1. CSC212 – Computer Organization and Design
Digital Electronics – William Kleitz, 9th Ed.
Chapter 01: Number Systems and Codes
Chapter 03: Basic Logic Gates
Chapter 05: Boolean Algebra
Chapter 06: Exclusive-OR and NOR Gates
Chapter 07: Arithmetic Operations & Circuits
Code Converters, Muxers and Demuxers

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

3. Chapter Objectives You should be able to:
Determine the weighting factor of each digit position in the decimal, binary, octal, and hexadecimal numbering systems.
Convert any number among the four number systems, and its equivalent value in any of the remaining three numbering systems.
Describe binary coded decimal (BCD) numbers.
Translate alphanumeric data to and from ASCII using the ASCII code translation table. 2

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

5. 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

6. Digital vs. Analog 5

7. 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

8. Analog Signal Voltages and Their Digital Equivalents

9. Digital-to-Analog and Back Again8

10. Why Digital Systems Are Immune to Analog Noise9

11. Digital Representations of Alternative Energy Sources
Energy technicians must keep track of the efficiency of their energy collection systems.
Naturally occurring quantities like solar, wind, and temperature are analog quantities and must be digitized before a computer can understand them.

12. A Solar Radiation Data-logger System

13. 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

14. 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

15. 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

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

17. 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

18. 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

19. Hexadecimal Numbering System

20. 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

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

22. 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

24. 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

25. 21

26. Numbering System Applications
Because digital systems must work with 1s and 0s, learning the different numbering systems is important.
Which system is used is determined by how the data were developed and how they are to be used.
Several numbering system applications follow. 22

27. Application 1-1
The four chemical storage tanks shown are monitored for temperature (T) and pressure (P). 22

28. Application 1-1 (continued)
Using the table shown below, interpret the following:
If the computer reads a binary string of what problems exist?
This indicates that the pressure in tanks C and B are too high. 22

29. Application 1-1 (continued)
Using the table shown below, interpret the following:
If the computer reads a hex value of 55H what problems exist?
Since 55H = This indicates that all tank temperatures are too high. 22

30. Application 1-1 (continued)
Using the table shown below, interpret the following:
If the temperature and pressure in tanks B and D are too high, what hex value is read by the computer?
This condition would produce a digital output of = CCH. 22

31. Application 1-1 (continued)
Using the table shown below, interpret the following:
Assume that tanks A and B are shut down and all sensors are tied high (1s). What is the lowest decimal value that indicates a problem in the other two tanks?
With the four low-order bits tied high, the lowest value that indicates a problem is or 3110. 22

32. Application 1-1 (continued)
Using the table shown below, interpret the following:
If only tanks A, B, and C are monitored, what octal value indicates tank B has both temperature and pressure problems?
The binary output would be = 148. 22

33. Application 1-2
A CD player converts 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?
The largest is FFF16 and the smallest is
How many different analog values can be represented?
FFF16 = , so including 0 the total is 4096 unique values. 22

34. Application 1-3
Typically, digital thermometers use BCD to drive their displays.
How many BCD bits are required to drive a 3-digit display?
12 bits are required; four for each digit.
What 12 bits represent 147°F?
0001 (1), 0100 (4), and 0111 (7). 22

35. 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?
Five hex characters are required since each hex character represents 4 bits. 22

36. Application 1-4 (continued)
What is the hex address of the 200th memory location?
000C8H = 20010, but 00000H is the first memory location, so we must subtract 1. The answer is C8 – 1 = C7.
If 50 memory locations are used for data storage starting at location 000C8H, what is the location of the last data item?
C8H gets the first data item, so the answer is = F9H. 22

37. Application 1-5
The part number 651-M is stored in ASCII in a computer memory. List the binary contents of its memory locations?
6 = = = = M =
Grouping the binary bits in eights, this string represents 5 hex memory locations: D D 22

38. 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 474F Assume that the leftmost bit of each ASCII string is padded with a zero.
The program segment is translated as GOT0 90.
The error is that a zero (0) was typed instead of the letter O. 22

39. 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 1s and 0s are easily represented by ON or OFF transistors, which output 0 V for 0 and +5 V for 1. 28

40. 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

41. 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