 # Number Systems and Codes

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Number Systems and Codes
Chapter 1 Number Systems and Codes 1

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

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

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

Figure 1-1 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

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

Figure 1-2 Figure 1-3 8

Why Digital systems are immune to analog noise
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Decimal Numbering System (Base 10)
10 different possible digits Least significant position Rightmost Most significant digit Leftmost Weighting factor of 10 10

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

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

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

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

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

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

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

Binary-Coded-Decimal System BCD
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

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

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

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Applications of the Numbering Systems
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Applications of the Numbering Systems
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

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

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

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

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

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

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