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

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CS101 Lecture 06: Logic Gates + Binary Addition = The Adder

Review Logic Gates Adding Binary Numbers

Aaron Stevens (azs@bu.edu)

30 January 2013

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What You’ll Learn Today

• 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

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

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

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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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

Hint: work right to left. Carry Values 1 1 0 0 1 1 1 + 0 0 1 0 1 0 0 1 14

Binary: Add with Carry

Sum Operands

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An 8-bit Adder

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

http://www.youtube.com/watch?v=GcDshWmhF4A

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Combinational Circuits

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

Three inputs require eight rows to describe all possible input combinations (23 = 8):

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Binary Number Addition

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

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

The full adder takes 3 inputs:

• A, B, and a carry-in value

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The Full Adder

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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An 8-bit Adder

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

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

We can abstract away the 1-bit adders, And summarize with this diagram: Notice the inputs and outputs.

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Number Overflow

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

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

• Basic logic gates (AND, OR, XOR)
• Combination gates
• Half-Adder
• Full Adder
• Overflow

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Announcements And To Do

• 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)