TOPIC: NUMBER SYSTEMS

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1 Ministry of Secondary Education Progressive Comprehensive High School PCHS Mankon Bamenda Department of Computer Studies Republic of Cameroon Peace Work - Fatherland TOPIC: NUMBER SYSTEMS Class: Comp. Sc. A/L By: DZEUGANG PLACIDE Since the early days of human civilization, people have been using their fingers, sticks and other things for counting. As daily activities became more complex, the numbers became more important in trade, time and distance, and in all other spheres of human life. It became apparent that we needed more than our fingers and toes to keep track of the number in our daily routine. In 3400 BC, the ancient Egyptians started using special symbols for writing the numbers. This was a major advancement, because it reduced the number of symbols required. However, it was difficult to represent large or small numbers by using such a graphical approach. Learning objectives After studying this lesson, student should be able to: Recognize different types of number systems as they relate to computers. Identify and define unit, number, base/radix, positional notation, and most and least significant digits as they relate to decimal, binary, octal, and hexadecimal number systems. Add and subtract in binary, octal, and hexadecimal number systems. Convert values from decimal, binary, octal, hexadecimal, and binary-code decimal number systems to each other and back to the other systems. Add in binary-coded decimal. Contents I. DEFINITIONS... 2 II. BASE (OR RADIX) OF SYSTEM... 2 III. TYPES OF NUMBER SYSTEM... 2 IV. CONVERSION BETWEEN NUMBER BASES... 4 V. ARITHMETIC OPERATIONS IN NUMBER SYSTEMS EXERCISES... 14

2 Topic: Number Systems 2 By DZEUGANG Placide I. DEFINITIONS A number system defines a set of values that is used to represent quantity. The number systems can be classified into two broad categories as follows: Nonpositional Number Systems: In ancient times, people used to count on their fingers. When the fingers became insufficient for counting, stones, pebbles or sticks were used to indicate the values. This method of counting is called the nonpositional number system. Positional Number Systems: In positional number systems, the value of each digit in a number is defined not only by the symbol, but also by the symbol's position. Positional number systems have a base or radix. The positional number system, which is currently used is called the decimal number system. This system is a base 10 system, that is, it contains 10 digits (0, 1, 2, 3,, 8, 9). Apart from the decimal number system, there are some other positional number systems such as binary number system, octal number system and hexadecimal number system, each having a radix of 2, 8 and 16, respectively. However, the principles, which are applied to the decimal number system, are also applicable for the other positional number systems. II. BASE (OR RADIX) OF SYSTEM In the number system, the base or radix tells the number of symbols used in the system. Radix of the system is always expressed in decimal numbers. The base or radix of the decimal system is 10. This implies that there are 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Similarly, the system using three symbols 0, 1, 2 will be of base 3, four symbols will be of base 4 and so on. The base of a number system is indicated by a subscript (decimal number) and this will be followed by the value of the number. The following are few examples: (7592) 10 is of base 10 number system. (214) 8 is of base 8 number system. (123) 16 is of base 16 number system III. TYPES OF NUMBER SYSTEM Eventually, the number systems that are generally used by the computers are as follows: Decimal number system Binary number system Octal number system Hexadecimal number system

3 Topic: Number Systems 3 By DZEUGANG Placide Number System Radix Value Set of Digits Example Decimal r = 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (25) 10 Binary r = 2 0, 1 (11001) 2 Octal r = 8 0, 1, 2, 3, 4, 5, 6, 7 (31) 8 Hexadecimal r = 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F (19) 16 Table 1. Types of Number Systems Factorisation. Let consider the number N of n digits in base b written as follow: (D n 1 D 2 D 1 D 0 ) b. N can be also written N = D n 1 b n D i b i + + D 2 b 2 + D 1 b 1 + D 0 = n 1 0 D i b i III.1 Decimal Number System The primary number system used is a base 10 number system or decimal number system, which we use every day while counting. This number system includes 10 digits from 0 to 9. Each digit in a base 10 number system represents units, which are ten times the units of the digit to its right. Example. Let s consider the number = III.2 Binary Number System The computers do not use the 10 digits of the decimal system for counting and arithmetic operations. Their CPU and memory are made up of millions of tiny switches that can be either in the ON or OFF states. Two digits, 0 and 1, are used to refer for the two states of ON and OFF, respectively. In the binary number system with base 2, the weight of n th bit of the number from the righthand side (RHS) is n th bit 2 n 1. The weighted values for each position are determined as follows: Table 2. Decimal Binary Comparison The problem with the binary system is that it takes a large number of digits to represent numerical values. This system is not efficient in representing fractional values. It cannot represent these values accurately and needs many digits to even come close to approximation III.3 Octal Number System

4 Topic: Number Systems 4 By DZEUGANG Placide The octal or base 8 number system is commonly used in computers. The octal number system, with its eight digits, '0', '1', '2', '3', '4', '5', '6' and '7', is a base 8 system. III.4 Hexadecimal Number System Hexadecimal is another number system that works exactly like the decimal and binary number systems, except that the base is 16. Similar to the decimal number that represents a power of 10, each hexadecimal number represents a power of 16. To represent the decimal numbers, this system uses numbers from 0 to 9 and characters from A to F to represent numbers from 10 to 15, respectively Decimal Binary Octal Hexadecimal A B C D E F IV. CONVERSION BETWEEN NUMBER BASES We have discussed earlier that internally the computer uses binary numbers for data representation, whereas externally it uses decimal numbers. However, any number in one number system can be represented in any other number system. Conversion between number systems can be classified into three types: Coding, decoding and encoding. IV.1. Coding Coding is the conversion from decimal base (base 10) to any non-decimal base b ( b 10). IV.1.1 Coding of whole numbers The method used for the conversion of a decimal number to a non-decimal base b is often called the remainder method. This method involves the following steps:

5 Topic: Number Systems 5 By DZEUGANG Placide 1. Begin by dividing the decimal number by b (the base of the number). 2. Note the remainder separately as the rightmost digit of the equivalent. 3. Continually repeat the process of dividing by b until the quotient is zero and keep writing the remainders after each step of division (these remainders will less than b). 4. Finally, when no more division can occur, write down the remainders in the reverse order (last remainder written first). Examples: a) Determine the binary equivalent of (36) 10. b) Convert (671)10 into base 6, then in Base Taking remainders in the reverse order, we have (36) 10 =(100100) 2 Remainders Taking the remainders in the reverse order, we have (670) 10 =(100105) 6 Remainders =F (670) 10 =(29F) 16 Taking the remainders in the reverse order, we have IV.1.2 Conversion of Decimal Fractions The method used for the conversion of decimal fractions is similar to the above technique except that instead of division, the mathematical process of multiplication is used. Moreover, instead of looking for a remainder, we will look for a whole number. This method involves the following steps: 1. Multiply the decimal fraction by the base b. 2. If a whole number is generated, place that integer in that position, if not then place Remove the whole number and continue steps 1 and 2 with the fraction value until it becomes Finally, when no more multiplication can occur, write down the remainders in the downward direction (as shown by the arrow mark). Examples: a) Determine the binary equivalent of (0.375) 10. Finally, (0.375) 10 = (0.011) x 2 = x 2 = x 2 = b) Determine hexadecimal conversion of ( ) 10 We have already noticed in the previous question that (671) 10 = (29F) 16. Now let convert (0.175) 10 in base 16

6 Topic: Number Systems 6 By DZEUGANG Placide x 16 = x 16 = 12.8 C 0.8 x 16 = 12.8 C... Finally, ( ) 10 = (29F.2CCC...) 16 IV. 2. Decoding To decode is convert from a non-decimal base b (b 10) to the decimal base (base 10). IV.2.1 Conversion of whole numbers In this case, each digit of the binary number is multiplied by its weighted position, and each of the weighted values is added together to get the decimal number. Examples: a) Convert into base 10. The Process: Above each of the digits in your number, list the power of the base that the digit represents. See the example on the left. It is now a simple process of multiplication and addition to determine your base 10 number. In this example you have 5 x 8 0 = 5 3 x 8 1 = 24 2 x 8 2 = 128 b) Convert to base 10. Now simply add these values together = 157 Answer: = **Remember: any number to the zero power equals one. 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 c) Convert 1C4 16 to base x 16 0 = 4 C x 16 1 = 12 x 16 1 = x 16 2 = = 11 Answer: = = 452 Answer: 1C4 16 = IV.2.2 Conversion of non-decimal fractions to decimal fraction The conversion of fractions is similar to the decimal numbers. The only difference is the negative exponents, which are used to denote the negative powers of b. Here, instead of a

$The exponential expressions of each fractional placeholder are b -1, b -2, and in this way the exponent notation proceeds. The steps involved in the conversion process are as follows: 1.$ $Write the weight value of each bit of the non decimal fractional number. 2. Multiply the weighted position with the respective bit of the non decimal fractional number. 3.$

7 Topic: Number Systems 7 By DZEUGANG Placide 'decimal' point we have a 'binary' point. The exponential expressions of each fractional placeholder are b -1, b -2, and in this way the exponent notation proceeds. The steps involved in the conversion process are as follows: 1. Write the weight value of each bit of the non decimal fractional number. 2. Multiply the weighted position with the respective bit of the non decimal fractional number. 3. Add all the weighted values to get the decimal number. Examples: a) Determine the decimal equivalent of ( ) 2. Thus, the decimal equivalent of ( ) 2 is ( ) 10. b) Determine the decimal equivalent of (237.04) 8. Sum of weight of all bits = = Thus, the decimal equivalent of (237.04) 8 is ( ) 10. c) Determine the decimal equivalent of (45C.8BE3) 16. Sum of weight of all bits = = Thus, the decimal equivalent of (45C.8BE3) 16 is ( ) 10. IV.3. Encoding Encoding is the conversion from a non-decimal base to a non-decimal base. Some special cases would be distinguished. IV.3.1 Conversion of Binary to Hexadecimal

8 Topic: Number Systems 8 By DZEUGANG Placide The conversion of an integer binary number to hexadecimal is accomplished by the following steps: 1. Break the binary number into four-bit sections starting from the LSB to the MSB. 2. Convert the four-bit binary number to its hexadecimal equivalent. For whole numbers, it may be necessary to add a zero to the MSB to complete a grouping of four bits. NOTE: By adding a zero, the MSB will not change the value of the binary number. Examples: a) Determine the hexadecimal equivalent of ( ) 2 The hexadecimal equivalent of ( ) 2 is (2BCD9) 16. b) Determine the hexadecimal equivalent of ( ) 2. The hexadecimal equivalent of ( ) 2 is (61.AF3) 16. IV.3.2 Conversion of Hexadecimal to Binary Converting a hexadecimal (base 16) number to a binary (base 2) number is a precise process. Since a single digit in a hexadecimal number corresponds directly to a four-digit binary number, to convert the hexadecimal number into its binary equivalent, the following steps are involved: 1. Convert each hexadecimal digit to its four-bit binary equivalent. 2. Combine the four-bit sections by removing the spaces to get the binary number. Examples: a) Determine the binary equivalent of (5AF) 16. Combining the four bits of the binary-coded values, we have Thus, the binary equivalent of (5AF) 16 is ( ) 2.

9 Topic: Number Systems 9 By DZEUGANG Placide b) Determine the binary equivalent of (2B.6C) 16. Combining the four bits of the binary-coded values, we have Thus, the binary equivalent of (2B.6C) 16 is ( ) 2. IV.3.3 Conversion of Octal to Hexadecimal Octal and hexadecimal have certain relations with binary, that is, the first digit in octal corresponds to the first three digits in its binary equivalent and so on. The same is true for hexadecimal and this time each digit represents four binary digits. This makes the conversion of octal to hexadecimal and vice versa quite easy. This conversion involves the following steps: 1. Convert each octal digit to three-bit binary form. 2. Combine all the three-bit binary numbers. 3. Divide the binary numbers into the four-bit binary form by starting the first number from the right bit to the first number from the left bit. 4. Finally, convert these four-bit blocks into their respective hexadecimal symbols. Examples: a) Determine the hexadecimal equivalent of (2327) 8. Combining the three-bit binary blocks, we have Dividing the group of binary numbers into the four-bit binary blocks and by converting these blocks into their respective hexadecimal symbols, we have: Thus, the hexadecimal equivalent of (2327) 8 is (4D7) 16. b) Determine the hexadecimal equivalent of (31.57) 8. Combining the three-bit binary blocks, we have Dividing the group of binary numbers into the four-bit binary blocks and by converting these blocks into their respective hexadecimal symbols, we have:

10 Topic: Number Systems 10 By DZEUGANG Placide Thus, the hexadecimal equivalent of (31.57) 8 is (19.BC) 16. IV.3.4 Conversion of Hexadecimal to Octal This conversion follows the same steps of octal to hexadecimal conversion except that each hexadecimal digit is converted into a four-bit binary form and then after grouping of all the four bit binary blocks, it is converted into the three-bit binary form. Finally, these three-bit binary forms are converted into octal symbols. Examples: a) Determine the octal equivalent of (5DE247) 16. Combining all the four-bit binary blocks, we have Dividing the group of binary numbers into the three-bit binary blocks and by converting these blocks into their respective octal symbols, we have: Thus, the octal equivalent of (5DE247) 16 is ( ) 8. b) Determine the octal equivalent of (7B.64D) 16. Combining all the four-bit binary blocks, we have Dividing the group of binary numbers into the three-bit binary blocks and by converting these blocks into their respective octal symbols, we have: Thus, the hexadecimal equivalent of (7B.64D)16 is ( )8 V. ARITHMETIC OPERATIONS IN NUMBER SYSTEMS V.1 Binary addition

Topic: Number Systems 11 By DZEUGANG Placide 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0, and carry 1 to the next more significant bit Examples, a) 00011010 + 00001100 = 00100110 1 1 carries 0 0 0 1 1 0

11 Topic: Number Systems 11 By DZEUGANG Placide = = = = 0, and carry 1 to the next more significant bit Examples, a) = carries = (26) = (12) = (38) 10 b) = carries = (19) = (62) = (81) 10 V.2 Binary Subtraction Rules of Binary Subtraction 0 0 = = 1, and borrow 1 from the next more significant bit 1 0 = = 0 Examples, a) = borrows = (37) = (17) = (20)10 b) = borrows = (51) = (22) = (29)10 V.3 Binary Multiplication The multiplication process for binary numbers is similar to that for decimal numbers. Partial products are formed, with each product shifted one place to the left. This is illustrated below. Example, a) 111 x 101 b) = = (41) = (6) Rules of Binary Multiplication

Topic: Number Systems 12 By DZEUGANG Placide 11110110 = (246) 10 V.4 Binary division Binary division follows a similar process to that of decimal division.

12 Topic: Number Systems 12 By DZEUGANG Placide = (246) 10 V.4 Binary division Binary division follows a similar process to that of decimal division. Example: Divide (a) (15) 10 by (5) 10 in binary form, and (b) (15) 10 by (6) 10 in binary form. V.5 Operation in octal and hexadecimal For the other bases, the process is the same. The following table below show table those various operations in different basis. Octal addition octal multiplication (Base 8) x Hexadecimal Addition Table A B C D E F A B C D E F A B C D E F A B C D E F A B C D E F A B C D E F A B C D E F

13 Topic: Number Systems 13 By DZEUGANG Placide A B C D E F A B C D E F A B C D E F A B C D E F A B C D E F A 1B C D E F A 1B 1C D E F A 1B 1C 1D E F A 1B 1C 1D 1E F A 1B 1C 1D 1E 1F A 1B 1C 1D 1E 1F 20 Hexadecimal Multiplication Table * A B C D E F A B C D E F A C E A 1C 1E C F B 1E A 2D C C C C A F E D C B C E 24 2A C E 54 5A E 15 1C 23 2A F 46 4D 54 5B B 24 2D 36 3F A 63 6C 75 7E 87 A 0 A 14 1E C A 64 6E C 96 B 0 B C D E F 9A A5 C 0 C C C C A8 B4 D 0 D 1A E 5B F 9C A9 B6 C3 E 0 E 1C 2A E 8C 9A A8 B6 C4 D2 F 0 F 1E 2D 3C 4B 5A A5 B4 C3 D2 E1

14 Topic: Number Systems 14 By DZEUGANG Placide EXERCISES MCQ 1. Choose the correct answer from below for the result of the binary multiplication (a) , (b) , (c) , (d) Choose the correct answer from below for the result of the binary division (a) 10, (b) 101, (c) 11, (d) Which of the following is the binary product ? (a) (b) , (c) , (d) Which of the following is the binary product ? (a) , (b) , (c) , (d) Which of the following is the binary division ? (a) 100, (b) 110, (c) 101, (d) Which of the following is the binary division ? (a) 10.1, (b) 11.11, (c) 11.01, (d) Exercise 1. (a) Convert the binary number 1011 into decimal form. (b) Convert the binary number into decimal form. (c) Convert the numbers 15 and 12 into binary form, add the two binary numbers together and convert the answer to decimal form to check that the sum is correct. (d) Convert the numbers 9 and 6 into binary form. Use this to find 9 6 in binary form. Check that the answer is correct by converting the binary answer into decimal form. Exercise 2. In each of the questions below, a product is written in decimal form. In each case, convert both numbers to binary form, multiply them in binary form and check that the solution is correct by converting the answer to decimal form. (Click on the green letters for solutions.) (a) 3 2, (b) 4 4, (c) 5 10, (d) 6 7, (e) 9 6, (f) 11 7 Exercise 3. In each of the questions below, a division is written in decimal form. In each case, convert both numbers to binary form, perform the division in binary form and check that the solution is correct by converting the answer to decimal form. (Click on the green letters for solutions.) (a) 6 2, (b) 8 2, (c) 9 3, (d) 10 4, (e) 21 7, (f) 18 8.

Number Systems CHAPTER Positional Number Systems

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