Chapter 2 Exercises and nswers nswers are in blue. For Exercises -5, match the following numbers with their definition.. Number. Natural number C. Integer number D. Negative number E. Rational number unit of an abstract mathematical system subject to the laws of arithmetic. natural number, a negative of a natural number, or zero. C The number zero and any number obtained by repeatedly adding one to it. n integer or the quotient of two integers (division by zero excluded). E value less than zero, with a sign opposite to its positive counterpart. D For Exercises 5-, match the solution with the problem. (binary + addition) 0 0 000+ (binary addition) 00 C + (binary addition) (binary subtraction) F 00 (binary subtraction) D 000 (binary subtraction) E For Exercises 2-7, mark the answers true and false as follows:. True. False inary numbers are important in computing because a binary number can be converted into every other base. inary numbers can be read off in hexadecimal but not in octal. Starting from left to right, every grouping of four binary digits can be read as one hexadecimal digit. byte is made up of six binary digits. Two hexadecimal digits can be stored in one byte. Reading octal digits off as binary produces the same result whether read from right to left as left to right.
Exercises 8-45 are problems or short answer questions. Distinguish between a natural number and a negative number. natural number is 0 and any number that can be obtained by repeatedly adding to it. negative number is less than 0, and opposite in sign to a natural number. lthough we usually do not consider negative 0. Distinguish between a natural number and a number. number is an integer or the quotient of integer numbers. (Division by 0 is excluded.) natural number is 0 and the positive integers. (See also definition in answer to Exercise.) Label the following numbers natural, negative, or. a..333333 b. /3 negative, c. 066 natural d. 2/5 e. 6.2 f. π (pi) not any listed If 89 is a number in each of the following bases, how many s are there? a. base 0 89 b. base 8 Can't be a number in base 8, c. base 2 26 d. base 3 470 e. base 6 293 Express 89 as a polynomial in each of the bases in Exercise. 8 * 0 2 + 9 * 0 + Can't be shown as a polynomial in base 8. 8 * 2 2 + 9 * 2 + 8 * 3 2 + 9 * 3 + 8 * 6 2 + 9 * 6 + Convert the following numbers from the base shown to base 0. a. (base 2) 7 b. 777 (base 8) 5 c. FEC (base 6) 4076 d. 777 (base 6) 9 e. (base 8) 73 Explain how base 2 and base 8 are related. ecause 8 is a power of 2, base-8 digits can be read off in binary and 3 base-2 digits can be read off in octal. Explain how base 8 and base 6 are related. 8 and 6 are both powers of two.
Expand Table 2. to include the numbers from 0 through 6. binary octal decimal 0 3 3 00 4 4 0 5 5 0 6 6 7 7 Expand the table in Exercise 26 to include hexadecimal numbers. binary octal decimal hexadecimal 0 3 3 3 00 4 4 4 0 5 5 5 0 6 6 6 7 7 7 Convert the following binary numbers to octal. Convert the following binary numbers to hexadecimal. Convert the following hexadecimal numbers to octal. a. 9 25 b. E7 347 C. 6E 56 Convert the following octal numbers to hexadecimal. a. 777 FF b. 605 85 c. 443 23 d. 52 5 e. Convert the following decimal numbers to octal. a. 90 605 b. 32 50 c. 492 2724 d. 066 2052 e. 200 372 Convert the following decimal numbers to binary. a. 45 00 b. 69 0000 c. 066 0000000 d. 99 000
e. Convert the following decimal numbers to hexadecimal. a. 066 42 b. 939 793 c. d. 998 3E6 e. 43 2 If you were going to represent numbers in base 8, what symbols might you use to represent the decimal numbers 0 through 7 other than letters? ny special characters would work or characters from another alphabet. Let's use # for 6 and @ for 7. Convert the following decimal numbers to base 8 using the symbols you suggested in Exercise 5. a. 066 354 b. 99099 #@F9 c. Perform the following octal additions a. 770 + 665 655 b. 0 + 707 00 c. 202 + 667 07 Perform the following hexadecimal additions a. 96 + 43 9F9 b. E9 + F F8 c. 066 + CD C33 Perform the following octal subtractions. a. 066 776 70 b. 234 765 247 c. 7766 5544 2222 Perform the following hexadecimal subtractions. a. C 9 b. 9988 98DD c. 9F8 492 9566 Why are binary numbers important in computing? Data and instructions are represented in binary inside the computer. byte contains how many bits? 8
How many bytes are there in a 64-bit machine? 8 Why do microprocessors such as pagers have only 8-bit machines? Pagers are not general-purpose computers. The programs in pagers are small enough to be represented in 8-bit machines. Why is important to study how to manipulate fixed-sized numbers? It is important to understand how to manipulate fixed-sized numbers because numbers are represented in a computer in fixed-sized format. How many ones are there in the number 98 in base 3? ((3*3*3*0) + (3*3*) + 3*9) + 8) = 23954 Describe how a bi-quinary number representation works. There are seven lights to represent ten numbers. The first two determine the meaning of the next five. If the first light is on, the next five represent 0,, 2, 3, and 4 respectively. If the second is on, the next five represent 5, 6, 7, 8, and 9 respectively.