Document Sample

INTRODUCTION TO COMPUTER LECTURE 02 NUMBER SYSTEM As we know every computer stores numbers, letters and other special characters in a code form. Before going into the details of these codes, it is essential to have a basic understanding of the number system. Number systems are basically of two types: i) Non positional number system ii) Positional number system. i) Non-Positional Number System In early day, human beings counted on fingers, when ten fingers were not adequate, stones, pebbles, or sticks we used to indicate values. This method of counting using an additive approach is called non-Positional Number System. In this system, we have symbols such as I for 1, II for 2, III for 3, IIII for 4, IIIII for 5 etc. Since it is very difficult to perform arithmetic operations with such a number system. ii) Positional Number System In a positional Number system there are only a few symbols called digits, and these symbols represents different values, depending on the position they occupy in the number. The value of each digit in such a number determined by three considerations. a) The digit itself. b) The position of the digit. c) The base of the number system. The number system that we use in our daily life is called the decimal number system. In this system the base is equal to 10 because there are altogether ten digits used in this number system. E.g. the decimal number 2586 written as (2586)10 consists of the digit 6 in the units position, 8 in the ten position, 5 in the hundreds position and 2 in the thousands position and its value can be written as : (2x1000) + (5x100) + (8x10) + (6x1) OR 2000 + 500 + 80 + +6 = 2586 Some of the number systems commonly used in computer design and by computer professionals are discussed bellow. 1. BINARY NUMBER SYSTEM The binary number system is exactly like the decimal system except that the base is 2 instead of 10. We have only two digits(0 and 1) that can be used in this number system. Each position in a binary number represents a power of the base (2). E.g. the decimal 21 is equivalent of the binary number 10101 written as (10101)2 is (1x 24 ) + (0x 23 ) + (1x22) + (0x21) + (1x20) =16 + 0 + 4 + 0 + 1 = 21 1 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Binary Decimal Equivalent 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7 2. OCTAL NUMBER SYSTEM In the octal number system the base is 8. So in this system there are only eight symbols or digits: 0,1,2,3,4,5,6 and 7 (8 and 9 do not exist in this system) again each position in an octal number represents a power of the base(8). The decimal equivalent of the octal number 2057 (written as 20578) is as follows: (2x83) + (0x82) + (5x81) + (7x80) =1024 + 0 + 40 + 7 = 1071 So we have 20578=107110 Observe that since there are only 8 digits in the octal number system, So 3 bits (23=8) are sufficient to represent any octal number in binary. 3. HEXADECIMAL NUMBER SYSTEM The hexadecimal number system is one with base of 16. The base of 16 suggests choices of 16 single character digits as follows: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F The symbols A, B,C,D,E,F representing the decimal values 10,11,12,13,14,15 respectively. Each position in a hexadecimal system represents a power of the base (16) thus the decimal equivalent of the hexadecimal number (1AF)16 (1 x 162 ) + (A x 161 ) + (F x 160) =(1x 256) + (10x 16) + (15 x 1) =256 + 160 + 15 =431 2 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Thus (1AF)16 =(431)10 CONVERTING FROM ONE NUMBER SYSTEM TO ANOTHER There are many methods or techniques that can be used to convert numbers from one base to another. We will see one technique used in converting to base 10 from any other base and second technique to be used in converting from base 10 to any other base. 1. Converting To Decimal From Another Base The following three steps are used to convert to a base 10 value from any other number system. Step1. Determine the column (positional) value of each digit ( this depends on the position of the digit and the base of the number system) Step2 Multiply the obtained column values (in Step1 ) by the digit in corresponding columns. Step3. Sum the products calculated in Step2. The total is the equivalent value in decimal. Example1: 110012 = ?10 Solution: Step1. Determine column value. Column No. from right Column Value 1 20=1 2 21=2 3 22=4 4 23=8 5 24=16 Step2. Multiply column values by corresponding column digit. 16 8 4 2 1 x x x x x 1 1 0 0 1 __________________________ 16 8 0 0 1 Step3. Sum the product 16 + 8 + 0 + 0 + 1=25 3 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Hence 110012=2510 Example 2. (4706)8=?10 Solution: 4x8 + 7x8 + 0x8 + 6x8 =4x512 + 7x64 +0x8 + 6x1 =2048 + 448 + 0 + 6 =2502 Hence (4706)8=(2502)10 Example 3. (1AC)16 =?10 Solution: (1AC)16=1x162 + 10x161 +12x160 =256 + 160 + 12 Hence =42810 Example 4. 40527=?10 Solution: 4x73 + 0x72 + 5x71 + 2x70 =4x343 + 0x49 + 5x7 + 2x1 =1372 + 0 + 35 + 2 Hence =(1409)10 Example 5. (4052)6=?10 Solution: 4x6 + 0x6 + 5x6 + 2x6 =4x216 + 0 + 5x6 + 2x1 =864 + 0 + 30 + 2 =89610 Hence (4052)6 =(896)10 2. Converting From Base 10 To A New Base The following four steps are used to convert a number from base 10 to another base. Step1. Divide the decimal number to be converted by the value of the new base. Step2. Record the remainder from Step1 as the rightmost digit ( least significant digit) of the new bas number. Step3. Divide the quotient of the previous divide by the new base. 4 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Step4. Record the remainder from Step3 as the next digit (to the left) of the new base number. Repeat Step3 and 4 recording remainders from right to left, until the quotient becomes zero in Step3. Note that the last remainder thus obtained will be the most significant digit(MSD) of the new base number. Example: 2510=?2 Step 1&2: 25/2= 12 and remainder 1 Step 3&4: 12/2= 6 and remainder 0 Step 3&4: 6/2 = 3 and remainder 0 Step 3&4: 3/2 = 1 and remainder 1 Step 3&4: 1/2 = 0 and remainder 1 As mentioned in step 2&4, the remainders have to be arranged in the reverse order so that the first remainder becomes the least significant (LSD) and the last remainder becomes the most significant digit (MSD). Hence (25)10 = (11001)2 Example 1. 4210 = ?2 2 42 2 21 0 2 10 1 2 5 0 2 2 1 1 0 Hence: (42)10= (101010)2 5 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Example 2. (952)10 =?8 8 952 8 119 0 8 14 7 2 1 6 Hence (952)10 = (1670)8 Example 3. (428)10 = ?16 16 428 16 26 12 1 10 Hence (428)10 = (1AC)16 3. Converting From A Base Other Than 10 To A Base Other Than 10 The following two steps are used to convert one number system to another number system. Step 1. Convert the original number to a decimal number (Base 10) Step 2. Convert the decimal so obtained to the new base. Example. (545)6 = ?4 Step 1. Conversion from base 6 to base 10 5x6 + 4x6 + 5x6 = 5x36 + 4x6 + 5x1 =180 + 24 + 5 = (209)10 Step 2. Convert (209)10 to base 4 4 209 4 52 1 4 13 0 3 1 Hence (545)6 = (3101)4 6 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 4. Shortcut Method For Binary To Octal Conversion The following steps are used in this method. Step 1. Divide the binary digits into groups of three (starting from right). Step 2. Convert each group of three binary digits into one octal digit by using weight 1,2 and 4 from right side. Example. (101110)2 = ?8 Step 1. divide the binary digits into groups of 3 starting from right (LSD) 101 110 Step 2. Convert each group into one digit of octal. (101)2 = 1x22 + 0x21 + 1x20 =4 + 0 + 1 =58 (110)2 = 1x22 + 1x21 + 0x20 =4 + 2 + 0 =68 Hence (101110)2 = (56)8 5. Shortcut Method For Octal To Binary Conversion The following steps are used in this method. Step 1. Convert each octal digit to a 3 digit binary number. Step 2. Combine all the resulting binary groups into a single binary number. Example. (562)8 = ?2 Solution. Step 1. Convert each octal digit to 3 binary digits. 58 = 1012 68 = 1102 28 = 0102 Step 2. Combine the binary groups. Hence: 5628 = (101110010)2 6. Shortcut Method For Binary To Hexadecimal Conversion The following steps are used in this method. Step 1. Divide the binary digits into groups of four. Step 2. Convert each group of four binary digits to one hexadecimal digit. 7 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Example. (11010011)2 = ?16 Solution. Step 1. Divide the binary digits into group of 4. Step 2. 1101 0011 Convert each group of 4 binary digits to 1 hexadecimal digit. 11012 = 1x23 + 1x22 + 0x21 + 1x20 =8+4+0+1 = 1310 = D16 00112 = 0x23 + 0x22 + 1x21 + 1x20 =0+0+2+1 = 316 Hence (1010011)2 = D316 7. Shortcut Method For Hexadecimal To Binary Conversion The following steps are used in this method Step 1. Convert the decimal equivalent of each hexadecimal digit to 4 binary digits. Step 2. Combine all the resulting binary groups into a single binary number. Example. (2AB)16 = ?2 Solution. Step 1. Convert the decimal equivalent of each hexadecimal digit to 4 binary digits. 216 = 210 = (0010)2 A16 = (10)10 = (1010)2 B16 = (11)10 = (1011)2 Step 2. Combine the binary groups. 2AB16 = (001010101011)2 8. Fractional Numbers In a binary number system, fractional numbers are formed in the same general way as in the decimal system. 0.235 = (2x10-1) + (3x10-2) + (5x10-3) and 68.53 = (6x101) +(8x100) + (5x10-1) + (3x10-2) 8 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA. INTRODUCTION TO COMPUTER LECTURE 02 Similarly in binary system 0.101 = (1x2-1) + (0x2-2) + (1x2-3) and 10.01 = (1x21) + (0x20) + (0x2-1) + (1x2-2) 9 PREPARED BY SADIQUE A. BUGTI LECTURER BUITMS QUETTA.

DOCUMENT INFO

Shared By:

Categories:

Stats:

views: | 4 |

posted: | 10/22/2011 |

language: | English |

pages: | 9 |

Description:
Introduction to Computer All 18 Lectures of My Class

OTHER DOCS BY coooolone

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.