Binary Arithmetic Adding Binary numbers Overflow conditions

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Presentation transcript:

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 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

Binary Addition Adding binary numbers employs the same procedures as adding decimal numbers: 00 +00 01 +00 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

11 +11 ??? 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 ???

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 ????????

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.

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.

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

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

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)