has the negative number range:

from -0.100 × 10-99 to -0.999 × 10+99 and positive number range

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Floating-Point Numbers

• Large negative number less than -0.999 × 1099

• Negative number between -0.999 × 1099 and -0.100 × 10-99

• Small negative number with magnitudes less than 0.100 × 10-99

• Zero

• Small positive number with magnitudes less than 0.100 × 10-99

• Positive numbers between 0.100 × 10-99 and 0.999 × 1099

• Large positive numbers greater than 0.999 × 10+99

Total of 358,201 distinct numbers (179,100 positive, 179,100 negative, and one 0) over the range between -0.999×1099 to 0.999×1099

What are these numbers?

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Floating-Point Numbers

• Overflow – the magnitude is too large to be represented by it’s positive exponent value (for both + and - numbers)

• Underflow – the magnitude is too small to be represented by it’s negative exponent value (for both + and - numbers) • In many cases the underflow error is less serious than the overflow error.

• Fraction (mantissa) represent the density between two real numbers. Increasing the fraction increases the accuracy. • Exponent represent the size (scope) of the number.

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IEEE 754 Floating-Point

• Single precision – 32 bits • Double precision – 64 bits • Extended precision – 80 bits • Starting from the leftmost bits

– Sign; Exponent, 8 for single (excess 127) and 11 for double (excess 1023); Fraction, 23 for single and 52 for double.

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Floating Point Operations

•

Addition/subtraction

–

Align the exponent

–

Add (subtract) the mantissa

–

Normalize the result

•

Multiplication (division)

–

Multiply (divide) the mantissa

–

Add the exponent (subtract)

–

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References

http://www.acs.ilstu.edu/faculty/cjong/Sprin

g2006/ITK225/ClassNotes

William Stallings, Computer Architecture

and Organization. 2000

Miles J. Murdocca and Vincent P. Heuring.

Principles of Computer Architecture. Class

Test Edition, 1999

http://www.acs.ilstu.edu/faculty/cjong/Spring2006/ITK225/ClassNotes