Computer Arithmetic: Understanding ALU, Integer Representation, and Floating Point Numbers, Slides of Microprocessor and Assembly Language Programming

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Chapter 9

Computer Arithmetic

Arithmetic & Logic Unit

• Does the calculations
• Everything else in the computer is there to service this unit
• Handles integers
• May handle floating point (real) numbers
• May be separate FPU (maths co-processor)
• May be on chip separate FPU (486DX +)

Integer Representation

• Only have 0 & 1 to represent everything
• Positive numbers stored in binary —e.g. 41=
• No minus sign
• No period
• Sign-Magnitude
• Two’s compliment

Sign-Magnitude

• Left most bit is sign bit
• 0 means positive
• 1 means negative
• +18 = 00010010
• -18 = 10010010
• Problems —Need to consider both sign and magnitude in arithmetic —Two representations of zero (+0 and -0)

Benefits

• One representation of zero
• Arithmetic works easily (see later)
• Negating is fairly easy —3 = 00000011 —Boolean complement gives 11111100 —Add 1 to LSB 11111101

Negation Special Case 1

• 0 = 00000000
• Bitwise not 11111111
• Add 1 to LSB +
• Result 1 00000000
• Overflow is ignored, so:
• • 0 = 0 

Range of Numbers

• 8 bit 2s compliment —+127 = 01111111 = 2^7 - — -128 = 10000000 = -2^7
• 16 bit 2s compliment —+32767 = 011111111 11111111 = 2^15 - 1 — -32768 = 100000000 00000000 = -2^15

Conversion Between Lengths

• Positive number pack with leading zeros
• +18 = 00010010
• +18 = 00000000 00010010
• Negative numbers pack with leading ones
• -18 = 10010010
• -18 = 11111111 10010010
• i.e. pack with MSB (sign bit)

Hardware for Addition and Subtraction

Multiplication

• Complex
• Work out partial product for each digit
• Take care with place value (column)
• Add partial products

Unsigned Binary Multiplication

Execution of Example

Multiplying Negative Numbers

• This does not work!
• Solution 1 —Convert to positive if required —Multiply as above —If signs were different, negate answer
• Solution 2 —Booth’s algorithm

Booth’s Algorithm