1. Number Systems and Codes
Chapter 1
Number Systems and Codes 1

2. Chapter Objectives You should be able to:
Determine the weight of each digit position in the decimal, binary, octal, and hexadecimal number systems.
Convert numbers among the four number systems.
Describe binary coded decimal (BCD) numbers.
Translate alphanumeric data to and from ASCII. 2

3. Digital versus Analog Digital Analog
OFF and ON states that can be represented using
0 and 1 (respectively).
Analog
Continuously varying
Examples: temperature, pressure, velocity 4

4. 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

5. Figure 1-1 5

6. Digital Representations of Analog Quantities
Audio Recording
Audio CD and MP3 players/recorders
Video Recording
DVDs store digital representations of analog video and audio signals 7

7. Analog signal voltages and their digital equivalents

8. Digital-to-Analog and Back Again8

9. Why Digital systems are immune to analog noise9

10. Decimal Numbering System (Base 10)
10 possible digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
Least-significant position is on the right end
Most-significant position is on the left end
Weighting factor of 10 10

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

12. Decimal-to-Binary Conversion
Subtracting weighting factors (Example 1-4)
Successive division (Example 1-5)
First remainder is the Least-Significant Bit (LSB)
Last remainder is the Most-Significant Bit (MSB) 12

13. Octal Numbering System (Base 8)
Eight allowable digits: 0, 1, 2, 3, 4, 5, 6, and 7
Weighting factor of 8 13

14. Octal Conversions Binary to octal Octal to binary Octal to decimal
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)
16 allowable digits.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F
Each hex digit represents a 4-bit group
See Table 1-3
Two hex digits are used to represent 8 bits
8 bits are called a byte
4 bits are called a nibble 15

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

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

18. Binary-Coded-Decimal System (BCD)
Each of the 10 decimal digits is represented by its 4-bit binary equivalent.
Decimal-to-BCD conversion
Convert each decimal digit to its 4-bit binary code
BCD-to-Decimal conversion
Reverse the process 18

19. The ASCII Code
American Standard Code for Information Interchange (ASCII)
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

20. 21

21. Numbering System Applications22

22. Applications of the Numbering Systems
A CD player can convert 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

23. Applications of the Numbering Systems
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

24. Applications of the Numbering Systems
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

25. Applications of the Numbering Systems
The part number 651-M is stored in ASCII in a computer memory. List the binary contents of its memory locations. 26

26. Applications of the Numbering Systems
A programmer uses a debugging utility to find an error in a BASIC program. The utility shows the ASCII code as hex 474F 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

27. Summary
Numerical quantities occur naturally 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 0 V for 0 and +5 V for 1. 28

28. 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

29. 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