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- 1. Digital Electronics Electronics Technology Landon Johnson Arithmetic Circuits
- 2. Arithmetic Competencies 18. Given an 8-bit binary signed number, state whether the number is positive or negative with 100% accuracy. 19. Given an 8-bit binary number, state the 1’s compliment of that number with 100% accuracy. 20. Given an 8-bit signed binary number, state the 2’s compliment of the number with 100% accuracy. 21. Given 2 decimal numbers, use two’s complement and show the steps to solve for the sum, and show the sum in binary and decimal with 100% accuracy. 22. Given 2 decimal numbers, use two’s complement and show the steps involved to solve for the difference, and show the difference in binary and decimal with 100% accuracy. 37. Without reference the student will explain the difference between a half adder and a full adder with 100% accuracy. 38. Without reference the student will draw a schematic showing how an eight-bit adder can be made using two four bit adders with 100% accuracy.
- 3. Binary Addition •In decimal, when we add two numbers and they exceed the place value of a digit, we carry over. 1 9 10 + 3 10 12 10 •The same thing works in binary 1 12 + 12 10 2
- 4. Binary Addition The four possible combinations of adding two binary numbers can be stated as follows: 0 0 + + 0 1 = = 0 1 carry carry 0 0 1 + 0 = 1 carry 0 1 + 1 = 0 carry 1
- 5. Binary Addition A B SUM CARRY 0 0 0 0 0 1 1 0 1 0 1 1 0 CARRY 0 1 SUM 1 A B HALF ADDER- used for LSB, adds two input numbers outputs sum and carry.
- 6. TEST Perform the following decimal additions. Convert the original decimal numbers to binary and add them. (A) 5 + 2 (B) 8 + 3 (C) 18 + 2 (D) 147 +75 (E) 31 + 7 We represent all binary numbers in groups of 8 because it’s the standard for most computers. 5 + 2 0 + 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 7 0 0 0 0 0 1 1 1 = 7 10
- 7. Two’s Complement Representation The most widely used method of representing binary numbers and performing arithmetic in computer systems. Both positive and negative numbers can be represented in the same format and binary subtraction is greatly simplified. Two’s complement uses the most significant bit (MSB) of the 8bit number to signify whether the number is positive or negative. The MSB called the sign bit and is defined as 0 for positive numbers and 1 for negative numbers. D7 D6 D5 D4 D3 D2 D1D0 SIGN BIT
- 8. Two’s Complement Representation A table of two’s-complement numbers can be developed by starting with some positive number and continuously subtracting 1
- 9. STEPS FOR DECIMAL-TO-TWO’S-COMPLEMENT CONVERSION 1. If the decimal number is positive, the two’s-complement number is the true binary equivalent of the decimal number. +18 = 0001 0010 2. If the decimal number is negative, the two’s-complement number is found by: the (a) Complementing each bit of the true binary equivalent of decimal number ( one’s complement ). (b) Adding 1 to the one’s complement number to get the magnitude bits. The sign bit will always be 1.
- 10. DECIMAL-TO-TWO’S-COMPLEMENT CONVERSION EXAMPLE Convert +35 to two’s complement SOLUTION: True Binary = 0010 0011 Two’s complement = 0010 0011
- 11. DECIMAL-TO-TWO’S-COMPLEMENT CONVERSION EXAMPLE Convert -35 to two’s complement SOLUTION: True Binary = 0010 0011 One’s complement = 1101 1100 Add 1 Two’s complement = 1101 1101 +1
- 12. STEPS FOR TWO’S-COMPLEMENT-TO-DECIMAL CONVERSION 1. If the two’s-complement number is positive (SIGN BIT = 0), do a regular binary-to-decimal conversion. 2. If the two’s-complement number is negative (SIGN BIT = 1), the decimal sign will be minus and the decimal number is found by: bit. (A) Complementing the entire two’s complement number, bit by (B) Adding 1 to arrive at the true binary equivalent. (C) Doing a regular binary-to-decimal conversion.
- 13. STEPS FOR TWO’S-COMPLEMENT-TO-DECIMAL CONVERSION EXAMPLE Convert 1101 1101 two’s complement back to decimal. SOLUTION: The sign bit is 1 so decimal result will be negative. Two’s complement = 1101 1101 Complement = 0010 0010 Add 1 True binary +1 = 0010 0011 Decimal equivalent = -35 Answer
- 14. Two’s Complement Arithmetic •All basic arithmetic functions involving positive and negative numbers can be dealt with simply by using 2’s complement. •Subtraction is done by adding the 2’s complement numbers. •Adding in 2’s complement, do regular binary addition. •Subtraction 2’s complement numbers, convert the number to be subtracted to a negative 2’s complement number and perform regular binary addition (5 - 3 = 5 + (-3). If the result is negative, the sign bit will be a 1.
- 15. Two’s Complement Addition EXAMPLE Add 19 + 27 using 8-bit two’s complement arithmetic SOLUTION: 19 = 0001 0011 + 27 = 0001 1011 Sum = 0010 1110 = 46 decimal
- 16. Two’s Complement Subtraction EXAMPLE Subtract 18 - 7 using 8-bit two’s complement arithmetic. 18 - 7 is the same as 18 + (-7), so add 18 plus negative 7. SOLUTION: Convert -7 to two’s complement True Binary +18 = 0001 0010 = 0000 0111 -7 = 1111 1000 One’s complement = 1111 1000 Add 1 +1 Sum = 0000 1011 = 11 Two’s complement = 1111 1001 Note: The carry-out of MSB is ignored. It will always occur for positive sums
- 17. PRACTICE Covert the following decimal numbers to two’s complement form and perform the operation indicated. 5 (a) + (e) - 7 - 28 + 38 12 (b) - + 6 125 (f) 66 32 (c) - - 18 36 (g) 48 32 (d) (h) 18 - 36 48
- 18. Binary Addition A B SUM CARRY 0 0 0 0 0 1 1 0 1 0 1 1 0 CARRY 0 1 SUM 1 A B HALF ADDER- used for LSB, adds two input numbers outputs sum and carry.
- 19. HALF ADDER • Logic device that adds two binary numbers • Only adds Least Significant Digit (LSD) column (1s column) in binary addition Input Logic Symbol: Logic Diagram: A B Output Half Adder Σ (sum) C0 (carry out)
- 20. FULL ADDER Used for adding binary place values other than the 1s place Input Logic Symbol: Logic Diagram: Cin A B Output Full Adder Σ (sum) C0 (carry out)
- 21. FULL ADDER FULL ADDER A B C IN SUM 0 0 0 0 0 1 0 1 C OUT 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 C OUT A B C IN SUM
- 22. PARALLEL ADDING • Use half adder for LSD • Use full adder for other digits A2 A1 A0 + B2 B1 B0
- 23. En to ter be bin ad ary de nu PARALLEL ADDER mb d h ere ers 0 00 1 + 1 0 01 1 0 1 1 1 0 + 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 SUM appears here Parallel adders are availablehalf-adderform. in IC 1s place uses 2s, 4s, 8s places use full adders
- 24. 8-BIT ADDER
- 25. WHAT IS THIS ? B A 0 0 4321 4321 B4 B3 B2 B1 74LS83A A4 A3 A2 A1 s4 B4 s3 B3 s2 B2 s1 B1 Cin Cout A 0 0 1 1 B 0 1 0 1 out 0 1 1 0
- 26. TEST 19. Given an 8-bit signed number state whether the number is positive or negative. 1001 1111 negative 20. Given an 8-bit signed number state the one’s complement of the number. 1001 1111 0110 0000 21. Given an 8-bit signed number state the two’s complement of the number. 1001 1111 0110 0000 +1 0110 0001
- 27. TEST 22. Given two decimal numbers, use two’s complement form and show the steps to solve for the sum, and show the sum in binary and decimal. 100 0110 0100 26 0001 1010 + 0111 1110 = 126 23. Given two decimal numbers, use two’s complement form and show the steps to solve for the difference, and show the difference in binary and decimal. - 78 32 0010 0000 0100 1110 1101 1111 1110 0000 +1 1110 0000 0010 1110 = 46

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