ITE102 – Computer Programming (C++) Number System ITE102 – Computer Programming (C++)

Topics 1-1 Introduction to Binary 1-2 Understanding Number System Problem Solving Topics 1-1 Introduction to Binary 1-2 Understanding Number System 1-3 Conversion of Number System Number Systems

Learning Objects After completing this module that student will be able to : Distinguish the different number systems Convert from one number system to another Number Systems

Introduction to data conversion Problem Solving Introduction to data conversion The study of binary system will help us gain better understanding of how computers perform computation. Number Systems

Understanding Number System Problem Solving Understanding Number System The radix, or base, of a number system is the total number of unique symbols available in that system. The largest valued symbol always has a magnitude of one less than the radix. Number Systems

Problem Solving Decimal Numbers Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called digits) for ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent numbers. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (plus) or − (minus) to indicate sign. Number Systems

Decimal Numbers (cont.) The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc. The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right. Number Systems

Binary Numbers (studied by Gottfried Leibniz in 1679) Problem Solving Binary Numbers (studied by Gottfried Leibniz in 1679) The binary system works in exactly the same way, except that its place value is based on the number two. In the binary system, we have the one's place, the two's place, the four's place, the eight's place, the sixteen's place, and so on. Each place in the number represents two times (2X's) the place to its right. Binary number system has a base, or radix, of 2. Binary numbers are composed of two symbols: 0 and 1. Number Systems

Binary Numbers (cont.) 1 0 1 0 8 + 2 = 10 Decimal Binary 0 0000 1 0001 Problem Solving Binary Numbers (cont.)  Decimal Binary  0 0000 1 0001  2  0010  3  0011  4  0100  5  0101  6  0110  7  0111  8  1000  9  1001  10  1010 8 + 2 = 10 8 4 2 1 1 0 1 0 Considering the digits that has a value of 1 and adding it number marker on the top of each digits Number Systems

Problem Solving Octal Numbers The octal number system has a base, or radix, of 8. Octal numbers are composed of eight symbols: 0, 1, 2, 3, 4, 5, 6, and 7. Number Systems

Problem Solving Hexadecimal Numbers The hexadecimal number system has a base, or radix, of 16. Hexadecimal numbers are composed of sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Symbols A to F correspond to decimal numbers 10 to 15. Number Systems

Conversion of the Number System Problem Solving Conversion of the Number System Binary to Decimal , Octal, Hexadecimal Decimal to Binary, Octal, Hexadecimal Octal to Decimal, Binary, Hexadecimal Hexadecimal to Binary, Octal, Decimal Number Systems

Problem Solving Binary to Decimal Remember that Binary numbers are based on the radix of 2 while Decimal numbers are based on the radix of 10. Remember also that binary will only be represented in 1s and 0s. Steps in converting Binary to Decimal: Place a number marker on the top of the given digits, starting from 0 up the last given digits—starting from the right to determine the exponent to use. Consider all the 1s in the given digits and multiply it with the base number of the given digits (which is base 2) and raised it with power of the number corresponded in the number marker you placed on the top of the given digits. Number Systems

Example: Binary to Decimal Problem Solving Example: Binary to Decimal Place the number marker on the top of the given digits, starting from the right, starting from 0 up to the last given digit on the left. 1010102 = ?10 = 101010 = 1*25 + 1*23 + 1*21 = 32 + 8 + 2 = 4210 5 4 3 2 1 0 Number Systems

Example: Binary to Decimal Problem Solving Example: Binary to Decimal 101.012 = ?10 = 101.01 = 1*22 + 1*20 + 1*2-2 = 4 + 1 + 0.25 = 5.2510 If dealing with fraction in binary the marker will start at the right side of the decimal point and starting with the -1 up to the last given digit. 1*2-2 will be converted into 1*2½ which results into 0.25 because you will be dividing 1 / 4 which comes from 1*2½ 2 1 0 -1 -2 101.012 = ?10 1 0 1 . 0 1 = 1*22 + 0*21 + 1*20 + 0*2-1 + 1*2-2 = 4 + 1 + 0.25 = 5.2510

Binary to Octal Since one octal digit is equivalent to three binary digits, just group three binary digits, starting from the least significant bit (right side). Append 0 to the most significant bit (left side), if the grouping does not have enough to form three binary digits. In short, you must complete the grouping of three digits. If you will be having a fraction (decimal point), append 0 to the least most significant bit (right side) of the given digits to complete the grouping of three bits. Number Systems

Binary to Octal (cont.) Steps in converting Binary to Octal: Group the given digits in three starting from the right side. If the grouping is not complete, place 0 to complete the grouping. Once you have grouped it into three digits, you starting converting the binary digits into decimal values following the concepts of binary digits (4, 2, 1) starting from the right. Number Systems

Example: Binary to Octal Problem Solving Example: Binary to Octal 11010102 = ?8 = 001 101 010 = 1 5 28 = 1528 3rd group 2nd group 1st group Add 0 to the left, to complete the grouping 4 2 1 4 2 1 4 2 1 Simply add all the number markers considering the binary digits that has 1 on it. Number Systems

Example: Binary to Octal Problem Solving Example: Binary to Octal 1101.012 = ?8 = 001 101 . 010 = 1 5 . 28 = 15.28 2nd group 1st group 1st group Add 0 to the right, to complete the grouping for the fractional part of the given binary 4 2 1 4 2 1 4 2 1 Simply add all the number markers considering the binary digits that has 1 on it. Number Systems

Binary to Hexadecimal Since one hexadecimal digit is equivalent to four binary digits, just group four binary digits, starting from the least significant bit (right side). Append 0 to the most significant bit (left side), if the grouping does not have enough to form four binary digits. In short, you must complete the grouping of four digits. If you will be having a fraction (decimal point), append 0 to the least most significant bit (right side) of the given digits to complete the grouping of three bits. Number Systems

Binary to Hexadecimal (cont.) Steps in converting Binary to Hexadecimal: Group the given digits in four starting from the right side. If the grouping is not complete, place 0 to complete the grouping. Once you have grouped it into three digits, you starting converting the binary digits into decimal values following the concepts of binary digits (8, 4, 2, 1) starting from the right. Number Systems

Example: Binary to Hexadecimal Problem Solving Example: Binary to Hexadecimal Simply add all the number markers considering the binary digits that has 1 on it. 11010102 = ?16 = 0110 1010 = 6 A16 = 6A16 2nd group 1st group Add 0 to the left, to complete the grouping 8 4 2 1 8 4 2 1 Number Systems

Example: Binary to Hexadecimal Problem Solving Example: Binary to Hexadecimal Simply add all the number markers considering the binary digits that has 1 on it. 1101.012 = ?16 = 1101 . 0100 = D . 416 = D.416 1st group 1st group Add 0 to the left, to complete the grouping 8 4 2 1 8 4 2 1 Number Systems

Decimal to Binary Converting a decimal number to a binary number is done by successively dividing the decimal number by 2 on the left side of the radix. If you will have a fractional part of the given decimal, successively multiplying the decimal number by 2 on the right side of the radix. Number Systems

Decimal to Binary (cont.) Steps in converting Decimal to Binary: Divide the given decimal number with the base number you are converting it to, which is 2. Whatever the answer you will get in the division will be divided again with the base (2) until you cannot divide the answer anymore with 2. The remainder that you will get will be the one you consider as your converted answer. Number Systems

Example: Decimal to Binary Problem Solving Example: Decimal to Binary Remainder of the division, will only have two values since it is in base 2, meaning you can only have 1 or 0 as a remainder. 610 = ?2 = 1102 = 6/2 0 = 3/2 1 = 1/2 1 = 0/2 remainder In reading the answer, you should read it upward. Number Systems

Example: Decimal to Binary Problem Solving Example: Decimal to Binary 6.62510 = ?2 = 110.1012 = .625*2 = 1.25*2 1 = 0.5*2 0 = 1.0 1 You will be multiplying only the decimal numbers with base 2 until you reach 0 in the decimal place. In reading the answer, you should read it downward. Stop here, because it is already 0 Number Systems

Decimal to Octal Converting a decimal number to an octal number is done by successively dividing the decimal number by 8 on the left side of the radix. If you will have a fractional part of the given decimal, successively multiplying the decimal number by 8 on the right side of the radix. Number Systems

Decimal to Octal (cont.) Steps in converting Decimal to Octal: Divide the given decimal number with the base number you are converting it to, which is 8. Whatever the answer you will get in the division will be divided again with the base (8) until you cannot divide the answer anymore with 8. The remainder that you will get will be the one you consider as your converted answer. Number Systems

Example: Decimal to Octal Problem Solving Example: Decimal to Octal 6610 = ?8 = 1028 = 66/8 2 = 8/8 0 = 1/8 1 = 0/8 Remainder of the division, will only have 0-7 values since it is in base 8 remainder In reading the answer, you should read it upward. Number Systems

Example: Decimal to Octal Problem Solving Example: Decimal to Octal 66.62510 = ?8 = 102.58 = .625*8 = 5. 0 5 You will be multiplying only the decimal numbers with base 8 until you reach 0 in the decimal place. In reading the answer, you should read it downward. Stop here, because it is already 0 Number Systems

Decimal to Hexadecimal Converting a decimal number to a hexadecimal number is done by successively dividing the decimal number by 16 on the left side of the radix If you will have a fractional part of the given decimal, successively multiplying the decimal number by 16 on the right side of the radix. Number Systems

Decimal to Hexadecimal (cont.) Steps in converting Decimal to Hexadecimal: Divide the given decimal number with the base number you are converting it to, which is 16. Whatever the answer you will get in the division will be divided again with the base (16) until you cannot divide the answer anymore with 16. The remainder that you will get will be the one you consider as your converted answer. Number Systems

Example: Decimal to Hexadecimal Problem Solving Example: Decimal to Hexadecimal Remainder of the division, will only have 0-9, A-F values since it is in base 16 28610 = ?16 = 11E16 = 286/16 14 = 17/16 1 = 1/16 1 = 0/16 In reading the answer, you should read it upward. The value 14 should be converted to hexadecimal format remainder Number Systems

Example: Decimal to Hexadecimal Problem Solving Example: Decimal to Hexadecimal 286.62510 = ?16 = 11E.A16 = .625*16 = 10.0 10 You will be multiplying only the decimal numbers with base 8 until you reach 0 in the decimal place. In reading the answer, you should read it downward. Stop here, because it is already 0 Number Systems

Problem Solving Octal to Decimal Remember that Octal numbers are based on the radix of 8 while Decimal numbers are based on the radix of 10. Remember also that Octal will only be represented with value 0-7. Steps in converting Octal to Decimal: Place a number marker on the top of the given digits, starting from 0 up the last given digits—starting from the right to determine the exponent to use. Considering all the given digits, multiply it with the base number of the given digits (which is base 8) and raised it with power of the number corresponded in the number marker you placed on the top of the given digits. Number Systems

Example: Octal to Decimal Problem Solving Example: Octal to Decimal Place the number marker on the top of the given digits, starting from the right, starting from 0 up to the last given digit on the left. 7618 = ?10 = 761 = 7*82 + 6*81 + 1*80 = 56 + 48 + 1 = 10510 2 1 0 Number Systems

Example: Octal to Decimal Problem Solving Example: Octal to Decimal 1*2-2 will be converted into 1*2½ which results into 0.25 because you will be dividing 1 / 4 which comes from 1*2½ 761.188 = ?10 = 761.15 = 7*82 + 6*81 + 1*80 + 1*8-1 + 8*8-2 = 448 + 48 + 1 +0.125+0.125 = 497.2510 If dealing with fraction in octal the marker will start at the right side of the decimal point and starting with the -1 up to the last given digit. 2 1 0 -1 -2 Number Systems

Problem Solving Octal to Binary Since one octal digit is equivalent to three binary digits, just convert the individual octal digit into three binary digits. Steps in converting Octal to Binary: Convert each of the given octal number by simply using the concept of (4, 2, 1). Place a binary 1 to correspond the given octal number. Number Systems

Example of Octal to Binary Problem Solving Example of Octal to Binary Simply add all the number markers considering the binary digits that has 1 on it, to get the octal number given. 7618 = ?2 = 7 111 = 6 110 = 1 001 = 111 110 0012 4 2 1 4 2 1 4 2 1 Number Systems

Example of Octal to Binary Problem Solving Example of Octal to Binary 761.258 = ?2 = 111 110 001 . 010 101 = 111 110 001.0101012 4 2 1 4 2 1 4 2 1 4 2 1 4 2 1 7 6 1 2 5 Number Systems

Octal to Hexadecimal When converting an octal digit to a hexadecimal digit, you must first convert the octal number to binary number and group it by four and convert the grouped digits to hexadecimal by using the concept of (8, 4, 2, 1). If the digits is not enough to form a grouping of four then append 0 on the left side of the digits. If the given octal have fraction, then append 0 on the right side of the given digits. Number Systems

Octal to Hexadecimal (cont.) Steps in converting Octal to Hexadecimal: Convert the octal digits in binary by considering each given digits and represent it in binary using the concept (4, 2, 1). Once it is in binary, that’s the time you convert the binary into hexadecimal, by grouping it into four. Number Systems

Example: Octal to Hexadecimal 7618 = ?16 = 111 110 001 = 0001 1111 0001 = 1F116 7 6 1 1 F 1 Number Systems

Example: Octal to Hexadecimal 761.758 = ?16 = 111 110 001 . 111 101 = 0001 1111 0001 . 1111 0100 = 1F1.F416 7 6 1 7 5 1 F 1 F 4 Number Systems

Problem Solving Hexadecimal to Binary Since one hexadecimal digit is equivalent to four binary digits, just convert the individual hexadecimal digit into four binary digit Since one hexadecimal digit is equivalent to four binary digits, just convert the individual hexadecimal digit into four binary digit Number Systems

Example: Hexadecimal to Binary 7AE316 = ?2 = 7 0111 A 1010 E 1110 3 0011 = 0111 1010 1110 00112 Number Systems

Example: Hexadecimal to Binary 7AE.316 = ?2 = 7 0111 A 1010 E 1110 3 0011 = 0111 1010 1110 . 00112 Number Systems

Hexadecimal to Octal When converting an hexadecimal digit to an octal digit, you must first convert the hexadecimal number to binary number and group it by three and convert the grouped digits to octal by using the concept of (4, 2, 1). If the digits is not enough to form a grouping of three then append 0 on the left side of the digits. If the given hexadecimal have fraction, then append 0 on the right side of the given digits. Number Systems

Hexadecimal to Octal (cont.) Steps in converting Hexadecimal to Octal: Convert the hexadecimal digits in binary by considering each given digits Once converted to binary group the binary into three and using the concept (4, 2, 1). Number Systems

Example: Hexadecimal to Octal 7AE316 = ?8 = 7 0111 A 1010 E 1110 3 0011 = 0 111 101 011 100 0112 = 753438 7 5 3 4 3 Number Systems

Example: Hexadecimal to Octal 7AE.316 = ?8 = 7 0111 A 1010 E 1110 3 0011 = 0111 1010 1110 . 00112 = 011 110 101 110 . 001 1002 = 3656.148 3 6 5 6 1 4 Number Systems

Hexadecimal to Decimal Each hexadecimal position is weighted with a power of 16. Digits on the left side of the radix point has a positive exponent while on the right side of the radix point has a negative exponent. Converting a hexadecimal number to a decimal number is done by successively multiplying the decimal number by 16 on the left side of the radix If you will have a fractional part of the given decimal, successively multiplying the decimal number by 16 on the right side of the radix. Number Systems

Hexadecimal to Decimal (cont.) Steps in converting Hexadecimal to Decimal: Place a number marker on the top of the given number for determining the exponent to be used. Get the individual digit and multiply it by the base number (16) and raised it with the exponent corresponds to the number marker you place on each digit, then to the addition operation. Number Systems

Introduction to Computer Programming 55

Example: Hexadecimal to Decimal Problem Solving Example: Hexadecimal to Decimal 286.A16 = ?10 = 286.A = 2*162 + 8*161 + 6*160 + A*16-1 = 646. 62510 2 1 0 -1 Number Systems

Example: Hexadecimal to Decimal Problem Solving Example: Hexadecimal to Decimal 286A16 = ?10 = 286A = 2*163 + 8*162 + 6*161 + A*160 = 1034610 3 2 1 0 Number Systems

Review of Number Systems

Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 Octal 8 0, 1, … 7 Hexa- decimal 16 0, 1, … 9, A, B, … F

Quantities/Counting (1 of 3) Decimal Binary Octal Hexa- decimal 1 2 10 3 11 4 100 5 101 6 110 7 111 p. 33

Quantities/Counting (2 of 3) Decimal Binary Octal Hexa- decimal 8 1000 10 9 1001 11 1010 12 A 1011 13 B 1100 14 C 1101 15 D 1110 16 E 1111 17 F

Quantities/Counting (3 of 3) Decimal Binary Octal Hexa- decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 10100 24 14 10101 25 15 10110 26 10111 27 Etc.

Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal

Quick Example 2510 = 110012 = 318 = 1916 Base

Decimal to Decimal Decimal Octal Binary Hexadecimal Next slide…

Weight 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base

Binary to Decimal Decimal Octal Binary Hexadecimal

Binary to Decimal Technique Multiply each bit by 2n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results

Example Bit “0” 1010112 => 1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 = 32 4310

Octal to Decimal Decimal Octal Binary Hexadecimal

Octal to Decimal Technique Multiply each bit by 8n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results

Example 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810

Hexadecimal to Decimal Octal Binary Hexadecimal

Hexadecimal to Decimal Technique Multiply each bit by 16n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results

Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810

Decimal to Binary Decimal Octal Binary Hexadecimal

Decimal to Binary Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.

Example 2 125 62 1 12510 = ?2 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 12510 = 11111012

Octal to Binary Decimal Octal Binary Hexadecimal

Octal to Binary Technique Convert each octal digit to a 3-bit equivalent binary representation

Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012

Hexadecimal to Binary Decimal Octal Binary Hexadecimal

Hexadecimal to Binary Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation

Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112

Decimal to Octal Decimal Octal Binary Hexadecimal

Decimal to Octal Technique Divide by 8 Keep track of the remainder

Example 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 8 0 2 123410 = 23228

Decimal to Hexadecimal Octal Binary Hexadecimal

Decimal to Hexadecimal Technique Divide by 16 Keep track of the remainder

Example 123410 = ?16 16 1234 77 2 16 4 13 = D 0 4 123410 = 4D216

Binary to Octal Decimal Octal Binary Hexadecimal

Binary to Octal Technique Group bits in threes, starting on right Convert to octal digits

Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278

Binary to Hexadecimal Decimal Octal Binary Hexadecimal

Binary to Hexadecimal Technique Group bits in fours, starting on right Convert to hexadecimal digits

Example 10101110112 = ?16 10 1011 1011 B B 10101110112 = 2BB16

Octal to Hexadecimal Decimal Octal Binary Hexadecimal

Octal to Hexadecimal Technique Use binary as an intermediary

Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16

Hexadecimal to Octal Decimal Octal Binary Hexadecimal

Hexadecimal to Octal Technique Use binary as an intermediary

Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148

Binary Addition

Introduction to Computer Programming 104 Binary Addition Introduction to Computer Programming 104

Binary Addition Two 1-bit values A B A + B 1 10 “two” pp. 36-38

Binary Addition Two n-bit values Add individual bits Propagate carries E.g., 1 1 10101 21 + 11001 + 25 101110 46

Introduction to Computer Programming 107 Binary Addition Introduction to Computer Programming 107

Introduction to Computer Programming 108 Binary Subtraction Introduction to Computer Programming 108

Introduction to Computer Programming 109 Binary Subtraction Introduction to Computer Programming 109

Fractions Decimal to decimal 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14 pp. 46-50

Fractions Binary to decimal 10.1011 => 1 x 2-4 = 0.0625 1 x 2-3 = 0.125 0 x 2-2 = 0.0 1 x 2-1 = 0.5 0 x 20 = 0.0 1 x 21 = 2.0 2.6875

Introduction to Computer Programming 112 Fractions Introduction to Computer Programming 112

Fractions

Introduction to Computer Programming 114 Fractions Introduction to Computer Programming 114

Introduction to Computer Programming 115 Fractions Introduction to Computer Programming 115

Introduction to Computer Programming 116 Fractions Introduction to Computer Programming 116

Fractions Decimal to binary 3.14579 11.001001... .14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. 3.14579 11.001001... p. 50

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