1. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 1 Digital Fundamentals CHAPTER 2 Number Systems, Operations, and Codes

2. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 2 Number Systems

3. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 3 Decimal Numbers The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 The decimal numbering system has a base of 10 with each position weighted by a factor of 10:The decimal numbering system has a base of 10 with each position weighted by a factor of 10:

4. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 4 Binary Numbers The binary number system has two digits: 0 and 1The binary number system has two digits: 0 and 1 The binary numbering system has a base of 2 with each position weighted by a factor of 2:The binary numbering system has a base of 2 with each position weighted by a factor of 2:

5. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 5 Decimal-to-Binary Conversion

6. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 6 Decimal-to-Binary Conversion Sum-of-weights methodSum-of-weights method Repeated division-by-2 methodRepeated division-by-2 method Conversion of decimal fractions to binaryConversion of decimal fractions to binary

7. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 7 Binary Arithmetic

8. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 8 Binary Arithmetic Binary additionBinary addition Binary subtractionBinary subtraction Binary multiplicationBinary multiplication Binary divisionBinary division

9. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 9 Complements of Binary Numbers

10. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 10 Complements of Binary Numbers 1’s complements1’s complements 2’s complements2’s complements

11. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 11 Complements of Binary Numbers 1’s complement1’s complement

12. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 12 Complements of Binary Numbers 2’s complement2’s complement

13. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 13 Signed Numbers

14. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 14 Signed Numbers Signed-magnitude formSigned-magnitude form 1’s and 2’s complement form1’s and 2’s complement form Decimal value of signed numbersDecimal value of signed numbers Range of valuesRange of values Floating-point numbersFloating-point numbers

15. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 15 Signed Numbers Signed-magnitude formSigned-magnitude form –The sign bit is the left-most bit in a signed binary number –A 0 sign bit indicates a positive magnitude –A 1 sign bit indicates a negative magnitude

16. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 16 Signed Numbers 1’s complement form1’s complement form –A negative value is the 1’s complement of the corresponding positive value 2’s complement form2’s complement form –A negative value is the 2’s complement of the corresponding positive value

17. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 17 Signed Numbers Decimal value of signed numbersDecimal value of signed numbers –Sign-magnitude –1’s complement –2’s complement

18. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 18 Signed Numbers Range of ValuesRange of Values 2’s complement form: – (2 n – 1 ) to + (2 n – 1 – 1)

19. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 19 Signed Numbers Floating-point numbersFloating-point numbers –Single-precision (32 bits) –Double-precision (64 bits) –Extended-precision (80 bits)

20. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 20 Arithmetic Operations with Signed Numbers AdditionAddition SubtractionSubtraction MultiplicationMultiplication DivisionDivision

21. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 21 Arithmetic Operations with Signed Numbers Addition of Signed Numbers The parts of an addition function are:The parts of an addition function are: –Addend –Augend –Sum Numbers are always added two at a time.

22. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 22 Arithmetic Operations with Signed Numbers Four conditions for adding numbers: Both numbers are positive.Both numbers are positive. A positive number that is larger than a negative number.A positive number that is larger than a negative number. A negative number that is larger than a positive number.A negative number that is larger than a positive number. Both numbers are negative.Both numbers are negative.

23. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 23 Arithmetic Operations with Signed Numbers Signs for Addition When both numbers are positive, the sum is positive.When both numbers are positive, the sum is positive. When the larger number is positive and the smaller is negative, the sum is positive. The carry is discarded.When the larger number is positive and the smaller is negative, the sum is positive. The carry is discarded.

24. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 24 Arithmetic Operations with Signed Numbers Signs for Addition When the larger number is negative and the smaller is positive, the sum is negative (2’s complement form).When the larger number is negative and the smaller is positive, the sum is negative (2’s complement form). When both numbers are negative, the sum is negative (2’s complement form). The carry bit is discarded.When both numbers are negative, the sum is negative (2’s complement form). The carry bit is discarded.

25. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 25 Arithmetic Operations with Signed Numbers Subtraction of Signed Numbers The parts of a subtraction function are:The parts of a subtraction function are: –Subtrahend –Minuend –Difference Subtraction is addition with the sign of the subtrahend changed.

26. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 26 Arithmetic Operations with Signed Numbers Subtraction The sign of a positive or negative binary number is changed by taking its 2’s complementThe sign of a positive or negative binary number is changed by taking its 2’s complement To subtract two signed numbers, take the 2’s complement of the subtrahend and add. Discard any final carry bit.To subtract two signed numbers, take the 2’s complement of the subtrahend and add. Discard any final carry bit.

27. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 27 Arithmetic Operations with Signed Numbers Multiplication of Signed Numbers The parts of a multiplication function are:The parts of a multiplication function are: –Multiplicand –Multiplier –Product Multiplication is equivalent to adding a number to itself a number of times equal to the multiplier.

28. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 28 Arithmetic Operations with Signed Numbers There are two methods for multiplication: Direct additionDirect addition Partial productsPartial products The method of partial products is the most commonly used.

29. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 29 Arithmetic Operations with Signed Numbers Multiplication of Signed Numbers If the signs are the same, the product is positive.If the signs are the same, the product is positive. If the signs are different, the product is negative.If the signs are different, the product is negative.

30. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 30 Arithmetic Operations with Signed Numbers Division of Signed Numbers The parts of a division operation are:The parts of a division operation are: –Dividend –Divisor –Quotient Division is equivalent to subtracting the divisor from the dividend a number of times equal to the quotient.

31. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 31 Arithmetic Operations with Signed Numbers Division of Signed Numbers If the signs are the same, the quotient is positive.If the signs are the same, the quotient is positive. If the signs are different, the quotient is negative.If the signs are different, the quotient is negative.

32. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 32 Hexadecimal Numbers

33. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 33 Hexadecimal Numbers Decimal, binary, and hexadecimal numbersDecimal, binary, and hexadecimal numbers

34. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 34 Hexadecimal Numbers Binary-to-hexadecimal conversionBinary-to-hexadecimal conversion Hexadecimal-to-decimal conversionHexadecimal-to-decimal conversion Decimal-to-hexadecimal conversionDecimal-to-hexadecimal conversion

35. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 35 Hexadecimal Numbers Binary-to-hexadecimal conversionBinary-to-hexadecimal conversion 1.Break the binary number into 4-bit groups 2.Replace each group with the hexadecimal equivalent

36. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 36 Hexadecimal Numbers Hexadecimal-to-decimal conversionHexadecimal-to-decimal conversion 1.Convert the hexadecimal to groups of 4-bit binary 2.Convert the binary to decimal

37. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 37 Hexadecimal Numbers Decimal-to-hexadecimal conversionDecimal-to-hexadecimal conversion –Repeated division by 16

38. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 38 Binary Coded Decimal (BCD)

39. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 39 Binary Coded Decimal (BCD) Decimal and BCD digits

40. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 40 Digital Codes

41. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 41 Digital Codes Gray codeGray code ASCII codeASCII code

42. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 42 Digital Codes Gray codeGray code

43. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 43 Digital Codes ASCII code (control characters)ASCII code (control characters)

44. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 44 Digital Codes ASCII code (graphic symbols 20h – 3Fh)ASCII code (graphic symbols 20h – 3Fh)

45. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 45 Digital Codes ASCII code (graphic symbols 40h – 5Fh)ASCII code (graphic symbols 40h – 5Fh)

46. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 46 Digital Codes ASCII code (graphic symbols 60h – 7Fh)ASCII code (graphic symbols 60h – 7Fh)

47. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 47 Digital Codes Extended ASCII code (80h – FFh) Non-English alphabetic charactersNon-English alphabetic characters Currency symbolsCurrency symbols Greek lettersGreek letters Math symbolsMath symbols Drawing charactersDrawing characters Bar graphing charactersBar graphing characters Shading charactersShading characters

48. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 48 Error Detection and Correction Codes

49. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 49 Error Detection and Correction Codes Parity error codesParity error codes Hamming error codesHamming error codes

50. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 50 Error Detection and Correction Codes Parity error codesParity error codes

51. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 51 Error Detection and Correction Codes Hamming error codesHamming error codes –Hamming code words –Hex equivalent of the data bits 00000000000111001101100111100101010010110101100110110100100101110011001010010101010111000011100110111100011111110123456789ABCDEF