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Chapter 7

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Chapter 7

Arithmetic Operations and Circuits

1

- Addition
- When the sum exceeds 1, carry a 1 over to the next-more-significant column.
- 0 + 0 = 0 carry 0
- 0 + 1 = 1 carry 0
- 1 + 0 = 1 carry 0
- 1 + 1 = 0 carry 1

5

- Addition
- General form A0 + B0 = 0 + Cout
- Summation symbol ()
- Carry-out (Cout)

- General form A0 + B0 = 0 + Cout

6

- Carry-out is added to the next-more-significant column as a carry-in.

7

- Subtraction
- 0 0 = 0 borrow 0
- 0 1 = 1 borrow 1
- 1 0 = 1 borrow 0
- 1 1 = 0 borrow 0

- General form A0B0 = R0 + Bout
- Remainder is R0
- Borrow is Bout

8

9

- Subtraction
- When A0 borrows from its left, A0 increases by 210.

10

- Multiplication
- Multiply the 20 bit of the multiplier times the multiplicand.
- Multiply the 21 bit of the multiplier times the multiplicand. Shift the result one position to the left.
- Repeat step 2 for the 22 bit of the multiplier, and for all remaining bits.
- Take the sum of the partial products to get the final product.

11

- Multiplication
- Very similar to multiplying decimal numbers.

12

- Division
- The same as decimal division.
- This process is illustrated in Example 7-4.

13

Example 7-4

14

Example 7-4 (Continued)

14

- Both positive and negative numbers can be represented
- Binary subtraction is simplified
- Groups of eight
- Most significant bit (MSB) signifies positive or negative

15

- Sign bit
- 0 for positive
- 1 for negative

- Range of positive numbers (8-bit)
- 0000 0000 to 0111 1111 (0 to 127)
- Maximum positive number: 2N-1-1

- Range of negative numbers (8-bit)
- 1111 1111 to 1000 0000 (-1 to -128)
- Minimum negative number: -2N-1

16

- If a number is positive,
- the two’s complement number is the true binary equivalent of the decimal number.

- If a number is negative:
- Complement each bit (one’s complement)
- Add 1 to the one’s complement
- The sign bit will always end up a 1.

18

18

- If the number is positive (sign bit = 0), convert directly
- If the number is negative:
- Complement the entire two’s-complement number
- Add 1
- Do the regular b-to-d conversion to get the decimal numeric value
- Result will be a negative number

19

- Convert the following numbers to two’s-complement form:
3510

-3510

- Convert the following two’s-complement number to decimal:
1101 1101

20

- Addition
- Regular binary addition

- Subtraction
- Convert number to be subtracted to a negative two’s-complement number
- Regular binary addition
- Carry out of the MSB is ignored

21

- Add the following numbers using two’s complement arithmetic:
19 + 27

18 – 7

21 – 13

59 – 96

22