Chapter 7
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Chapter 7. Arithmetic Operations and Circuits. 1. 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. Addition

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Chapter 7

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Chapter 7

Chapter 7

Arithmetic Operations and Circuits

1


7 1 binary arithmetic

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

Binary Arithmetic

  • Addition

    • General form A0 + B0 = 0 + Cout

      • Summation symbol ()

      • Carry-out (Cout)

6


Binary arithmetic1

Binary Arithmetic

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

7


Binary arithmetic2

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 A0B0 = R0 + Bout

    • Remainder is R0

    • Borrow is Bout

8


Chapter 7

9


Binary arithmetic3

Binary Arithmetic

  • Subtraction

    • When A0 borrows from its left, A0 increases by 210.

10


Binary arithmetic4

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 arithmetic5

Binary Arithmetic

  • Multiplication

    • Very similar to multiplying decimal numbers.

12


Binary arithmetic6

Binary Arithmetic

  • Division

    • The same as decimal division.

    • This process is illustrated in Example 7-4.

13


Chapter 7

Example 7-4

14


Chapter 7

Example 7-4 (Continued)

14


7 2 two s complement representation

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

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

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 representation1

Two’s-Complement Representation

18


Two s complement to decimal conversion

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

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

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 point1

Discussion Point

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

    19 + 27

    18 – 7

    21 – 13

    59 – 96

22


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