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Digital Electronics: Understanding Numbering Systems and Algebraic Switching

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Learn the basics of decimal, binary, hexadecimal, and octal numbering systems, arithmetic operations, number conversions, and digital vs. analog quantities. Understand fundamental principles in digital electronics.

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Digital Electronics: Understanding Numbering Systems and Algebraic Switching

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  1. EET209/4 DIGITAL ELECTRONICS CO1: Ability to apply the basic principles of numbering system and Algebraic Switching for complex digital electronics system. CHAPTER 1

  2. NUMBER & CODES • Digital vs. analog quantities • Decimal numbering system (Base 10) • Binary numbering system (Base 2) • Hexadecimal numbering system (Base 16) • Octal numbering system (Base 8) • Number conversion • Binary arithmetic • 1’s and 2’s complements of binary numbers • Signed numbers • Arithmetic operations with signed numbers • Digital codes • Logic Gates

  3. DIGITAL vs analog • Digital signals are represented by only two possible - 1 (binary 1) or 0 (binary 0) • Sometimes call these values “high” and “low” or “ true” and “false” • Example: light switch , it can be in just two position – “ on ” or “ off ”

  4. DIGITAL vs analog • More complicated signals can be constructed from 1s and 0s by stringing them end-to-end. • Example: 3 binary digits, have 8 possible combinations 000,001,010,011,100,101,110 and 111 The diagram shows an example a typical digital signal, represented as a series of voltage levels that change as time goes on.

  5. DIGITAL vs analog • Analog electronics can be any value within limits. • Example: Voltage change simultaneously from one value to the next, like gradually turning a light dimmer switch up or down. The diagram shows an analog signal that changes with time.

  6. DIGITAL waveform (a) Positive - going pulse (b) Negative - going pulse

  7. Numbering systems • We are all familiar with decimal number systems use everyday : calculator, calendar, phone or any common devices use this numbering system : Decimal = Base 10 ( )10 • Some other number systems: Binary = Base 2 ( )2 Octal = Base 8 ( )8 Hexadecimal = Base 16 ( )16

  8. Numbering systems • Decimal  0 ~ 9 • Binary  0 ~ 1 • Octal  0 ~ 7 • Hexadecimal  0 ~ F

  9. Dec Hex Octal Binary 0123456789101112131415 0123456789ABCDEF 000001002003004005006007010011012013014015016017 00000000000000010000001000000011000001000000010100000110000001110000100000001001000010100000101100001100000011010000111000001111 N U M B E R S Y S T E M S

  10. Significant digits Binary: 1 1 1 0 1 1 0 1 Most significant bit (MSB) Least significant bit (LSB) Hexadecimal: 1 D 6 3 A 7 A Most significant digit (MSD) Least significant digit (LSD)

  11. Decimal number (base 10) • Base 10 system - (0,1,2,3,4,5,6,7,8,9) • Example : 39710 3 9 7 Weights for whole numbers are positive power of ten that increase from right to left , beginning with 100 + (7 X 100) = (3 X 102) (9 X 101) + 300 + 90 + 7 = = 39710

  12. binary number (base 2) • Base 2 system – (0 , 1) • used to model the series of electrical signals- computers use to represent information • ‘0’represents the no voltage or an off state • ‘1’represents the presence of voltage or an on state • Example: 1012 • 10 1 Weights in a binary number are based on power of two, that increase from right to left, beginning with 20 = + (0 X 21) + (1 X 20) (1X 22) = 4 + 0 + 1 = 510

  13. octal number (base 8) • Base 8 system – (0,1,2,3,4,5,6,7) • multiplication and division algorithms for conversion to and from base 10 (decimal) • Example: 7568convert to decimal: • 7 5 6 Weights in a binary number are based on power of eight that increase from right to left, beginning with 80 = + + (6 X 80) (7 X 82) (5 X 81) = 448 + 40 + 6 49410 =

  14. octal number (base 8) • Readily converts to binary • Groups of three (binary) digits can be used to represent each octal number • Example: 7568convert to binary 7 6 5 111 101 1102

  15. hexadecimal number (base 16) • Base 16 system • Uses digits 0 ~ 9 & • letters A,B,C,D,E,F • Groups of four bits • represent each • base 16 digit

  16. hexadecimal number (base 16) • Base 16 system • multiplication and division algorithms for conversion to and from base 10 (decimal) • Example: A9F16convert to decimal • A 9 F Weights in a hexadecimal number are based on power of sixteen that increase from right to left, beginning with 160 = (10 X 162) (9 X 161) + (15 X 160) + 2560 + 144 + 15 = 271910 =

  17. hexadecimal number (base 16) • Readily converts to binary • Groups of four (binary) digits can be used to represent each hexadecimal number • Example:A9F16convert to binary A 9 F 1010 1001 11112

  18. NUMBER & CODES • Digital vs. analog quantities • Decimal numbering system (Base 10) • Binary numbering system (Base 2) • Hexadecimal numbering system (Base 16) • Octal numbering system (Base 8) • Number conversion • Binary arithmetic • 1’s and 2’s complements of binary numbers • Signed numbers • Arithmetic operations with signed numbers • Digital codes • Logic Gates DONE!

  19. NUMBER CONVERSION • Any Radix (base) to Decimal Conversion

  20. NUMBER CONVERSION • Binary (Base 2) to Decimal (Base 10) Conversion • Example: • Convert (10101101)2 to its decimal equivalent: • Binary 1 0 1 0 1 1 0 1 • Positional Values • Products x x x x x x x x 27 26 25 24 23 22 21 20 128 + 0 + 32 + 8 + 0 + 4 + 0 + 1 17310

  21. NUMBER CONVERSION • Octal (Base 8) to Decimal (Base 10) Conversion • Example: • Convert 6538 to its decimal equivalent: • Octal • Positional Values • Products 6 5 3 x x x 82 81 80 384 + 40 + 3 42710

  22. NUMBER CONVERSION • Hex (Base 16) to Decimal (Base 10) Conversion • Example: • Convert 3B4F16 to its decimal equivalent: • Hex • Positional Values • Products 3 B 4 F x x x x 163 162 161 160 12288 + 2816 + 64 + 15 1518310

  23. NUMBER CONVERSION • Decimal to Any Radix (Base) Conversion • INTEGER DIGIT: • Repeated division by the radix & record the remainder • FRACTIONAL DECIMAL: • Multiply the number by the radix until the answer is in integer • Example: • Convert 25.312510 to binary

  24. NUMBER CONVERSION • Decimal (Base 10) to Binary(Base 2) Conversion • Example: Convert 25.312510 to binary 2 5 = 12 + 1 2 1 2 = 6 + 0 2 6 = 3 + 0 2 3 = 1 + 1 2 1 = 0 + 1 2 remainder MSB LSB 2510 = 1 1 0 0 1 2

  25. NUMBER CONVERSION • Decimal (Base 10) to Binary(Base 2) Conversion • Example: Convert 25.312510 to binary • . 0 1 0 1 • Carry • 0.3125 x 2 = 0.625 0 • 0.625 x 2 = 1.25 1 • 0.25 x 2 = 0.50 0 • 0.5 x 2 = 1.00 1 • Answer: 1 1 0 0 1.0 1 0 12 MSB LSB

  26. NUMBER CONVERSION • Decimal (Base 10) to Octal (Base 8) Conversion • Example: Convert 42710 to octal • 427 / 8 = 53 R3 Divide by 8; R is LSB • 53 / 8 = 6 R5 Divide Q by 8; R is next digit • 6 / 8 = 0 R6 Repeat until Q = 0 6538

  27. NUMBER CONVERSION • Decimal (Base 10) to Hex (Base 16) Conversion • Example: Convert 83010 to hexadecimal • 830 / 16 = 51 R14 • 51 / 16 = 3 R3 • 3 / 16 = 0 R3 14 = E in Hex 33E16

  28. NUMBER CONVERSION • Binary (Base 2) to Octal (Base 8) Conversion (vice versa) • Grouping the binary position in groups of three starting at the least significant position. • Each octal number converts to 3 binary digits • Example: • To convert 6538 to binary, • just substitute code: 6 5 3 110 101 011

  29. NUMBER CONVERSION • Binary (Base 2) to Octal (Base 8) Conversion (vice versa) • Example: • Convert the following binary numbers to their octal equivalent (vice versa). • 1001.11112 • 47.38 • 1010011.110112 • Answer: • 11.748 • 100111.0112 • 123.668

  30. NUMBER CONVERSION • Binary (Base 2) to Octal (Base 8) Conversion (vice versa) • Example: • Substitution code can also be used to convert binary to octal by using 3-bit groupings: • 010 101 101 010 111 001 101 010 2 5 5 2 7 1 5 2 255271528

  31. NUMBER CONVERSION • Binary (Base 2) to Hexadecimal (Base 16) Conversion (vice versa) • Grouping the binary position in 4-bit groups, starting from the least significant position. • The easiest method for converting binary to hexadecimal is to use a substitution code. • Each hex number converts to 4 binary digits.

  32. NUMBER CONVERSION • Binary (Base 2) to Hexadecimal (Base 16) Conversion (vice versa) • Example: • Convert the following binary numbers to their hexadecimal equivalent (vice versa). • 10000.12 • 1F.C16 • Answer: • 10.816 • 00011111.11002

  33. NUMBER CONVERSION • Binary (Base 2) to Hexadecimal (Base 16) Conversion (vice versa) • Example: • Convert 0101011010101110011010102 to hex using the 4-bit substitution code : 0101 0110 1010 1110 0110 1010 5 6 A E 6 A 56AE6A16

  34. BINARY ARITHMETIC • Binary Addition • 0 + 0 = 0 Sum of 0 with a carry of 0 • 0 + 1 = 1 Sum of 1 with a carry of 0 • 1 + 0 = 1 Sum of 1 with a carry of 0 • 1 + 1 = 10 Sum of 0 with a carry of 1 • Example: • 11001 111 • + 1101 + 11 • 100110 ???

  35. BINARY ARITHMETIC • Simple Arithmetic • Addition • Example: • 100011002 • + 1011102 • 101110102

  36. BINARY ARITHMETIC • Binary Subtraction • 0 - 0 = 0 • 1 - 1 = 0 • 1 - 0 = 1 • 10 -1 = 1 0 - 1 with a borrow of 1 • Example: • 1011 101 • - 111 - 11 • 100 ???

  37. BINARY ARITHMETIC • Simple Arithmetic • Subtraction • Example: • 10001002 • - 1011102 • 101102

  38. BINARY ARITHMETIC • Binary Multiplication • 0 x 0 = 0 Example: • 0 x 1 = 0 • 1 x 0 = 0 100110 • 1 x 1 = 1 x 101 • 100110 • 000000 • + 100110 • 10111110

  39. BINARY ARITHMETIC • Binary Division • Use the same procedure as decimal division. • Example: • Perform the following binary divisions: • (a) 110 ÷ 11 • (b) 110 ÷ 10

  40. BINARY ARITHMETIC Solution:

  41. COMPLEMENTS OF BINARY NUMBERS • 1’s complement • Changing all the 1s to 0s and all the 0s to 1s • Example: • 1 1 0 1 0 0 1 0 1 Binary number • 0 0 1 0 1 1 0 1 0 1’s complement

  42. COMPLEMENTS OF BINARY NUMBERS • 2’s complement • Step 1: Find 1’s complement of the number • Binary # 11000110 • 1’s complement 00111001 • Step 2: Add 1 to the 1’s complement • 00111001 • + 00000001 • 00111010

  43. Signed magnitude numbers 110010.. …00101110010101 Sign bit 31 bits for magnitude Please remember that 0 bit indicates POSITIVE number, and a 1 sign bit indicates a NEGATIVE number. 0 = positive 1 = negative

  44. Signed magnitude numbers • Left most is the sign bit; • ‘0’ is for positive, and ‘1’ is for negative • Sign-magnitude (always 8 bit numbers) • 0 0 0 1 1 0 0 1 = +25 • sign bit magnitude bits • 1’s complement • The positive number is same as sign magnitude. The negative number is the 1’s complement of the corresponding positive number. • Example: • +25 is 00011001 -25 is 11100110

  45. Signed magnitude numbers • 2’s complement • The positive number – same as sign magnitude and 1’s complement • The negative number is the 2’s complement of the corresponding positive number. • Example : • Express +19 and -19 in • sign magnitude • 1’s complement • 2’s complement

  46. Signed magnitude numbers • 2’s complement • The positive number – same as sign magnitude and 1’s complement • The negative number is the 2’s complement of the corresponding positive number. • Example : • Express +19 and -19 in • sign magnitude • 1’s complement • 2’s complement

  47. Exercise • Convert the following numbers: • a) 62.812510 to binary • b) 19810 to binary • Perform the following binary arithmetic operations: • a) 1112 + 1012 c) 1112 x 1102 • b) 11012 – 1012 d) 10102 / 102 • a) Determine the 1’s complement of 1001010 and 1100 • b) Determine the 2’s complement of 11001100 and 1001 • Express each pair of decimal numbers below to 8-bit binary using sign magnitude, 1’s compliment and 2’s compliment. • a) +28 and -18 • b) +45 and -55

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