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Binary Arithmetic

Binary Arithmetic. Adding Binary numbers Overflow conditions How does it all work? AND, OR and NOT. Decimal Addition. You are probably already familiar with adding decimal numbers. 7 +2 9. 5 +1 6. 2 +3 5.

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Binary Arithmetic

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  1. Binary Arithmetic • Adding Binary numbers • Overflow conditions • How does it all work? • AND, OR and NOT

  2. Decimal Addition • You are probably already familiar with adding decimal numbers 7 +2 9 5 +1 6 2 +3 5 • You also know that when the addition of two numbers exceeds the base, a value is “carried” over to the next column 1 Carry the “1” 7 + 5 1 2

  3. Binary Addition • Adding binary numbers employs the same procedures as adding decimal numbers: 00 +00 00 01 +00 01 00 +01 01 • As with decimal addition, when the addition of two numbers exceeds the base value, the value is “carried” over to the next column 1 Carry the “1” 01 +01 10

  4. Binary Addition Examples 01001100 01101010 10110110 11001100 00110011 11111111 01111111 00000001 10000000 • What happens when we have the following case? 1 Carry the “1” 11 +11 ???

  5. More Binary Addition Examples 01111111 01000001 11000000 10001100 00111011 11000111 01111111 01001001 11001000 • What happens when we get the following case? (note: assume that we are limited to 8 bits) 11111111 00000001 ????????

  6. Overflow • In the following case, we have a condition called “overflow”. The result may be somewhat unexpected 1 Carry the “1” 11111111 00000001 00000000 • When adding two numbers, it is possible that the result will not fit within the space we have provided. The addition completes, but part of the value is lost. • Luckily, overflow is often considered to be an error condition, so the program “knows” that this has happened and can take appropriate action.

  7. How does it all work? • In the first week of lectures, we discussed vacuum tubes, relays, and transistors • As we went through the discussion, I asked, “How could you add two numbers using switches?” • I didn’t expect an answer, but I did show how switches could be set up to implement the AND function and the OR function.

  8. AND, OR and NOT • You may recall the “truth” tables for the AND and OR functions: A A A 0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 B B AND OR NOT (~) • The AND, OR and NOT functions are the basic functions for “Boolean” Algebra

  9. Binary Arithmetic/AND and OR • All binary arithmetic can be represented using boolean algebra • Each 1-bit adder has 3 inputs: A, B and CarryIn • Each 1-bit adder has 2 output: C and CarryOut • CIN 0 0 0 0 1 1 1 1 • A 0 0 1 1 0 0 1 1 • B 0 1 0 1 0 1 0 1 • C 0 1 1 0 1 0 0 1 • COUT 0 0 0 1 0 1 1 1

  10. Boolean Equations for Addition CIN 0 0 0 0 1 1 1 1 A 0 0 1 1 0 0 1 1 B 0 1 0 1 0 1 0 1 C 0 1 1 0 1 0 0 1 COUT 0 0 0 1 0 1 1 1 C = (~A and B and ~CIN) or (A and ~B and ~CIN) or (~A and ~B and CIN) or (A and B and CIN) COUT= (A and B) or (A and CIN) or (B and CIN)

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