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.
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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.
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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.
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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.
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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
!
Operands !
!
!
1
0
+ 0
1
Hint: work right to left.
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1
0
0
0
1
1
0
!
1!
0 !
1!
Carry Values
Sum
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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
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A binary Adding Machine
Computer Science
http://www.youtube.com/watch?v=GcDshWmhF4A
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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.
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Combinational Circuits
Computer Science
Three inputs require eight rows
to describe all possible input
combinations (23 = 8):
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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?
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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.)
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Full Adder Circuit
Computer Science
The full adder takes 3 inputs:
  A, B, and a carry-in value
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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
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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
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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.
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An 8-bit Adder
Computer Science
We can abstract away the 1-bit adders,
And summarize with this diagram:
Notice the inputs and outputs.
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Number Overflow
Computer Science
If each value is stored using eight bits, consider
adding 250 + 9:
11111010!
!(8 bits)!
+ 00001001!
!(8 bits)
00000011!
!(8 bits)
How do we interpret the value 00000011?
250 + 9 = 3. WTF?
This is called overflow.
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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?
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What You Learned Today
Computer Science
  Basic logic gates (AND, OR, XOR)
  Combination gates
  Half-Adder
  Full Adder
  Overflow
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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)
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