Goals. Unary Numbers. Decimal Numbers. 3,148 is s 100 s 10 s 1 s. Number Bases 1/12/2009. COMP370 Intro to Computer Architecture 1

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1 Number Bases //9 Goals Numbers Understand binary and hexadecimal numbers Be able to convert between number bases Understand binary fractions COMP37 Introduction to Computer Architecture Unary Numbers Decimal Unary Decimal Numbers Decimal numbers work well for humans. Each position is times the one to the right 3,48 is s s s s COMP37 Intro to Computer Architecture

2 Number Bases //9 Binary Numbers Computers usually use binary numbers, base Binary numbers only have two digits, and Each position is times the one to the right (9 ) is ( ) 3 s 6 s 8 s 4 s s s Hexadecimal Numbers It can be convenient to represent values in base 6 or hexadecimal. Hex numbers have the digits A B C D E F Each position is 6 times the one to the right 3E8 6 (,776 ) is 496 s 56 s 6 s s 3 E 8 Indicating the Number Base The number or radix base can be specified by a subscript after the number. 47 F 6 Computer languages use a prefix 47 B XF Binary starts with zero B (C++ only) Hexadecimal starts with zero X Converting Decimal to Binary Start with the largest even power of bigger than the decimal number For each power of two if the number >= power of { put a bit to the right; Subtract power of from the number; else { put a to the right; Repeat for all powers of COMP37 Intro to Computer Architecture

3 Number Bases // Decimal to Binary Example Convert 37 to binary The answer is Decimal to Binary Conversion Convert the following decimal numbers to binary: = = = = = = = = Repeated Division Conversion To convert a decimal number to any base, R Generated digits are right to left, least to most number = decimal number; while (number > ) { rem = number % R; Next digit is rem; number = number / R; // integer division Decimal to Binary Conversion Repeated Division by Method: Convert to = 6 binary 6 = 3 Remainder 3 = = MSB LSB Stop when the whole number quotient is. COMP37 Intro to Computer Architecture 3

4 Number Bases //9 Decimal to Binary Example 9 Convert 9 to 9 binary 4 = 9 = 4 = = = Remainder MSB LSB Converting Decimal to Hex Start with the largest even power of 6 bigger than the decimal number Repeat for all powers of 6, big to small if the number >= power of 6 { quotient = number / power of 6; append quotient to result; number = number % power of 6; Add the remainder to the result; Decimal to Hex Example /56 =, rem= /6=3, rem=3 D Convert 37 to Hex The answer is 3D Decimal to Hex Conversion Repeated Division by 6 Method: 99 Convert 99 to 6 = 8 hex. 8 = 6 6 = Remainder or B Stop when the whole number quotient is. MSB B LSB COMP37 Intro to Computer Architecture 4

5 Number Bases //9 Binary to Decimal Conversion For each bit, add the value of that bit. Convert the binary whole number to decimal Weight: Binary number: = = = 9 Decimal Binary Hexadecimal Decimal Binary Hexadecimal A B C 3 D 4 E 5 F Online Quiz Complete the online quiz on base number conversion on the assignments page of the web site, Due by 5:pm on Friday, January 6, 9 Your score is the percentage right minus the number of seconds over 9. You may take the exam as often as you like. Your grade will be determined by your best score. This is not easy. COMP37 Intro to Computer Architecture 5

6 Number Bases //9 Converting Hex to Binary Converting Binary to Hex Each hexadecimal digit converts to four bits. Convert A7C to binary A 7 C Group the binary bits into groups of four. Each group of four bits converts to a single hexadecimal digit. Convert to hexadecimal A D Decimal Fractions To the right of the decimal or radix point, each digit it is / th of the digit it to the lft left. s s. / s / s / s Binary Fractions To the right of the radix point, each digit is ½ of the digit it to the left. The decimal number.375 is 8 s 4 s s s. / s /4 s /8 s. COMP37 Intro to Computer Architecture 6

7 Number Bases //9 Example Binary Fractions Decimal Binary Binary to Decimal Conversion Convert the Fractional binary number. to decimal Weight: 3 4 Binary number:.. = = =.6875 Decimal to Binary Conversion number = decimal fraction; while (number > ) { number = number * ; if ( number >= ) { append to result; result = result ; else { append to result; Decimal to Binary Example Convert.35 to binary 3.5 * =.65 result =..65 * =.5 result =..5 * =.5 result =..5 * =. result =. COMP37 Intro to Computer Architecture 7