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Cosc 235: Computer Organization. Binary Arithmetic. Objectives. Add with binary numbers Represent negative binary numbers in one’s and two’s compliments Perform 3 methods of subtraction with binary numbers Multiply and divide with binary numbers

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objectives
Objectives
  • Add with binary numbers
  • Represent negative binary numbers in one’s and two’s compliments
  • Perform 3 methods of subtraction with binary numbers
  • Multiply and divide with binary numbers
  • Subtract decimal numbers by nine’s compliment
addition
Addition

Add 2 numbers, A and B, to get the sum, S.

A A3A2A1A0

+ B +B3B2B1B0

S C S3S2S1S0

Any base: add the digits corresponding to the same power

  • Carry
addition1
Addition
  • The sum of A0 and B0 produce 2 outputs:
    • Sum, S, the portion of A0 + B0 that will fit in the 0th position
    • Carry, C, the portion of A0 + B0 that will not fit in the 0th position and must be considered in the 1st position
truth table for binary addition
Truth Table for Binary Addition

A0 B0 C1 S0

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

non lsb addition
Non-LSB Addition
  • If we are not at the LSB of the numbers, we have an additional input: the carry from the previous digit
  • How many digits do we need to handle a 3-input binary addition?
truth table for binary addition1
Truth Table for Binary Addition

Ci Ai Bi Ci+1 Si

0 0 0 0 0

0 0 1 0 1

0 1 0 0 1

0 1 1 1 0

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 1 1 1 1

fractional addition
Fractional addition

0.01012 = 5/1610

0.01102 = 6/1610

--------- -------

0.10112 = 11/1610

storing negative numbers
Storing negative numbers
  • Sign + magnitude
  • For base n, we have (n-1)-complement
    • Representation scheme
    • Subtract each digit in the negative number from (n-1)
    • Example: 9’s compliment for base 10
    • Example: 1’s complement for base 2
  • Binary case: 2’s complement
nine s compliment
Nine’s Compliment
  • The Nine’s Compliment is sometimes used in operations with BCD
  • 9’s compliment is obtained by subtracting a decimal number from all nines

A = 764 A = 23.675

A = 999 A = 99.999

-764-23.675

235 76.324

one s compliment
One’s compliment

Pad with leading 0’s for a specific word size

  • Compliment each bit
    • Complimenting: change 0 to 1, 1 to 0
binary subtraction algorithm
Binary Subtraction Algorithm
  • Obtain the 1’s compliment of the subtrahend
  • Add
  • Take the MSB carry and add back into the results at the LSB
    • “End-around carry”
  • If EAC==0, B>A and result is negative
    • Compliment result to verify value

_

  • A - B => A + B + EAC
one s compliment example
One’s Compliment Example

A 13 1101

-B-10-1010

A 1101

+B +0101

(1)0010

+ 1 (EAC)

0011

fractional one s compliment
Fractional One’s Compliment

A = 28/32 = .11100

-X = 17/32 = .10001

A .11100

+X+ .01110

(1).01010

+ 1 (EAC)

.01011 = 11/32

two s compliment
Two’s Compliment
  • Adding the EAC is cumbersome
  • Skip this step: use a different storage structure
  • Two’s Compliment is the One’s compliment + 1
two s compliment methods for obtaining
Two’s Compliment: methods for obtaining
  • A = 2110 = 0101012

A1 = 101010

+ 1..

1010112 = A2

  • If X = next power of 2 > A, then

set all bits left of and including X.

right of X, set bits to X – A

  • Compliment only bits to the left of the least significant 1
subtraction using two s compliment
Subtraction using Two’s Compliment
  • If the end carry = 1, that indicates a positive result, and the end carry is ignored
  • If the end carry = 0, that indicates a negative result, and the end carry is ignored
    • Negative result is in Two’s Compliment form
    • To verify, take the 2’s compliment of the result
fractional two s compliment subtraction
Fractional Two’s Compliment Subtraction

10/1610 = .10102

- 7/1610 = -.01112

.1010

+.1001

(1).0011 = 3/1610

binary multiplication
Binary Multiplication
  • Simpler than decimal multiplication
  • Only need to shift the multiplicand and add
example binary multiplication
Example Binary multiplication

00101 multiplicand

x 00101 multiplier

-------

00101 multiply by 1

00000 shift and multiply by 0

+00101 shift and multiply by 1 (etc)

--------

0011001 = 24 + 23 + 20 = 25

another example
Another example

011010 multiplicand = 2610

x 001010 multiplier = 1010

--------

000000 multiply by 0

011010 shift & multiply by 1

000000 shift & multiply by 0

011010 shift & multiply by 1

----------

0100000100 28 + 22 = 26010

binary division
Binary division
  • Like multiplication, binary division is easier than decimal division
  • Our quotient bits can either be a 0 or a 1, not a multiple of the divisor
summary
Summary
  • How the arithmetic operations are performed in binary
  • Addition
  • Subtraction
  • Multiplication
  • Division
glossary
Glossary
  • One’s Compliment

The One’s compliment of a binary number is accomplished by converting all 1’s to 0 and all 0’s to 1

  • Two’s Compliment

One’s compliment + 1

  • Nine’s Compliment

Decimal number resulting from the subtraction of a number from all 9’s.

glossary1
Glossary
  • End Carry

A carry having significance greater than the MSB of either term in an addition

  • End Around Carry

End carry generated during one’s compliment subtraction is taken around and added to the LSB of the sum

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