1. CHAPTER 2 Number Systems, Operations, and Codes
Digital Fundamentals
CHAPTER 2 Number Systems, Operations, and Codes

2. Decimal Numbers
The 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:

3. Binary Numbers Two digits called bits 0 or 1
Right most bit is least significant bit
Left most bit is most significant bit
Each position is weighted by a factor of 2.
Decimal point

4. Counting in Binary Decimal Number Binary Number

5. Figure 2–1 Illustration of a simple binary counting application.
Thomas L. Floyd Digital Fundamentals, 9e
Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

6. Binary Numbers The 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:

7. Binary to Decimal Add the weights of all the bits that are 1.
Ignore the bits that are 0.
Convert to decimal.
1st determine the weights = =
= 109

8. Decimal-to-Binary Conversion
Sum-of-weights method
Repeated division-by-2 method
Conversion of decimal fractions to binary

9. Repeated Division by 2
Divide the number by 2 and write down the remainder.
Continue the process until the whole-number quotient is 0.
Convert decimal 19 to binary.
Convert decimal 45 to binary.

10. Decimal Fractions Repeated multiplication by 2
Multiply the decimal fraction by 2
Write down the carry bit if more than 1
Continue process until you have the desired number of bits or when the fractional part is all zeros.

11. Convert to binary
Carry MSB LSB
X 2 =
X 2 =
X 2 =
X 2 =

12. Binary Arithmetic Binary addition Binary subtraction
Binary multiplication
Binary division

13. Complements of Binary Numbers
1’s complements
2’s complements

14. Complements of Binary Numbers
1’s complement
Change all 1s to 0s and all 0s to 1s

15. Complements of Binary Numbers
2’s complement
Find 1’s complement and then add 1

16. Signed Numbers

17. Topics for Signed Numbers
Signed-magnitude form
1’s and 2’s complement form
Decimal value of signed numbers (How to convert)
Range of values (max and min)
Floating-point numbers

18. Signed Numbers Signed-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

19. Signed Numbers 1’s complement form 2’s complement form
A negative value is the 1’s complement of the corresponding positive value
2’s complement form
A negative value is the 2’s complement of the corresponding positive value

20. Signed Numbers Decimal value of signed numbers Sign-magnitude
1’s complement
2’s complement

21. Signed Numbers Range of Values Total combinations = 2n
2’s complement form:
– (2n – 1) to + (2n – 1 – 1)
Range for 8 bit number: n = 8 -(28-1) = -27 = minimum +(28-1) – 1 = = maximum
Total combination of numbers is 28 = 256.

22. Signed Numbers
Range for 16 bit number: n = 16 -(216-1) = = minimum +(216-1) - 1 = +215 = maximum Total combinations is 216 = (64K)
8 bit examples: =
-128 = -1 =
-127 =
+127

23. Signed Numbers Floating-point numbers Two Parts Three forms
Can represent very large or very small numbers based on scientific notation. Binary point “floats”.
Two Parts
Mantissa represents magnitude of number
Exponent represents number of places that binary point is to be moved
Three forms
Single-precision (32 bits) float
Double-precision (64 bits) double
Extended-precision (80 bits) long double
Also have Quadruple and Quadruple extended!

24. Single Precision IEEE 754 standard
32 bits
S Exponent (E) Mantissa (fraction, F)
1 bit bits bits
IEEE 754 standard
Mantissa (F) has hidden bit so actually has 24 bits. Gives 7 significant figures.
1st bit in mantissa is always a one
Exponent (E) is biased by 127 called Excess-127 Notation
Add 127 to exponent so easier to compare
Range of exponents is -126 to +128
Sign (S) bit tells whether number is negative or positive

25. Single Precision
Example: Convert to Floating Point
1st, convert to binary using divide by 2 method =
Positive number, so sign bit (S) equals 0.
2nd, count number of places to move binary point
= x 212
Add 127 to 12 = =
S E F
Fill in with trailing zeroes
Mantissa is fractional part,
Finally, put everything together

26. Special Cases Zero and infinity are special cases Not a Number (NaN)
Can have +0 or -0 depending on sign bit
Can also have +∞ or -∞
Not a Number (NaN)
if underflow or overflow
Type
Exponent
Mantissa
Zeroes
Denormalized numbers
non zero
Normalized numbers
1 to 2e − 2
any
Infinities
2e − 1
NaNs

27. Examples Type Exponent Mantissa Value Zero 0000 0000
0.0
One
1.0
Denormalized number
5.9×10-39
Large normalized number
3.4×1038
Small normalized number
1.18×10-38
Infinity
NaN

28. Double Precision Exponent has 11 bits so uses Excess-1023 Notation
Mantissa has 53 bits (one hidden)
53 bits gives 16 significant figures

29. Arithmetic Operations with Signed Numbers
Addition
Subtraction
Multiplication
Division

30. Arithmetic Operations with Signed Numbers
Addition of Signed Numbers
The parts of an addition function are:
Augend - The first number
Addend - The second number
Sum The result
Numbers are always added two at a time.

31. Arithmetic Operations with Signed Numbers
Four conditions for adding numbers:
1. Both numbers are positive.
2. A positive number that is larger than a negative number.
3. A negative number that is larger than a positive number.
4. Both numbers are negative.

32. Arithmetic Operations with Signed Numbers
Signs for Addition
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.

33. 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 both numbers are negative, the sum is negative (2’s complement form). The carry bit is discarded.

34. Examples (8 bit numbers)
Add 7 and 4 (both positive)
Add 15 and -6 (positive > negative)
Add 16 and -24 (negative > positive)
Add -5 and -9 (both negative)
Discard carry
Sign bit is negative so negative number in 2’s complement form
Discard carry

35. Overflow
Overflow occurs when number of bits in sum exceeds number of bits in addend or augend.
Overflow is indicated by the wrong sign.
Occurs only when both numbers are positive or both numbers are negative
_________ ____
Sign Incorrect Magnitude Incorrect

36. Arithmetic Operations with Signed Numbers
Subtraction of Signed Numbers
The parts of a subtraction function are:
Minuend - The first number
Subtrahend - The second number
Difference - The result
Subtraction is addition with the sign of the subtrahend changed.

37. Arithmetic Operations with Signed Numbers
Subtraction
The 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.

38. Subtraction Examples Find 8 minus 3. Find 12 minus -9.
Discard carry
Minuend Subtrahend Difference
Discard carry

39. Arithmetic Operations with Signed Numbers
Multiplication of Signed Numbers
The parts of a multiplication function are:
Multiplicand - First number
Multiplier - Second number
Product - Result
Multiplication is equivalent to adding a number to itself a number of times equal to the multiplier.

40. Arithmetic Operations with Signed Numbers
There are two methods for multiplication:
Direct addition
add multiplicand multiple times equal to the multiplier
Can take a long time if multiplier is large
Partial products
Similar to long hand multiplication
The method of partial products is the most commonly used.

41. Arithmetic Operations with Signed Numbers
Multiplication of Signed Numbers
If the signs are the same, the product is positive. (+ X + = + or - X - = +)
If the signs are different, the product is negative. (+ X - = or - X + = -)

42. Multiplication Example
Both numbers must be in uncomplemented form
Multiply 3 by -5.
Opposite signs, so product will be negative.
= = ’s complement of
Multiplicand X Multiplier First partial product Second partial product Sum of 1st and 2nd
Third partial product Sum and Final Product
Final result is negative, so take 2’s complement is the result which in decimal is -15.

43. Arithmetic Operations with Signed Numbers
Division of Signed Numbers
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.

44. Arithmetic Operations with Signed Numbers
Division of Signed Numbers
If the signs are the same, the quotient is positive. (+ ÷ + = + or - ÷ - = +)
If the signs are different, the quotient is negative. (+ ÷ - = - or - ÷ + = -)

45. Division Example Both numbers must be in uncomplemented form
Divide by
Both numbers are positive so quotient will be positive. Set the quotient to zero initially.
quotient:
Dividend ’s complement of Divisor First partial remainder
Add 1 to quotient: =
Subtract the divisor from the dividend by using 2’s complement addition. ( ) Ignore the carry bit.
First partial remainder ’s complement of Divisor zero remainder
Add 1 to quotient: =
Subtract the divisor from the 1st partial remainder using 2’s complement addition.
So final quotient is and final remainder is

46. Hexadecimal Numbers

47. Hexadecimal Numbers Decimal, binary, and hexadecimal numbers
4 bits is a nibble
FF16 = 25510

48. Hexadecimal Numbers Binary-to-hexadecimal conversion
Hexadecimal-to-decimal conversion
Decimal-to-hexadecimal conversion

49. Hexadecimal Numbers Binary-to-hexadecimal conversion
Break the binary number into 4-bit groups
Replace each group with the hexadecimal equivalent
Convert to Hex
C A = CA5716
Convert 10A416 to binary =

50. Hexadecimal Numbers Hexadecimal-to-decimal conversion
Convert the hexadecimal to groups of 4-bit binary
Convert the binary to decimal

51. Hexadecimal Numbers Decimal-to-hexadecimal conversion
Convert to binary by repeated division by 2 and then convert binary to hex
Repeated division by 16

52. Figure 2–4 Getting the 2’s complement of a hexadecimal number, Method 1.

53. Figure 2–5 Getting the 2’s complement of a hexadecimal number, Method 2.

54. Octal Not used as frequently Convert binary to octal
Can group in 3 bits instead of 4 bits like Hex.
Symbols range from 0 to 7
Example. Convert to octal. =
Example. Convert to binary. =

55. Binary Coded Decimal (BCD)

56. Binary Coded Decimal (BCD)
Decimal and BCD digits

57. Digital Codes

58. Digital Codes
Gray code
ASCII code

59. Gray Code Unweighted - no weights associated to the bit positions
Only one bit change from one code word to the other.
Minimizes chance of error

60. Conversion Binary to Gray code Example. Convert 10110 to Gray Code
MSB of Gray is set to the MSB of binary
Going from left to right, add each adjacent pair of binary code bits. Discard carries.
Example. Convert to Gray Code
MSB of binary number is 1, so set the MSB of the Gray Code to 1.
Binary
Gray code
Add adjacent pairs of binary numbers
1+0 1
1+1
0+1 1
1+0 1

61. Conversion Gray code to Binary Example. Convert 11011 to binary
MSB of binary is set to the MSB of Gray code
Going from left to right, add the binary bit to the next Gray code bit. Discard carries.
Example. Convert to binary
MSB of Gray Code is 1, so set the MSB of the binary number to 1.
Gray Code
Binary + + + 1 +

62. Figure 2– A simplified illustration of how the Gray code solves the error problem in shaft position encoders.

63. American Standard Code for Information Interchange (ASCII)
ASCII code (1st 32 are control characters)
128 characters
Uses 7 bits

64. ASCII code (graphic symbols 20h – 3Fh)

65. Digital Codes
ASCII code (graphic symbols 40h – 5Fh)

66. Digital Codes
ASCII code (graphic symbols 60h – 7Fh)

67. Digital Codes Extended ASCII code (80h – FFh)
Non-English alphabetic characters
Currency symbols
Greek letters
Math symbols
Drawing characters
Bar graphing characters
Shading characters

68. Error Detection and Correction Codes

69. Error Detection and Correction Codes
Parity error codes
Odd parity – odd sum of 1’s
Even parity – even sum of 1’s
The sum includes the parity bit
Parity bit can be at beginning or end of the code.
Can detect single bit errors, but not 2 or more errors.
Hamming error codes

70. Error Detection and Correction Codes
Parity error codes

71. Error Detection and Correction Codes 1 2 3 4 5 6 7 8 9 A B C D E F
Hamming error codes
Hamming code words
Hex equivalent of the data bits