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Number system and computer codes

Chapter 2. Number system and computer codes. Prelude. Fingers, sticks, and other things for counting were not enough! Counting large numbers Count in groups. Evolution of the number system. Number systems. A set of values used to represent quantity Non-Positional Number Systems

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Number system and computer codes

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  1. Chapter 2 Number system and computer codes

  2. Prelude • Fingers, sticks, and other things for counting were not enough! • Counting large numbers • Count in groups Evolution of the number system

  3. Number systems • A set of values used to represent quantity • Non-Positional Number Systems • count with their fingers, stones and pebbles • difficult to perform arithmetic operations • No zero, difficult to calculate large numbers • E.g. the Roman number system • Positional Number Systems • Finite number of symbols to represent any numbers • Symbol and it’s position defines a number • Decimal, binary, octal, hexadecimal

  4. ASCII- American standard for Information Interchange

  5. Base or radix • Number of unique digits

  6. Number Systems - Decimal • The decimal system is a base-10 system. • There are 10 distinct digits (0 to 9) to represent any quantity. • For an n-digit number, the value that each digit represents depends on its weight or position. • The weights are based on powers of 10. 1024 = 1*103 + 0*102 + 2*101 + 4*100= 1000 + 20 + 4

  7. Number Systems - Binary • The binary system is a base-2 system. • There are 2 distinct digits (0 and 1) to represent any quantity. • For an n-digit number, the value of a digit in each column depends on its position. • The weights are based on powers of 2. 10112 = 1*23 + 0*22 + 1*21 + 1*20 =8+2+1 =1110

  8. Number Systems - Octal • Octal and hexadecimal systems provide a shorthand way to deal with the long strings of 1’s and 0’s in binary. • Octal is base-8 system using the digits 0 to 7. • To convert to decimal, you can again use a column weighted system • 75128 = 7*83 + 5*82 + 1*81 + 2*80 = 391410 • An octal number can easily be converted to binary by replacing each octal digit with the corresponding group of 3 binary digits 75128 = 1111010010102

  9. Number Systems - Hexadecimal • Hexadecimal is a base-16 system. • It contains the digits 0 to 9 and the letters A to F (16 digit values). • The letters A to F represent the unit values 10 to 15. • This system is often used in programming as a condensed form for binary numbers (0x00FF, 00FFh) • To convert to decimal, use a weighted system with powers of 16.

  10. Example- Value of 2001 in Binary, Octal and Hexadecimal

  11. Example- Conversion: Binary  Octal  Hexadecimal

  12. Converting decimal to binary, octal and hexadecimal • To convert from decimal to a different number base such as Octal, Binary or Hexadecimal involves repeated division by that number base • Keep dividing until the quotient is zero • Use the remainders in reverse order as the digits of the converted number Repeated Divide by 2

  13. BaseN to Decimal Conversions • Multiply each digit by increasing powers of the base value and add the terms • Example: 101102 = ??? (decimal) CPE1002 (c) Monash University

  14. Binary Addition • Similar to decimal operation • Leading zeroes are frequently dropped. 4 Possible Binary Addition Combinations: (1) 0 (2) 0 +0 +1 00 01 (3) 1 (4) 1 +0 +1 01 10 Ex 1,2,3 For Exam Carry Sum

  15. Binary Subtraction • Just like subtraction in any other base • Minuend 10110 • Subtrahend - 10010 • Difference 00100 • And when a borrow is needed. Note that the borrow gives us 2 in the current bit position. Ex 1,2 For Exam

  16. And a full example • And more ripple -

  17. Octal/Hex addition/subtraction • Octal Addition 1 1 1 Carries 5 4 7 1Augends + 3 7 5 4 Addend 11445 Sum • Octal Subtraction 6 10 4 10Borrows 7 4 5 1 Minuend - 5 6 4 3 Subtrahend 1 6 0 6 Difference • Hexadecimal Addition 1 0 1 1 Carries 5 B A 9 Augend + D 0 5 8 Addend 1 2 C 0 1 Sum • Hexadecimal Subtraction 9 10 A 10 Borrows A 5 B 9 Minuend + 5 8 0 D Subtrahend 4 D A C Difference

  18. BCD • Binary-coded decimal, or BCD, is a method of using binary digits to represent the decimal digits 0 through 9. A decimal digit is represented by four binary digits … The binary combinations 1010 to 1111 are invalid and are not used.

  19. ASCII Code • "ask-key“- common code for microcomputer • Standard ASCII character set • 128 decimal numbers ranging (0-127) • Assigned to letters, numbers, punctuation marks, and the most common special characters. • The Extended ASCII Character Set • also consists of 128 decimal numbers (128-255) • representing additional special, mathematical, graphic, and foreign characters. • Groups of 32 characters

  20. EBCDIC - Extended Binary Coded Decimal Interchange Code • It is an 8 bit character encoding • Used on IBM mainframes and AS/400s. • It is descended from punched cards • The first four bits are called the zone • category of the character • Last four bits are the called the digit • identify the specific character • There are a number of different versions of EBCDIC, customized for different countries.

  21. Assignments IOA, IA, GA, Case !@#$

  22. Binary • Multiplication Division 1 1 0 1 0 Multiplicand x 1 0 1 0 Multiplier 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 Product Chapter 1

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