Homework on  Binary Signals and Logic Devices (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material.

The first part of this assignment looks at the requirements for adding two 8-digit decimal numbers. As you do the assignment, remember that the circuitry for adding two numbers is only an infinitesimal portion of modern computers.

1. How many binary digits are needed to express an arbitrary 8-digit decimal number?  (If you are to express an arbirtrary number, you must have enough bits to express the largest possible 8-digit decimal number).
The largest 8-digit decimal number is 99,999,999. This number is smaller than 2²⁷ so it can be expressed as a sum of 2²⁶ and smaller powers. This then means 27 binary digits are required (including a digit  for 2⁰).

2. How many full-adders and half-adders are needed to add two of these numbers?
When adding numbers all but the first digit require the use of a carry digit. Half-adders give the sum of two numbers and output a carry digit while full-adders add the sum of two numbers, taking into account a carry digit and output the same as a half-adder. Therefore we require1 half-adder for the first bit and 26 full-adders for the remaining digits.

3. How many XOR gates, how many AND gates, and how many OR gates would this take?

26(2 XOR) + 1 XOR

26(2 AND) + 1 AND

26 OR

for a total of 132 gates

4. If current processing techniques can fit about 5 million gates per cm² on a chip, how much space does the addition of two 8-digit decimal numbers occupy?

5.  If a chip is 1 cm², how many additions of 8-digit decimal numbers could it perform simultaneously?
If we divide 1 cm² by the number of cm² required for a 8 digit decimal addition we find the number of additions possible on a 1 cm² chip.