Chapter 1

1. Chapter 1 Number Systems and Codes 1

2. Objectives • You should be able to: • Explain the difference between analog and digital. • Determine the weighting of digit positions in decimal, binary, octal, and hexadecimal numbering systems. • Convert numbers among the four numbering systems. 2

3. Objectives • You should be able to: • Describe binary coded decimal (BCD) numbers. • Translate alphanumeric data to and from ASCII. 3

4. Digital versus Analog • Digital • ON and OFF • 0 and 1 • Analog • Continuously varying • Examples: temperature, pressure, velocity • See Figure 1-1 4

5. Figure 1-1 5

6. Discussion Points • Explain the difference between analog and digital signals. • Describe some applications for digital technology. • What are the benefits of using digital systems? • Are there any problems associated with digital systems? 6

7. Digital Representations of Analog Quantities • Audio Recording • CD, DAT, and MP3 • Conversions • Digital-to-analog • Analog voltage to 8-bit Digital equivalent • See Figures 1-2 and 1-3 7

8. Figure 1-2 Figure 1-3 8

9. Why Digital systems are immune to analog noise 9

10. Decimal Numbering System (Base 10) • 10 different possible digits • Least significant position • Rightmost • Mostsignificant digit • Leftmost • Weighting factor of 10 10

11. Binary Numbering System (Base 2) • Only 0 and 1 • Weighting factor of 2 • Conversion techniques • Digit times weighting factor • Successive division 11

12. Decimal-to-Binary Conversion • Subtracting weighting factors • Successive division • Least Significant Bit (LSB) • Most Significant Bit (MSB) 12

13. Octal Numbering System(Base 8) • Allowable digits • 0,1,2,3,4,5,6,7 • Weighting factor of 8 13

14. Octal Conversions • Binary to octal • Group binary positions in groups of three • Write the octal equivalent • Octal to binary • Reverse the process • Octal to decimal • Multiply by weighting factors • Decimal to octal • Successive division 14

15. Hexadecimal Numbering System(Base 16) • 4-bit groupings • See Table 1-3 in your text • Two hex digits are used to represent 8 bits • A byte • 4 bits are a nibble 15

16. Hexadecimal Conversions • Binary to hexadecimal • Group the binary in groups of four • Write the equivalent hex digit • Hexadecimal to binary • Reverse the process 16

17. Hexadecimal Conversions • Hexadecimal to decimal • Multiply by weighting factors • Decimal to hexadecimal • Successive division 17

18. Binary-Coded-Decimal SystemBCD • Each of the 10 decimal digits has a 4-bit binary code • Conversion • Convert each decimal digit to its 4-bit binary code • BCD to decimal - reverse the process 18

19. Comparison of Numbering Systems • See Table 1-4 in your text 19

20. The ASCII Code • Represents alphanumeric data • Uses 7 bits • 128 different code combinations • See Table 1-5 • 3-bit group is most significant • 4-bit group is least significant 20

21. 21

22. Applications of the Numbering Systems • Application 1-1 22

23. Applications of the Numbering Systems • Application 1-2 • A CD player is capable of converting 12 bit signals from a CD into equivalent analog values. • What are the largest and smallest hex values that can be used in this system? • How many different analog values can be represented? 23

24. Applications of the Numbering Systems • Application 1-3 • Typically, digital thermometers use BCD to drive their displays. • How many BCD bits are required to drive a 3 digit thermometer display? • What bits are sent to the display for 147 degrees? 24

25. Applications of the Numbering Systems • Application 1-4 • Most PC-compatible computer systems use a 20-bit address code to identify each of over 1 million memory locations. • How many hex characters are required to identify the address of each memory location? • What is the hex address of the 200th memory location? • If 50 memory locations are used for data storage starting at location 000C8H, what is the location of the last data item? 25

26. Applications of the Numbering Systems • Application 1-5 • If the part number 651-M is stored in ASCII in a computer memory, list the binary contents of its memory locations. 26

27. Applications of the Numbering Systems • Application 1-6 • A programmer uses a debugging utility to find an error in a BASIC program. The utility shows the ASCII code as hex 474F5430203930. Assume that the leftmost bit of each ASCII string is padded with a zero. • Translate the program segment that is displayed. • Try to determine what the error is. 27

28. Summary • Numerical quantities occur in analog form but must be converted to digital form to be used by computers or digital circuitry. • The binary numbering system is used in digital systems because the 1’s and 0’s are easily represented by ON or OFF transistors, which output 0V for 0 and 5V for 1. 28

29. Summary • Any number system can be converted to decimal by multiplying each digit by its weighting factor. • The weighting factor for the least significant digit in any number system is always 1. • Binary numbers can be converted to octal by forming groups of 3 bits and to hexadecimal by forming groups of 4 bits. 29

30. Summary • The successive division procedure can be used to convert from decimal to binary, octal or hexadecimal • The binary-coded-decimal system uses groups of 4 bits to drive decimal displays such as those in a calculator. • ASCII is used by computers to represent all letters, numbers and symbols in digital form. 30