Expert Answers

1. Table 2A contains four base-10 addition problems, for which you are to use complement arithmetic to obtain the solution. Above each of the four problems, there is a pair of operands. In the first problem, both operands are positive. Transfer the first operand to the first now, the second operand to the second row, and compute the sum. For the remaining 3 problems in table 2A, one or both operands is negative. Enter negative operands in their10’s complement form (i.e., obtain the 9’s complement, then add 1). Now perform the addition in the usual manner to obtain the correct result. If you result is 500 or greater, then the correct answer is a negative number, whose magnitude just the 10’s complement of the result you calculated.

 

b. Table 2B, contains four base-2 addition problems, for which you are to use complement arithmetic to obtain the solution. The operands are given in base 10. In the first problem, both operands are positive. Convert the first operand to base 2, and enter the result the first row. Convert the second operand to base 2, and enter the result in the second row. Then perform binary addition (0+0 is 0, 1 + 0 is 1, 0 + 1 is 1, 1 + 1 is 0 carry 1, and 1 + 1 + 1 is 1 carry 1).

For the remaining 3 problems in table 2B, one or both operands is negative. Enter negative operands in their 2’s complement from (i.e., obtain the 1’s complement, then add 1). Now perform  the  additional in the usual manner to obtain the correct result. If your result is 1000 00002 or greater, then the correct answer is  negative number, whose magnitude is just the 2’s complement of the result you calculated.

c. Table 2C, contains four base 16 additional problems, for which you are to use to complement arithmetic to obtain the solution. The operands are given in base 10. in the first problem, both are operands positive. Convert the first operand to base 16, and enter the result the first row. Convert the second operand to base 16, and enter the result in the second row. Then perform hexadecimal addition. Use 16’s complement for negative operands. If your result is 1016 or greater, then the correct answer is a negative number, whose magnitude is just the 16’s complement of the result you cancelled.

 

 

2. Conversion between bases comer up over and over again throughout the course, as you will see. You should strive to becoming very facile with changing bases, especially binary integers to/from hexadecimal integers. Octal base is now little-used (it was used more frequently in the past), but a little practice using octal will help you learn bit manipulations.

a. Convert 41.4B816  from hexadecimal to binary. Enter each binary digit in a separate box. A straightforward way to get the correct answer is to work ‘outwards’ from the hexadecimal point, writing groups of 4 binary digits has been set for in this problem, you will find that you must drop leading 0’s of the integer part, and trailing 0’s in the fractional part. Note the dropping these 0’s does not change the value of your result.

b. Convert 60.92816 from hexadecimal to binary. Enter your entire answers in the box, including the binary point.

c. Convert 335.4328 from octal to binary. Enter each binary digit in a separate box. Again you will find that you must drop leading 0’s of the integer part, and trailing 0’s in the factional part.

d. Convert 371.6468  from octal to binary. Enter your entire answer  in the box including the binary point.

e. Convert 11 1111 0011.1110 1000 1012 from binary to hexadecimal. Enter each hexadecimal digit in a separate box. A straightforward way to get the correct answer is to pad with the whole number part of the binary value (i.e., the part to the of the binary point) with 0’s on the left to make even groups of 4, and pad the fractional part (to the right of the binary point) with 0’s on the right to make even groups of 4. Then convert groups of 4 bits to their corresponding hexadecimal digits.

f. Convert 10 1110 0101.1000 0000 0012 from binary to hexadecimal. Enter your entire answer in the box, including the hexadecimal point.

g. Convert 1 111 011.010 101 101 112 from binary to octal. Enter each octal digit in a separate box. Analogous to part E, make groups of 3 (instead of group of 4).

h. Convert 1 010 111 101.100 101 110 112 from binary to octal. Enter your entire answer in the box, including the octal point.

i. Convert 81C.E316 from the hexadecimal to octal. The easiest approach is to use binary representation as an intermediate step. First convert from hex to binary (as in problem B above), them from binary to octal (as in problem H).

 

 

3. The algorithms we are taught in grade school for multiplying and dividing base-10 numbers can be extended to other base in a relatively straightforward way. Doing so is outside the scope of this course; however, we will consider the important special case of multiplying and dividing number by the power of the base the number. In this case, we simply move the radix point to the right for multiplication and to the left for division.

For example, 123.4510 * 10 = 1234.510, and 123.4510 / 10 = 12.34510

Solve the following problems by moving the radix point:

a. Multiple 85892.563710 by 100010 Enter your entire answer (in base 10, including the decimal point) in the box provided.

b. Divide 85892.563710 by100010

c. Multiply 1111.01012 by 810 Enter your  entire answer (in base 2, including the binary point) in the box provided. Hint: first express 810 as a power of 2.

d. Divide 1111.01012 by 410.

e. Multiply CIB96B.BD813A16 by 6553610. Entire your entire answer(in base 16, including the hexadecimal point)in the box provided.

f. Divide CIB96B.BD813A16 by409610.

 

4. An important standard for encoding of characters in language with Latin alphabets is ASCII conversion charts at any of a number of source on the internet. Just enter ‘ASCII conversion’ into any search engine. See, for example, AsciiTable.com.

This exercise will give you a little practice representing ASCII character in binary and hexadecimal.

a. Use an ASCII code chart to convert THuRs[JuLY:4^1776 from ASCII to hexadecimal. Complete Table 5.1 by representing each character with two hexadecimal digits (you can use uppercase to lowercase for your hexadecimal values).

b. Decode an exciting secret message in Table 5.2 by first converting from binary to hexadecimal, then using an ASCII code chart to convert to ASCII. Each blank in the hexadecimal, column should be two hexadecimal digits (case-insensitive). Each blank in the ASCII column should be a single character (case-sensitive).

 

5. Learning to represent number in IEEE-754 format helps you to understand how floating point data is stored in computer systems. Converting a number to IEEE-754 format combines techniques we have studied previously, thus reinforcing your understanding of these techniques.

In the following problems, you will convert a floating point number to IEEE-754 32-bit floating format. The IEEE-754 standard also defines other formats, most notably 34-bit floating point. The concepts for converting to64-bit format are the same as for 32-bits, only with more bits. Therefore, we will focus on the shorter 32-bit format in order to learn the principle involved. When you need that extra push to 64 bits, see Spinal Tap video at right.

a. Solve the conversion in Table 6.1 I strongly recommend that you work on and grade only 1 row at a time, so that any mistakes you make on one row will not propagate to the next row. Simply convert the integer and fractional parts separately to binary (see Question1, parts A and C), and join the two parts together with a binary point.

b. Solve the conversion in Table 6.2. The procedure is the same as for part A, but you will not have each bit of your result checked individually, putting more of the burden on you to be carefully with your calculations.

 

6. A. Unicode, a vast extension of ASCII, represents most any imaginable character. What is the Unicode representation of the lower case Greek letter pi (π)? Your answer should be a 4-digits hexadecimal code (case-insensitive). Just enter ;Unicode pi’ in your favorite Internet search engine to find the answer.

B. What is the Unicode representation if this chess piece:? Your answer should be a 4-digits hexadecimal code (case-insensitive). Again, an Internet search engine should be helpful.

C. which of the following images appear in the Week 1 lecture videos?

a. _______ A cartoon with a man in a swimming pool

b. _______ A cartoon with a woman receiving a compliment from her boss.

c. _______ A penguin colony.

d. _______ Roman soldiers.

e. _______ A binary start system.

f. _______ A photography of the famous mathematician, George Boole

 

D. _________ All computer data is stored internally as sequences of binary digits (aka ‘bits’).

E. IEEE-754 format is used to store

F. A special relationship between base 2 and this base makes conversion between bases especially simple.

G. ________ 2’s complement notation is useful for representing Boolean data.

H. ________ An algorithm is a step-by-step procedure for solving a problem.

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