1. Topic 3 Number Representations and Computer Arithmetics
Introduction to Computer Systems Engineering
(CPEG 323)
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2. Outline Introduction Number representation and 2’s complement
ALU design
Integer multiplication/division
Real Numbers
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3. Recap Review Steps in executing an instruction?
Decoded opcode
Fields of an instruction?
Performance?
Base of the operands?
ALU
ALU 32 a 32
result 32 b
What's up ahead:
Number representation
Arithmetic algorithms
Logic implementation
Instructions
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4. Reading Up to now, you should have read (or are reading) New material
Patterson and Hennessy, chapters 1-2 and appendix A
New material
Patterson and Hennessy, chapter 3
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5. 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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6. 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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7. 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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8. Unsigned Integers Why unsigned integers? How to represent?
Memory access, PC, SP, RA
In C, unsigned int
How to represent?
Number your fingers!
Radix number system
Decimal: 10 different symbols:
number=
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9. 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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10. Unsigned Integers(Radix number system, Cont.)
How to convert binary to octal and hexadecimal?
Example:
010100=(0 *22+1 *21+0*20)*23+(1*22+0*21+0*20)
=2*81+ 4 *80= 248
Hexadecimal:
B 6
Binary:
Octal:
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11. 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 >10” vs. “If ai+ bi >2”
Which one is more efficient for computer, decimal or binary?
Also, packed BCD
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12. Signed Integers
How to distinguish a negative number from a positive number ?
How to do so efficiently ?
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13. 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 ?)
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