Contents:

• Overview
• Internet Requirements
• Practice Problems
• Problems to be Submitted
• Extra Credit

Topic: Number Systems and Arithmetic
Related Reading: Ch. 2
Due: 8pm Thursday, 9 September 2004

You will not need an Internet connection for completing the assignment, other than for submission.

• If you want to give yourself a way to practice and then check your answers, there are two ways.

  • Many calculators will do some conversions for you. In particular, the standard calculator application that comes as part of Windows will convert numbers between decimal, binary, octal or hexadecimal. This calculator can usually be started by looking in the Start -> Programs -> Accessories menu. When it begins, it generally looks like a very simple calculator. However, there is a menu title "View" at the top of the calculator that lets you select between "Standard" and "Scientific" views. Select the Scientific view.

    In scientific mode, you will see a box that allows you to pick between one of four number systems: Hex(adecimal), Dec(imal), Oct(al), or Bin(ary). If you want it to convert from decimal to binary, select decimal mode, then type in a decimal number, then click on binary mode. In the same way you can convert from binary to octal, and so on. You can also perform all of the arithmetic operations while working in any of the number systems. Please note, however, that this calculator does not handle fractional values in any base other than decimal.

    One warning: When working outside of decimal notation, the calculator can only handle values which fit in 32 bits or less (limit is slightly less than 4.3 billion). Also, I strongly recommend that you keep the selection on the right as "Dword" when working in one of the other bases.

    The calculator which ships with Macintosh "OS X" can partly work in such alternate bases. Namely it can convert from decimal to alternate bases, by selecting the menu choice: View -> Display Format. Unfortunately, this is not as graceful of a solution because the input keys continue to be interpreted as decimal digits even as the underlying value is displayed in an alternate base.

  • Laboratory 2 of Meyer's book gives an introduction to two applets which may help you.

    Number Systems is software which also lets you convert from one base to another. This has several features not available in the Windows calculator, namely that it shows you the steps of the conversion algorithm, and that it works with several other bases which are not powers of two.

    Binary Addition is software which shows you the steps of a binary addition.

• Exercise 21 of Ch. 2 (p. 48); answer in back of text
• Exercise 23 of Ch. 2 (p. 48); answer in back of text
• Exercise 29 of Ch. 2 (p. 48); answer in back of text
• Exercise 30 of Ch. 2 (p. 49)
• Exercise 31 of Ch. 2 (p. 49); answer in back of text
• Exercise 32 of Ch. 2 (p. 49)
• Exercise 33 of Ch. 2 (p. 49); answer in back of text
• Exercise 37 of Ch. 2 (p. 49); answer in back of text

First, we wish to verify that you were successful in learning to use the Windows (or similar) calculator and/or the Meyer software. These first problems are based on numbers that are intentionally too large to do by hand in a timely manner.

1. (2 points)

   Convert 4048891811_{(base 10)} to hexadecimal.

2. (2 points)

   Convert 2114112_{(base 8)} to decimal.

3. (2 points)

   Perform the following hexadecimal addition:
   52E4F03B + 902DC47

Next, we wish to verify that you were successful in learning to do these calculations by hand. For this problems, we intentionally use various bases that cannot be checked using the Windows calculator or Meyer software. (If you have some other calculator which works with various bases, please do not use it! You need to be able to do this by hand)

4. (2 points)

   Convert the number 2532_{(base 6)} into its base 10 representation.

5. (2 points)

   Convert the number 349_{(base 10)} into its base 6 representation.

6. (3 points)

   Express in base 16, the sum
   FEDCBA092_(base 16) + 32154AAAA_(base 16)

7. (3 points)

   Express in base 7, the sum
   4302463_(base 7) + 513460_(base 7)

8. (4 points)

   At the end of Ch. 2 there are a series of "Thought Questions." Pick any one question to answer. The length of your answer should be appropriate for the question, however I envision answers in the range of 1/2-page to 1-page.

Overall, please type your answers to all of the problems in a single document to be submitted electronically. Please see details about the submission process.

Converting from binary to octal is easy because 8¹ = 2³ (thus a correspondance exists between 1 octal digit and 3 binary bits).

Similarly, converting from binary to hexadecimal is easy because 16¹ = 2⁴ (thus a correspondance exists between 1 hexadecimal digit and 4 binary bits).

It so happens that converting from base four to hexadecimal is straightforward because 16¹ = 4² (thus a correspondance exists between 1 hexadecimal digit and 2 digits in base four).

Convert the number 6F9C3_{(base 16)} into its base 4 representation.