ࡱ> lnk !bjbj *~rr"8 8 8L*,VVlllGGG~$!m$GGGGGllGdll~G~*lp\j0,$9$*$*@GGGGGGGGGG$GGGGGGGGG8 X: Chapter 2 Exercises and Answers Answers are in blue. For Exercises 1-5, match the following numbers with their definition. A. Number B. Natural number C. Integer number D. Negative number E. Rational number 1.A unit of an abstract mathematical system subject to the laws of arithmetic. A2.A natural number, a negative of a natural number, or zero. C3.The number zero and any number obtained by repeatedly adding one to it. B4.An integer or the quotient of two integers (division by zero excluded). E5.A value less than zero, with a sign opposite to its positive counterpart. D For Exercises 6-11, match the solution with the problem. A. 10001100 B. 10011110 C. 1101010 D. 1100000 E. 1010001 F. 1111000 6.1110011 + 11001 (binary addition) A7.1010101 + 10101 (binary addition) C8.1111111 + 11111 (binary addition) B9.1111111 111 (binary subtraction) F10.1100111 111 (binary subtraction) D11.1010110 101 (binary subtraction) E For Exercises 12 -17, mark the answers true and false as follows: A. True B. False 12.Binary numbers are important in computing because a binary number can be converted into every other base. B13.Binary numbers can be read off in hexadecimal but not in octal. B14.Starting from left to right, every grouping of four binary digits can be read as one hexadecimal digit. B15.A byte is made up of six binary digits. B16. Two hexadecimal digits can be stored in one byte. A17.Reading octal digits off as binary produces the same result whether read from right to left as left to right. A Exercises 18- 45 are problems or short answer questions. 18.Distinguish between a natural number and a negative number. A natural number is 0 and any number that can be obtained by repeatedly adding 1 to it. A negative number is less than 0, and opposite in sign to a natural number. Although we usually do not consider negative 0.19.Distinguish between a natural number and a rational number. A rational number is an integer or the quotient of integer numbers. (Division by 0 is excluded.) A natural number is 0 and the positive integers. (See also definition in answer to Exercise 1.) 20.Label the following numbers natural, negative, or rational. A. 1.333333 rational B. 1/3 negative, rational C. 1066 natural D. 2/5 rational E. 6.2 rational F. ( (pi) not any listed21. If 891 is a number in each of the following bases, how many 1s are there? A. base 10 891 B. base 8 Can't be a number in base 8, C. base 12 1261 D. base 13 1470 E. base 16 219322.Express 891 as a polynomial in each of the bases in Exercise 1. A. 8 * 102 + 9 * 10 + 1 B. Can't be shown as a polynomial in base 8. C. 8 * 122 + 9 * 12 + 1 D. 8 * 132 + 9 * 13 + 1 E. 8 * 162 + 9 * 16 + 123.Convert the following numbers from the base shown to base 10. A. 111 (base 2) 7 B. 777 (base 8) 511 C. FEC (base 16) 4076 D. 777 (base 16) 1911 E. 111 (base 8) 7324.Explain how base 2 and base 8 are related. Because 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.25.Explain how base 8 and base 16 are related. 8 and 16 are both powers of two. 26.Expand Table 2.1 to include the numbers from 10 through 16. binary octal decimal 000 0 0 001 1 1 010 2 2 011 3 3 100 4 4 101 5 5 110 6 6 111 7 7 1000 10 8 1001 11 9 1010 12 10 1011 13 11 1100 14 12 1101 15 13 1110 16 14 1111 17 15 10000 20 1627.Expand the table in Exercise 26 to include hexadecimal numbers. binary octal decimal hexadecimal 000 0 0 0 001 1 1 1 010 2 2 2 011 3 3 3 100 4 4 4 101 5 5 5 110 6 6 6 111 7 7 7 1000 10 8 8 1001 11 9 9 1010 12 10 A 1011 13 11 B 1100 14 12 C 1101 15 13 D 1110 16 14 E 1111 17 15 F 10000 20 16 2028. Convert the following binary numbers to octal. A. 111110110 766 B. 1000001 101 C. 10000010 202 D. 1100010 14229. Convert the following binary numbers to hexadecimal. A. 10101001 A9 B. 11100111 E7 C. 01101110 6E D. 01121111 This is not a binary number30.Convert the following hexadecimal numbers to octal. A. A9 251 B. E7 347 C. 6E 15631.Convert the following octal numbers to hexadecimal. A. 777 1FF B. 605 185 C. 443 123 D. 521 151 E. 1 132.Convert the following decimal numbers to octal. A. 901 1605 B. 321 501 C. 1492 2724 D. 1066 2052 E. 2001 372133.Convert the following decimal numbers to binary. A. 45 101101 B. 69 1000101 C. 1066 10000101010 D. 99 1100011 E. 1 134.Convert the following decimal numbers to hexadecimal. A. 1066 42A B. 1939 793 C. 1 1 D. 998 3E6 E. 43 2B35.If you were going to represent numbers in base 18, what symbols might you use to represent the decimal numbers 10 through 17 other than letters? Any special characters would work or characters from another alphabet. Let's use # for 16 and @ for 17.36.Convert the following decimal numbers to base 18 using the symbols you suggested in Exercise 15. A. 1066 354 B. 99099 #@F9 C. 1 137.Perform the following octal additions A. 770 + 665 1655 B. 101 + 707 1010 C. 202 + 667 107138.Perform the following hexadecimal additions A. 19AB6 + 43 19AF9 B. AE9 + F AF8 C. 1066 + ABCD BC3339.Perform the following octal subtractions. A. 1066 776 70 B. 1234 765 247 C. 7766 5544 222240.Perform the following hexadecimal subtractions. A. ABC 111 9AB B. 9988 AB 98DD C. A9F8 1492 956641.Why are binary numbers important in computing? Data and instructions are represented in binary inside the computer.42.A byte contains how many bits? 843.How many bytes are there in a 64-bit machine? 844.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.45.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.46.How many ones are there in the number AB98 in base 13? ((13*13*13*10) + (13*13*11) + 13*9) + 8) = 2395447.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, 1, 2, 3, and 4 respectively. 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