Number System Conversions PDF

NUMBER SYSTEM Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system. Computer architecture supports following number systems.  Binary number system  Oct...

NUMBER SYSTEM Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system. Computer architecture supports following number systems.  Binary number system  Octal number system  Decimal number system  Hexadecimal (hex) number system BINARY NUMBER SYSTEM A Binary number system has only two digits that are 0 and 1. Every number (value) represents with 0 and 1 in this number system. The base of binary number system is 2, because it has only two digits. OCTAL NUMBER SYSTEM Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits. DECIMAL NUMBER SYSTEM Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits. HEXADECIMAL NUMBER SYSTEM A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 14, E is 15 and F is 16. Number system Base(Radix) Used digits Example Binary 2 0,1 (11110000)2 Octal 8 0,1,2,3,4,5,6,7 (360)8 Decimal 10 0,1,2,3,4,5,6,7,8,9 (240)10 0,1,2,3,4,5,6,7,8,9, Hexadecimal 16 (F0)16 A,B,C,D,E,F CONVERSIONS DECIMAL TO OTHER 1. DECIMAL TO BINARY Decimal Number System to Other Base To convert Number system from Decimal Number System to Any Other Base is quite easy; you have to follow just two steps: A) Divide the Number (Decimal Number) by the base of target base system (in which you want to convert the number: Binary (2), octal (8) and Hexadecimal (16)). B) Write the remainder from step 1 as a Least Signification Bit (LSB) to Step last as a Most Significant Bit (MSB). Decimal to Binary Conversion Result Decimal Number is : (12345)10 Binary Number is (11000000111001)2 2. DECIMAL TO OCTAL Decimal to Octal Conversion Result Decimal Number is : (12345)10 Octal Number is (30071)8 3. DECIMAL TO HEXADECIMAL Decimal to Hexadecimal Conversion Result Example 1 Decimal Number is : (12345)10 Hexadecimal Number is (3039)16 Example 2 Decimal Number is : (725)10 Hexadecimal Number is (2D5)16 Convert 10, 11, 12, 13, 14, 15 to its equivalent... A, B, C, D, E, F BINARY TO OTHER A) Multiply the digit with 2(with place value exponent). Eventually add all the multiplication becomes the Decimal number. 1. BINARY TO DECIMAL 2. BINARY TO OCTAL An easy way to convert from binary to octal is to group binary digits into sets of three, starting with the least significant (rightmost) digits. Binary: 11100101 = 11 100 101 011 100 101 Pad the most significant digits with zeros if necessary to complete a group of three. Then, look up each group in a table: Binary: 000 001 010 011 100 101 110 111 Octal: 0 1 2 3 4 5 6 7 Binary = 011 100 101 Octal = 3 4 5 = 345 oct 3. BINARY TO HEXADECIMAL An equally easy way to convert from binary to hexadecimal is to group binary digits into sets of four, starting with the least significant (rightmost) digits. Binary: 11100101 = 1110 0101 Then, look up each group in a table: Binary: 0000 0001 0010 0011 0100 0101 0110 0111 Hexadecimal: 0 1 2 3 4 5 6 7 Binary: 1000 1001 1010 1011 1100 1101 1110 1111 Hexadecimal: 8 9 A B C D E F Binary = 1110 0101 Hexadecimal = E 5 = E5 hex OCTAL TO OTHER 1. OCTAL TO BINARY Converting from octal to binary is as easy as converting from binary to octal. Simply look up each octal digit to obtain the equivalent group of three binary digits. Octal: 0 1 2 3 4 5 6 7 Binary: 000 001 010 011 100 101 110 111 Octal = 3 4 5 Binary = 011 100 101 = 011100101 binary 2. OCTAL TO HEXADECIMAL When converting from octal to hexadecimal, it is often easier to first convert the octal number into binary and then from binary into hexadecimal. For example, to convert 345 octal into hex: (from the previous example) Octal = 3 4 5 Binary = 011 100 101 = 011100101 binary Drop any leading zeros or pad with leading zeros to get groups of four binary digits (bits): Binary 011100101 = 1110 0101 Then, look up the groups in a table to convert to hexadecimal digits. Binary: 0000 0001 0010 0011 0100 0101 0110 0111 Hexadecimal: 0 1 2 3 4 5 6 7 Binary: 1000 1001 1010 1011 1100 1101 1110 1111 Hexadecimal: 8 9 A B C D E F Binary = 1110 0101 Hexadecimal = E 5 = E5 hex Therefore, through a two-step conversion process, octal 345 equals binary 011100101 equals hexadecimal E5. 3. OCTAL TO DECIMAL The conversion can also be performed in the conventional mathematical way, by showing each digit place as an increasing power of 8. 345 octal = (3 * 82) + (4 * 81) + (5 * 80) = (3 * 64) + (4 * 8) + (5 * 1) = 229 decimal OR Converting octal to decimal can be done with repeated division. 1. Start the decimal result at 0. 2. Remove the most significant octal digit (leftmost) and add it to the result. 3. If all octal digits have been removed, you’re done. Stop. 4. Otherwise, multiply the result by 8. 5. Go to step 2. Octal Digits Operation Decimal Result Operation Decimal Result 345 +3 3 ×8 24 45 +4 28 ×8 224 5 +5 229 done.   (345)8 =(229)10 HEXADECIMAL TO OTHER 1. HEXADECIMAL TO BINARY Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits. Hexadecimal: 0 1 2 3 4 5 6 7 Binary: 0000 0001 0010 0011 0100 0101 0110 0111 Hexadecimal: 8 9 A B C D E F Binary: 1000 1001 1010 1011 1100 1101 1110 1111 Hexadecimal = A 2 D E Binary = 1010 0010 1101 1110 = 1010001011011110 binary 2. HEXADECIMAL TO OCTAL 1's complement The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is called as taking complement or 1's complement. Example of 1's Complement is as follows. Binary Addition It is a key for binary subtraction, multiplication, division. There are four rules of binary addition. In fourth case, a binary addition is creating a sum of (1 + 1 = 10) i.e. 0 is written in the given column and a carry of 1 over to the next column. Example − Addition 2's complement The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number. 2's complement = 1's complement + 1 Example of 2's Complement is as follows. Rules of Binary Addition  0 + 0 = 0  0 + 1 = 1  1 + 0 = 1  1 + 1 = 0, and carry 1 to the next more significant bit For example, 00011010 + 00001100 = 00100110 1 1 Carries 0 0 0 1 1 0 1 0 = 26(base 10) + 0 0 0 0 1 1 0 0 = 12(base 10) 0 0 1 0 0 1 1 0 = 38(base 10) 00010011 + 00111110 = 01010001 1 1 1 1 1 carries 0 0 0 1 0 0 1 1 = 19(base 10) + 0 0 1 1 1 1 1 0 = 62(base 10) 0 1 0 1 0 0 0 1 = 81(base 10) Rules of Binary Multiplication  0 x 0 = 0  0 x 1 = 0  1 x 0 = 0  1 x 1 = 1, and no carry or borrow bits For example, 00101001 × 00000110 = 11110110 0 0 1 0 1 0 0 1 = 41(base 10) × 0 0 0 0 0 1 1 0 = 6(base 10) 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 = 246(base 10) Binary Division Binary division is the repeated process of subtraction, just as in decimal division. For example, 00101010 ÷ 00000110 = 1 1 1 = 7(base 10)  00000111 1 1 1 0 ) 0 0 1 0 1 0 1 0 = 42(base 10) - 1 1 0 = 6(base 10) 1 borrows 1 1 0 0 1 - 1 1 0 1 1 0 - 1 1 0 0 10000111 ÷ 00000101 = 1 1 0 1 1 = 27(base 10) 00011011 1 135(base 1 0 1 ) 1 0 0 0 0 1 1 1 = 10) - 1 0 1 = 5(base 10) 1 1 1 0 - 1 0 1 1 1 - 0 1 1 1 - 1 0 1 1 0 1 - 1 0 1 0 Example − Division  HTML Tags are containers. The html tag indicates that everything between and is code that conforms to the standards of the type of HTML. Everything between the opening tag and closing tag are inside that tag and therefore have the attributes that tag gives them. Those attributes can be modified. SR. TAG ATTRIBUT DESCRIPTION EXAMPLE NO NAME ES 1. The The element is a container for followin all the head elements. g elements The element can include a title Title of the document can go for the document, scripts, styles, meta inside information, and more. the The content of the document...... element: 2. href The tag defines a hyperlink, target=_b which is used to link from one page to lank another. Title of the document _parent The most important attribute of the element is the href attribute, _self which indicates the link's destination. Visit yahoo.com! _top 3. align The tag defines a paragraph. Title of the document This is some text in a paragraph. 4. color The tag specifies the font face, face font size, and color of text. Title of the document size This is some text! This is some text! This is some text! 5. - align The to tags are used to define HTML headings. Title of the document defines the most important heading. defines the least This is heading 1 important heading. This is heading 2 This is heading 3  This is heading 4 This is heading 5 This is heading 6 6. align The tag defines an HTML bgcolor table. Title of the document border cellpaddi An HTML table consists of the ng element and one or more , cellspacin , and elements. g Month width The element defines a table row, Savings the element defines a table header, and the element defines a table cell. January $100 7. compact The tag defines an unordered Title of the type (bulleted) list. document Coffee Use the tag together with the Tea Milk tag to create unordered lists. 8. align The tag defines an image in an alt HTML page. Title of the document border height The tag has two required