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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
- Subtract decimal numbers by nine’s compliment

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

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

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 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

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

- 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

Pad with leading 0’s for a specific word size

- Compliment each bit
- Complimenting: change 0 to 1, 1 to 0

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

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

- 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

- 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

- 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

10/1610 = .10102

- 7/1610 = -.01112

.1010

+.1001

(1).0011 = 3/1610

Binary Multiplication

- Simpler than decimal multiplication
- Only need to shift the multiplicand and add

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

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

- 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

- How the arithmetic operations are performed in binary
- Addition
- Subtraction
- Multiplication
- Division

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.

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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