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7-1 Binary Arithmetic

- 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

Binary Arithmetic

- 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

Binary Arithmetic

- 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

Binary Arithmetic

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

13

7-2 Two’s-Complement Representation

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

15

Two’s-Complement Representation

- 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

Decimal-to-Two’s-Complement Conversion

- 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

Two’s-Complement-to-Decimal Conversion

- 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

Discussion Point

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

3510

-3510

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

1101 1101

20

7-3 Two’s-Complement Arithmetic

- 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

Discussion Point

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

19 + 27

18 – 7

21 – 13

59 – 96

22

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