Topic 3 Number Representations and Computer Arithmetics

1
Topic 3Number Representations and Computer
Arithmetics

• Introduction to Computer Systems Engineering
• (CPEG 323)

2
Outline

• Introduction
• Number representation and 2s complement
• ALU design
• Integer multiplication/division
• Real Numbers

3
Recap

• Review
• Steps in executing an instruction?

• Fields of an instruction?
• Performance?
• Base of the operands?

• What's up ahead
• Number representation
• Arithmetic algorithms
• Logic implementation
• Instructions

4
Reading

• Up to now, you should have read (or are reading)
• Patterson and Hennessy, chapters 1-2 and appendix
  A
• New material
• Patterson and Hennessy, chapter 3

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Numbers Representation

• It may get complicated

• How to represent negative numbers?

• What is the range of numbers?

• What if a number is out of the range?

• How about rational and irrational numbers?

• How does hardware add/sub/div/mul these numbers?

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Bits and Numbers

• Bits are just bits NO inherent meaning
• conventions define relationship between bits and
  numbers

• How does the hardware know which convention is to
  use?
• Add s1, s2, s3
• Addu s1, s2, s3
• What is the sign of s1, s2, and s3?

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Number Representations

• Unsigned integers
• Signed integers
• BCD (Binary Coded Decimal)
• Fixed point
• Floating point
• Other types of data
• Characters (ASCII, Unicode)
• Pixels (graphics)
• Groups of bits

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Unsigned Integers

• Why unsigned integers?
• Memory access, PC, SP, RA
• In C, unsigned int

• How to represent?
• Number your fingers!
• Radix number system
• Decimal 10 different symbols 0 1 2 3 4 5 6 7 8
  9
• number

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Unsigned Integers(Radix number system)

• In general, k-radix number system
• number

• How many different di?
• What is the biggest number?
• What is the smallest one?
• Special cases
• 2-radix (binary)
• 8 (octal)
• 16 (hexademical)
• How many different symbols do thy have?

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Unsigned Integers(Radix number system, Cont.)

• How to convert binary to octal and hexadecimal?
• Example
• 010100(0 221 21020)23(122021020)
• 281 4 80 248

1 9 4 8 B
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Hexadecimal

• Binary 000110010100100010110110

Octal
0 6 2 4 4 2 6 6
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Unsigned Integers(BCD binary coded decimal)

• Representation 12710
• How many bits necessary for BCD? For binary?
• Storage space used ?
• Addition
• How to determine a carry
• If ai bi gt10 vs. If ai bi gt2
• Which one is more efficient for computer, decimal
  or binary?

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Signed Integers

• How to distinguish a negative number from a
  positive number ?
• How to do so efficiently ?

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Signed Integers (Sign and magnitude)

• -12 -(sign) 12(absolute value)
• A separate sign bit
• A magnitude

• For hardware
• Where to put the sign bit? Right or Left?
• How to know the sign of addition? (e.g. one extra
  step ?)
• What is the sign of 0? (positive and negative 0
  ?)