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Topic 3 Number Representations and Computer Arithmetics

This topic covers number representation, including signed and unsigned integers, BCD, fixed point, and floating point, and explores how they are manipulated in computer hardware.

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Topic 3 Number Representations and Computer Arithmetics

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  1. Topic 3Number Representations and Computer Arithmetics Introduction to Computer Systems Engineering (CPEG 323) cpeg323-08F\Topic3-05F

  2. Outline • Introduction • Number representation and 2’s complement • ALU design • Integer multiplication/division • Real Numbers cpeg323-08F\Topic3-05F

  3. Decoded opcode a ALU ALU 32 result b 32 32 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 cpeg323-08F\Topic3-05F

  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 cpeg323-08F\Topic3-05F

  5. It may get complicated: Numbers Representation • 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? cpeg323-08F\Topic3-05F

  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? cpeg323-08F\Topic3-05F

  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 cpeg323-08F\Topic3-05F

  8. 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= cpeg323-08F\Topic3-05F

  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? cpeg323-08F\Topic3-05F

  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 Binary: 000110010100100010110110 1 9 4 8 B 6 Hexadecimal: Octal: 0 6 2 4 4 2 6 6 cpeg323-08F\Topic3-05F

  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? cpeg323-08F\Topic3-05F

  12. Signed Integers • How to distinguish a negative number from a positive number ? • How to do so efficiently ? cpeg323-08F\Topic3-05F

  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 ?) cpeg323-08F\Topic3-05F

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