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Binary Addition

Binary Addition. CSC 103 September 17, 2007. Recap: Binary Numbers. Physical representation Transistor Concept of “on” and “off” for physical manufacturing of computers  T/F… Abstract representation Logic: NOT, AND, OR Truth tables

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Binary Addition

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  1. Binary Addition CSC 103 September 17, 2007

  2. Recap: Binary Numbers • Physical representation • Transistor • Concept of “on” and “off” for physical manufacturing of computers  T/F… • Abstract representation • Logic: NOT, AND, OR • Truth tables • ANY Boolean expression can be built with transistors – wired as AND, OR or NOT

  3. Recap: Transistors = 0 = 1 = 0 or = 1 = 1 or = 0 = 1

  4. Logic Functions: NOT • The ‘NOT’ function • A A’ • 0 1 • 1 0

  5. Recap Logic Gate: AND Function (=1) 1 (=1) 0 0 0 1 0 0

  6. Logic Gate: OR Function 1 0 1 1 0 1 0

  7. Onto Addition and the Adder Circuit...

  8. Binary Addition • Add • 0 + 0 = • 0 + 1 = • 1 + 0 = • 1 + 1 = • Add these numbers c: 1000111 1011010 01001100111001 s:

  9. Binary Addition: Half Adder • We need a circuit to add two bits • Either bit can be ‘0’ or ‘1’ • The function in the truth table is • Sum = A’B + AB’  Exclusive-OR function • Carry = AB

  10. The Half-Adder and Exclusive OR Gate • A’B + AB’ = Exclusive OR • Typically abbreviated to XOR • Simulator uses EOR A B’ A’ B A B A B A B | S C 0 0 | 0 0 0 1 | 1 0 1 0 | 1 0 1 1 | 0 1

  11. Recap Logic Gates: Symbols AB, AB A+B A, A’ XOR

  12. Summary: The Half-Adder and Exclusive OR Gate • Exclusive OR • Typically abbreviated to XOR • Simulator uses EOR A B

  13. Binary Addition: Half Adder

  14. Half-Adder  Full-Adder

  15. The Full Adder • A full adder is a circuit with three inputs (including a ‘carry-in’) and two outputs (the sum and carry-out) • What is the third input? • Exercise: Add 111+ 101 (carry) 1 1 1 ( ‘A’ ) 1 0 1 ( ‘B’ ) (sum) • For adding two numbers, we need three inputs

  16. The Full Adder • Cascade two half-adders to get a full adder A B Cin

  17. HW: Cascade 2 Full Adders for a 2-Bit Adder A2A1 1 1 + B2B1+1 0 B2 A2 B1 A1 Full Adder Full Adder Cout2 Cin2 = Cout1 Cin1 S2 S1

  18. Summary • Binary addition • Concept of ‘sum’ and ‘carry’ • Half adder and full adder circuits • Cascading circuits to make larger ones

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