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Addition and Subtraction

Addition and Subtraction. Outline. Arithmetic Operations (Section 1.2) Addition Subtraction Multiplication Complements (Section 1.5) 1’s complement 2’s complement Signed Binary Numbers (Section 1.6) 2’s complement Addition Subtraction. Addition. 101101+100111 Rules

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Addition and Subtraction

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  1. Addition and Subtraction

  2. Outline • Arithmetic Operations (Section 1.2) • Addition • Subtraction • Multiplication • Complements (Section 1.5) • 1’s complement • 2’s complement • Signed Binary Numbers (Section 1.6) • 2’s complement • Addition • Subtraction

  3. Addition • 101101+100111 • Rules • Any carry obtained in a given significant position is used by the pair of digits one significant position higher

  4. Subtraction • 101101-100111 • The borrow in a given significant position adds 2 to a minuend digit

  5. Multiplication • 1011 X 101

  6. Complement • 1’s complement • 2’s complement

  7. 1’s complement • Rule: 1’s complement of a binary number is formed by changing • 1’s to 0’s • 0’s to 1’s • 1011000→0100111 • 0101101 →

  8. 2’s Complement • Alternative Method • Write the 1’s complement • Add 000…1 to 1’s complement • Example • 1101100 • 0010011 (1’s complement) • 0010100 (2’s complement)

  9. Unsigned Subtraction • X-Y • Determine Y’s 2’s complement • X+(2’s complement of Y) • If X is larger or equal to Y, an end carry will result. Discard the end carry. • If X is less than Y, no end carry will result. To obtain the answer in a familiar form, take the 2’s complement of the sum and place a negative sign in front.

  10. Subtraction of Unsigned Number • Example 1.7 (2’s complement) • X=1010100 • Y=1000011 • X-Y (Discard end carry) • Y-X (No end carry)

  11. Signed Binary Number • Signed-magnitude representation • Signed 1’s complement representation • Signed 2’s complement representation

  12. Signed Magnitude Representation • Rules • Represent the sign in the leftmost position • 0 for positive • 1 for negative • Example • 01001↔(+)9 • 11001↔(-)9 • Used in ordinary arithmetic, but not in computer arithmetic

  13. Interpretation • The user determines whether the number is signed or unsigned • 01001 • 9 (unsigned binary) • +9 (signed binary) • 11001 • 25 (unsigned binary) • -9 (signed binary)

  14. Signed Complement System • A signed complement system negates a number by taking its complement • Example • 00001001 (9) • 11110110 (-)9 in signed 1’s complement • 11110111 (-)9 in signed 2’s complement • Usage: • 1’s complement: Seldom used • 2’s complement: Most common

  15. Arithmetic Addition • The rule for adding signed numbers in 2’s complement form is obtained from addition of two numbers • A carry out of the sign bit is discarded • In order to obtain correct answer, we must ensure that the result has a sufficient number of bits to accommodate the sum • Useful Facts • Positive numbers have 0 in the leftmost bit • Negative numbers have a 1 in the leftmost bit

  16. Negative Number • Determine the value of a negative number in signed 2’s complement by converting the number to a positive number to place it in a more familiar form • 11111001 is negative because the left most bit is 1. • 2’s complement: 00000111 (+7) • Therefore, 11111001 is -7

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