CS101 Lecture 06: Logic Gates + Binary Addition = The Adder Review - PDF document

1/30/13 CS101 Lecture 06: Logic Gates + Binary Addition = The Adder Review Logic Gates Adding Binary Numbers Aaron Stevens (azs@bu.edu) 30 January 2013 Computer Science What You ’ ll Learn Today Computer Science  Review: logic gates  Review: How do we add in binary?  Combining logic gates in a single circuit.  Logic gates + binary addition = digital computing Why should you care? This is what a computer actually does! WTF? Exactly. 2 1

1/30/13 Review: AND Gate Computer Science The AND gate accepts two input signals. The output is 1 if and only if both inputs are 1. Otherwise the output is 0. Just like wiring in series. 3 Review: OR Gate Computer Science The OR gate accepts two input signals. The output is 1 if either input is 1. The output is 0 if both inputs are 0. Just like wiring in parallel. 4 2

1/30/13 Review: XOR Gate Computer Science An XOR gate accepts two input signals. When the 2 inputs differ, the output is 1 When both inputs are the same, the output is 0 Just like wiring with a 3-way switch. 5 Binary: Add with Carry Computer Science 0 + 0 is a sum of 0 with a carry of 0 1 + 0 is a sum of 1 with a carry of 0 0 + 1 is a sum of 1 with a carry of 0 1 + 1 is a sum of 0 with a carry of 1 Carry Values � 1 1 0 � � 0 1 1 1 � Operands � + 0 0 1 0 � � 1 0 0 1 � Sum Hint: work right to left. 6 14 3

1/30/13 An 8-bit Adder Computer Science Consider this 8-bit adder: Notice how the carry out from one bit’s adder becomes the carry-in to the next adder. http://faculty.mc3.edu/pvetere/Applets/APPLETS/BINARYADD/applet_frame.htm 7 A binary Adding Machine Computer Science http://www.youtube.com/watch?v=GcDshWmhF4A 4

1/30/13 Combinational Circuits Computer Science Combines some basic gates (AND, OR, XOR, NOT) into a more complex circuit.  Outputs from one circuit flow into the inputs of another circuit.  The input values explicitly determine the output values. 9 Combinational Circuits Computer Science Three inputs require eight rows to describe all possible input combinations (2 3 = 8): 10 5

1/30/13 Binary Number Addition Computer Science Look closely at the values for Sum and Carry… Do they look like any of the gates we’ve seen? 11 A Circuit for Binary Addition Computer Science Sum = A XOR B Carry = A AND B This circuit is called a half-adder. (It doesn’t take a carry-in.) 12 6

1/30/13 Full Adder Circuit Computer Science The full adder takes 3 inputs:  A, B, and a carry-in value 13 The Full Adder Computer Science Here is the Full Adder, with its internal details hidden (an abstraction). What matters now are:  inputs are A, B, and CI.  outputs are S and CO 14 7

1/30/13 An 8-bit Adder Computer Science Consider this 8-bit adder: Notice how the carry out from one bit’s adder becomes the carry-in to the next adder. http://faculty.mc3.edu/pvetere/Applets/APPLETS/BINARYADD/applet_frame.htm 15 An 8-bit Adder Computer Science To add two 8-bit numbers together, we need an 8-bit adder: Notice how the carry out from one bit’s adder becomes the carry-in to the next adder. 16 8

1/30/13 An 8-bit Adder Computer Science We can abstract away the 1-bit adders, And summarize with this diagram: Notice the inputs and outputs. 17 Number Overflow Computer Science If each value is stored using eight bits, consider adding 250 + 9: � (8 bits) � 11111010 � � (8 bits) + 00001001 � 00000011 � � (8 bits) How do we interpret the value 00000011 ? 250 + 9 = 3. WTF? This is called overflow. 18 9

1/30/13 Number Overflow Computer Science Adding two 8-bit numbers can result in an 9 bit result.  More generally, adding 2 n-bit numbers can give a n+1 bit result. Can we trust the sum? What does one do about overflow? 19 What You Learned Today Computer Science  Basic logic gates (AND, OR, XOR)  Combination gates  Half-Adder  Full Adder  Overflow 20 10

1/30/13 Announcements And To Do Computer Science  Readings:  Reed chapter 7, pp121-131  Wikipedia: http://en.wikipedia.org/wiki/VLSI (introductory remarks about gates on integrated circuits)  HW02 due tonight  HW03 due next week  Quiz 1 on Friday 2/1  Covers material from lectures 1-4 (i.e., not including this week) 21 11