Homework 1

Homework 1
Numbering Systems and Data Types

Due: (in class) Tuesday, October 30, 2007 at 9am

Outcomes

After successfully completing this assignment, you will be able to...

convert numbers between the binary, hexadecimal, and decimal numbering systems
do simple arithmetic problems in binary and hexadecimal
show how signed integers are represented in binary
show how floating point numbers are represented in binary

Before Starting

Read Chapters 1 and 2.

Format and Submission of Assignment

The solution for this assignment must be submitted as hard-copy at the beginning of class (9am) on the due date. All papers must be stapled together, and your name, login name, and section number must appear at the top of every sheet. Handwriting must be legible - if we can't read it, the grade is zero (you may use a word processor or an editor to write your solution if you wish; please turn in a printed copy).

Show your work for each problem, and/or explain the steps you are taking to arrive at the answer. Put a circle around your answer. If you just give the answer to a problem, without explaining or showing the steps in obtaining the answer, that problem will be marked wrong.

Problems (each problem is worth 10 points)

Perform the following conversions on unsigned numbers. SHOW YOUR WORK.
- a). 1000011100 binary to decimal
- b). 1110000110100101 binary to hexadecimal (no need to show your work for this part)
- c). DAB hexadecimal to decimal
- d). 715 decimal to hexadecimal
- e). 227 decimal to binary
In the base 6 numbering system there are 6 digits: 0, 1, 2, 3, 4, 5. Perform the following conversions on unsigned numbers. SHOW YOUR WORK.
- a). 255 base 6 to decimal
- b). 997 decimal to base 6
Perform the following conversions. Assume 16-bit words. SHOW YOUR WORK.
- a). Write -114 decimal as a hexadecimal number in sign-magnitude form
- b). Write -114 decimal as a hexadecimal number in one's complement form
- c). Write -114 decimal as a hexadecimal number in two's complement form
An 8-bit word contains the bit pattern 10000110. What decimal number does this represent if the word is
- a). an unsigned number
- b). a signed number in two's complement form
To convert a floating point number from decimal to binary, the integer and fractional parts must be considered separately. In class, we saw how to convert the integer portion by doing repeated divisions by 2. The fractional part involves repeated multiplications by 2. At each step, the fractional part of the decimal number is multiplied by 2. The digit to the left of the decimal point in the product will be 0 or 1 and contributes to the binary representation of the fraction, starting with the most significant bit. The fractional part of the product is used as the multiplicand in the next step. Continue until the product contains a fractional part equal to 0, for exact representations.
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   EXAMPLE:  Convert 4.625 decimal to binary

             Integer portion:  4 decimal = 100 binary

             Fractional portion:
                               0.625 x 2 = 1.25   ==>      .1
                               0.25  x 2 = 0.5    ==>       0
                               0.5   x 2 = 1.0    ==>       1
                               fractional portion of product is zero, so stop

             4.625 decimal = 100.101 binary
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Note, however, that the process will not necessarily yield an exact representation. Just as some fractions cannot be represented exactly in the decimal numbering system (e.g. 1/3), neither can some fractions be represented exactly in the binary numbering system (e.g. 1/10). In cases that do not yield an exact result, continue until you find a repeating fraction, or for twenty-three binary places, whichever comes first.
- a). Convert -100.375 decimal to IEEE single-precision format (see pp 37 - 40 in the textbook). Give the results as 8 hexadecimal digits. SHOW YOUR WORK.
- b). Convert 41D00000 hex (which is in IEEE single- precision format) to a decimal floating point number. SHOW YOUR WORK.
Compute the sum of the following IEEE single-precision floating point numbers and express the result as a hexadecimal number. The numbers to be added are given in hexadecimal: 41280000, 40700000. (Hint: the exponents must be the same when you add two numbers in floating point format, same as you would do if you added two numbers that were specified using scientific notation).
- a). Represent the values 12 and -11 as 8-bit 1's complement integers. Add the two binary representations. What is the binary result? What decimal value does that 1's complement result represent?
- b). Represent the values 12 and -11 as 8-bit signed magnitude integers. Add the two binary representations. What is the binary result? What decimal value does that signed magnitude result represent?
- c). Represent the values 12 and -11 as 8-bit 2's complement integers. Add the two binary representations. What is the binary result? What decimal value does that 2's complement result represent?
Perform the following additions on 16-bit hexadecimal values. Provide your answers as 16-bit hexadecimal values.
- a). x034A + x2FF6 (notice the addition of two two's-complement positive values gives a positive result)
- b). xFFD4 + xFFF1 (notice the addition of two two's-complement negative values gives a negative result)
- c). xB123 + xC777 (explain why the addition of two two's-complement negative numbers results in a two's-complement positive number)
- a). Do problem 2.22 from the textbook.
- b). Do problem 2.24 from the textbook.
- a). Do problem 2.34 from the textbook.
- b). Do problem 2.54 from the textbook.