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Number Systems

Number Systems. Decimal, Binary, and Hexadecimal. Positional Notation. Positional Notation. A positive number N can be written as:. Where:. Definitions. Where:. . = radix point r = radix or base of number system n = number of integer digits to the left of radix point

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Number Systems

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  1. Number Systems Decimal, Binary, and Hexadecimal

  2. Positional Notation

  3. Positional Notation A positive number N can be written as: Where:

  4. Definitions Where: • . = radix point • r = radix or base of number system • n = number of integer digits to the left of radix point • m = number of fractional digits to the right of radix point • ai = integer digit i • aj = fractional digit j • an-1 = most significant digit • a -m = least significant digit

  5. Examples • (1920)10  1920 • Base 10 or decimal number • Unless otherwise noted, assume base 10 • (10010101)2  100101012 • Base 2 or binary number • (95)16  9516 • Base 16 or Hex number

  6. Online HW Assignments • (1920)10  1920 • Base 10 or decimal number • (10010101)2  %10010101 • Base 2 or binary number • (95)16  $95 • Base 16 or Hex number

  7. Polynomial Notation A positive number N can also be written as: Example:

  8. Common Number Systems

  9. Decimal Number System • Radix or Base 10 • Digits: 0, 1,2,3,4,5,6,7,8,9 • Example: 104510 • First 17 positive integers • 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16

  10. Binary Number System • Radix or Base 2 • Digits: 0, 1 • Example: 10101102 • First 17 positive integers • 0,1,10,11,100,101,110,111,1000,1001,1010,1011,1100,1101,1110,1111,10000

  11. Hexadecimal Number System • Radix or Base 16 • Digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F • Example: EF5616 • First 17 positive integers • 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10

  12. Summary

  13. Decimal Arithmetic Addition Subtraction Multiplication Division

  14. Decimal Addition Table

  15. Example Add the following decimal numbers 199 + 73

  16. Solution Add the following decimal numbers 1 1 199 + 73 2 7 2

  17. Solution Subtract the following decimal numbers 123 - 77

  18. Solution Subtract the following decimal numbers 1 123 - 77 6

  19. Solution Subtract the following decimal numbers 1 1 113 - 77 4 6

  20. Solution Subtract the following decimal numbers 1 1 013 - 77 0 4 6

  21. Decimal Multiplication Table

  22. Example Multiply the following decimal numbers 72 x 23

  23. Partial Products Solution Multiply the following decimal numbers 72 x 23 Multiplicand Multiplier 216 1440 1656 Product

  24. ExampleDivide the following decimal numbers 3 23

  25. ExampleDivide the following decimal numbers 0 7 Quotient 3 23 Divisor Dividend 0 23 21 2 Remainder

  26. Binary Arithmetic

  27. Binary Addition • Single Bit Addition Table 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 Note “carry”

  28. Example Add the following binary numbers 1 1 0 1 +0 1 1 1

  29. Solution: Add the following binary numbers 1 1 1 1 0 1 +0 1 1 1 1 0 1 0 0

  30. Binary Subtraction • Single Bit Subtraction Table 0 - 0 = 0 1 - 0 = 1 1 - 1 = 0 0 - 1 = 1 with a “borrow”

  31. ExampleSubtract the following binary numbers 1 1 0 1 - 0 1 1 1

  32. 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 Solution: Subtract the following binary numbers 1 1 0 1 - 0 1 1 1 0 1 1 0

  33. Collaborative Learning Learning methodology in which students are not only responsible for their own learning but for the learning of other members of the group.

  34. Think - Pair - Share (TPS) Quizzes • Think – Pair – Share • Think individually for one time units • Pair with partner for two time units • Share with group for one and half time units • Report results

  35. TPS Quiz THINK One Unit (e.g. 30 Seconds)

  36. TPS Quiz PAIR Two Units (e.g. 60 Seconds)

  37. TPS Quiz SHARE 1.5 units (e.g. 45 Seconds)

  38. TPS QUIZ Report

  39. TPS Quiz #1-2-3

  40. Binary Multiplication • Single Bit Multiplication Table

  41. Binary Multiplication • Single Bit Multiplication Table 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1

  42. ExampleMultiply the following binary numbers 1 1 0 1 x 0 1 1 1

  43. SolutionMultiply the following binary numbers 1 1 0 1 x 0 1 1 1 1 1 0 1 1 1 0 1 0 Partial Products 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1

  44. Binary Division • Single Bit Division Table 0 / 0 = N/A 0 / 1 = 0 1 / 0 = N/A 1 / 1 = 1

  45. ExampleDivide the following binary numbers 1 1 1 1 1 0 1

  46. 1 1 1 1 1 0 1 SolutionDivide the following binary numbers 1 0 0 0 Quotient Dividend Divisor 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 Remainder 1 1 0

  47. Hexadecimal Arithmetic

  48. Hexadecimal Addition Table

  49. Example Add the following hexadecimal numbers AF + 1B

  50. Solution Add the following hexadecimal numbers 1 AF + 1B C A

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