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Numbers & Logic

Learn about numeric systems like decimal, binary, octal, and hexadecimal, along with logical expressions and operators in this informative chapter.

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Numbers & Logic

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  1. Numbers & Logic Bits & Pieces Chapter 0

  2. Base 10 example • Decimal Number 9 7 0 1 • Place 4 3 2 1 • Place - 1 3 2 1 0 • 10(place - 1) 103 102 101 100 • =============================== • = 9*1000 + 7*100 + 0*10 + 1*1 • = 9701 Chapter 0

  3. Numeric Values • The numeric value of a set of digits is determined as: • The sum of the products of each digit and its corresponding place value, • where the place value is the numeric-base raised to the place - 1. Chapter 0

  4. Base 2 example • Binary Number 0 1 0 1 • 2(place - 1) 23 22 21 20 • =============================== • = 0*8 + 1*4 + 0*2 + 1*1 • = 5 Chapter 0

  5. A general exampleBase n • Binary Number 0 1 0 1 • n(place - 1)n3n2n1n0 • =============================== • 0*(n * n* n) + 1*(n*n) + 0* n + 1*1 Chapter 0

  6. Commonly Used Systems • Binary Base 2 • Octal Base 8 • Decimal Base 10 • Hexadecimal Base 16 Chapter 0

  7. Legal Digits • What are the legal digits? • Start at zero and stop at the base - 1 • Binary 0, 1 • Octal 0, 1, 2, 3, 4, 5, 6, 7 • Decimal 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Hex 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Chapter 0

  8. What is the decimal value of? • 10101 base 2 • 10101 base 8 • 10101 base 10 • 10101 base 16 Chapter 0

  9. 0000 00 00 0 0001 01 01 1 0010 02 02 2 0011 03 03 3 0100 04 04 4 0101 05 05 5 0110 06 06 6 0111 07 07 7 1000 10 08 8 1001 11 09 9 1010 12 10 A 1011 13 11 B 1100 14 12 C 1101 15 13 D 1110 16 14 E 1111 17 15 F Counting in Binary, Octal, Decimal and Hexadecimal A single Hex digit can be used to represent the value of four binary digits Chapter 0

  10. Hex = Binary Shorthand • Hexadecimals are often used as a shorthand for large binary values. • This shorthand is useful for specifying memory locations, e.g. • Decimal - 16,274,482 • Binary - 111110000101010000110010 • Hex - F85432 Chapter 0

  11. Binary to Hex • Each Hexadecimal digit represents four binary digits • 1111 1000 0101 0100 0011 0010 • F 8 5 4 3 2 Chapter 0

  12. Binary to Octal • Each Octal digit represents three binary digits • 111 110 000 101 010 000 110 010 • 7 6 0 5 2 0 6 2 Chapter 0

  13. ASCII & EBCDIC • American Standard Code for Information Interchange • ASCII @ Wikipedia • Binary Coded Decimals • Binary Coded Decimals • Extended Binary Coded Decimals • EBCDIC @ Wikipedia • Unicode • Unicode @ Wikipeda Chapter 0

  14. Boolean Logic • A two valued logic often used in computers and information systems. • The only legal values in Boolean Logic are • TRUE • FALSE Chapter 0

  15. Logical Values • Logical values can only be True or False • Similar to numeric values, logical values can be combined into logical expressions using logical operators. • The logical operators are: not and or < <= = >= > Chapter 0

  16. Logical Expressions • a logical-expression is any expression that evaluates to False or True • False • True • notlogical-expression • logical-expression and logical-expression • logical-expressionorlogical-expression Chapter 0

  17. Logical Expressionsare not unlikeNumerical Expressions • A numerical-expression is any expression that evaluates to a legal numerical value. • Examples of numerical expressions: • 3 • -4 • 3 + 8 / 2 • (3 + 8) / 2 Chapter 0

  18. Numerical Operators • Unary operators have only one argument • the positive and negative signs are the unary numerical operators. • +- • Binary operators require two arguments • addition, subtraction, multiplication, division, and exponentiation are the binary operators • + - * / ^ Chapter 0

  19. You’ve probably already used logical expressions • The relational operators > >= = < <= evaluate to logical results. • Example • the expression 3 + 5 <= 8 - 4 evaluates to a value of False, so it is a logical expression. • Note that here we have combined numerical expressions with relational operators to form a logical expression. Chapter 0

  20. Logical Operators • Unary operators have only one argument • not is the only unary logical operator. • not • Binary operators require two arguments • conjunction and disjunction are the binary operators • and or Chapter 0

  21. A NOT A F T T F A B A AND B F F F F T F T F F T T T A B A OR B F F F F T T T F T T T T Truth Tables Chapter 0

  22. Operator Precedence • Higher precedence evaluate first, • Equal precedence evaluate left to right • Parenthesis can be used to modify the order of precedence, expressions inside parenthesis are evaluated first. Chapter 0

  23. - (unary) * / div mod numerical operators + - < = >= > <= relational operators not and logical operators or Operator Precedence Chapter 0

  24. Evaluation of Logical Expressions • A = True • B = False • Given the above evaluate the following: • A or B => True • A and B => False • 3 > 7 or A => TRUE • (3 < 7) and not A => False Chapter 0

  25. Complex Logical Expression • A = True B = False C = True D =False • A or not B and not (3 + 7 <= 10 / 2) or C and D Chapter 0

  26. Complex Logical Expression • A = True B = False C = True D =False • A or not B and not (3 + 7 <= 10 / 2) or C and D • T or not F and not (3 + 7 <= 10 / 2) or T and F Chapter 0

  27. Complex Logical Expression • A = True B = False C = True D =False • A or not B and not (3 + 7 <= 10 / 2) or C and D • T or not F and not (3 + 7 <= 10 / 2) or T and F • T or not F and not ( 10 <= 5 ) or T and F • T or not F and not ( F ) or T and F Chapter 0

  28. Complex Logical Expression • A = True B = False C = True D =False • A or not B and not (3 + 7 <= 10 / 2) or C and D • T or not F and not ( F ) or T and F • T or T and T or T and F Chapter 0

  29. Complex Logical Expression • A = True B = False C = True D =False • A or not B and not (3 + 7 <= 10 / 2) or C and D • T or T and T or T and F • T or T or F Chapter 0

  30. Complex Logical Expression • A = True B = False C = True D =False • A or not B and not (3 + 7 <= 10 / 2) or C and D • T or T or F • T or F • T Chapter 0

  31. Normal Forms • Conjunctive Normal Form • A and B and C and D and E • any false value makes the expression false • Disjunctive Normal Form • A or B or C or D or E • any true value makes the expression true Chapter 0

  32. Computer Time • millisecond 10-3 = 1/1,000 • microsecond 10-6 = 1/1,000,000 • nanosecond 10-9 = 1/1,000,000,000 • picosecond 10-12 = 1/1,000,000,000,000 • femtosecond 10-15 = 1/1,000,000,000,000,000 • Conversion of Time Units Chapter 0

  33. Computer Units • Thousand 103 = 1,000 • Kilobyte 210 = 1,024 • Million 106 = 1,000,000 • Megabyte 220 = 1,048,576 • Billion 109 = 1,000,000,000 • Gigabyte 230 = 1,073,741,824 • Trillion 1012 = 1,000,000,000,000 • Terabyte 240 = 1,099,511,627,776 Chapter 0

  34. Bigger Units • Trillion 1012 = 1,000,000,000,000 • Terabyte 240 = 1,099,511,627,776 • Quadrillion 1015 =1,000,000,000,000,000 • Petabyte 250 = 1,125,899,906,842,624 • Quintillion 1018 = 1,000,000,000,000,000,000 • Exabyte 260 = 1,152,921,504,606,846,976 • Sextillion 1021 = 1,000,000,000,000,000,000,000 • Zettabyte 270 = 1,180,591,620,717,411,303,424 • Septillion 1024 = 1,000,000,000,000,000,000,000,000 • Yottabyte 280 = 1,208,925,819,614,629,174,706,176 • Byte Converter – File Size Calculator • Million, Billion, Trillion Chapter 0

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