W1017 Binary Adders

From Coder Merlin
Boy in sailor suit with blackboard with math

Prerequisites[edit]

Introduction[edit]

One of the most fundamental operations performed by computers, aside from the logical operations that we've already discussed, is the arithmetic operation of addition.

Half-Adder[edit]

Let's consider what's required to add two, single-bit binary integers. We'll need one bit to represent the sum of the integers, and another to handle the carry. Representing this in the form of a truth table yields:

Single-bit half-adder
Inputs Outputs
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0

This is formally termed a half-adder, a logic circuit capable of adding two bits.



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Observe, Ponder, and Journal Section 1
  1. What truth table do you recognize that produces the output of the Carry column?
  2. What truth table do you recognize that produces the output of the Sum column?

Full-Adder[edit]

In order to add two single-bit binary integers PLUS a carry, we need an adder capable of adding three single-bit binary numbers. Again, we'll need one bit to represent the sum of the integers, and another to handle the carry. Representing this in the form of a truth table yields:

Single-bit full-adder
Inputs Outputs
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1

This is formally termed a full-adder, a logic circuit capable of adding three bits.

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Observe, Ponder, and Journal Section 2
  1. What do you notice about the relationship between the first-half (top four rows) of the full-adder as compared to all of the rows of the half-adder?
  2. Why is this true?

Ripple Carry Adder[edit]

Four-bit Ripple Carry Adder

We've learned that a half-adder can add two bits and full-adder can add three bits. How can we add a multi-bit number such as a 16-bit word? By cascading four adders such that the carry output of the prior adder feeds the carry input of the subsequent adder we can add two four-bit numbers. This concept can be easily extended to an arbitrary number of bits.





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Observe, Ponder, and Journal Section 3
  1. Why does the least significant bit position use a half-adder rather than a full-adder?
  2. Assume that proper inputs are applied for all bits in numbers A and B. Will the correct output from S be available instantaneously? If not, why not?
  3. Assume that we have a standard (non-scientific calculator) capable of adding two 16-bit words. Two numbers, A and B, are added together. After the addition, it is noted that is high. What can we infer? What is this state commonly called?

Key Concepts[edit]

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Key Concepts
  • A half-adder is a logic circuit capable of adding two bits and output a carry bit and a sum bit.
  • A full-adder is a logic circuit capable of adding three bits and output a carry bit and a sum bit.
  • A ripple-carry-adder is a logic circuit constructed of adders, cascaded in such a manner that the carry output of each adder feeds the carry input of the subsequent adder. Using this method we are able to add an arbitrary number of bits.

Exercises[edit]

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Exercises
  •  J1017  Create a journal and answer all questions in this experience. Be sure to:
    • edit your journal using emacs within your ~/Journals directory
    • properly name your journal as J1017.txt
    • include all sections of the journal, properly formatted
    • push your changes to GitHub
    • properly tag your journal as J1017.Final
    • push your tag to GitHub

  • Construct your work using Falstad's Editor
  • Begin each exercises with a Blank Circuit
  • Label each circuit diagram (using Text) with:
    • Your name
    • The date
    • The name of the logic gate (e.g. "NOT")
    • Each output (e.g. "sum", "carry out")
  • Save each circuit using the Save As... option from the File submenu and then click on the link presented
  • Each file contains your work for the exercise. Create a new subdirectory, J1017, in your Journals directory. Upload each file to the J1017 directory via SFTP. Be sure to push the files to your GitHub repository.
  • You may use any of the following logic gates in your implementation: AND, OR, NOT, XOR, NAND, and NOR

  1. Construct a half-adder
  2. Construct a full-adder

  •  M1017-31  Complete  Merlin Mission Manager  Mission M1017-31

References[edit]

  • Adder (Wikipedia)
  • Schocken, Simon and Nisan, Noam. The Elements of Computing Systems. MIT Press, 2005.


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