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# Module 13 - Introduction to Number Systems and LogicNavy Electricity and Electronics Training Series (NEETS)Chapter 1:  Pages 1-61 through 1-69

Module 13 - Introduction to Number Systems and Logic

In this case, the higher order group is invalid, but the lower order group is valid. Therefore, the correction factor is added only to the higher order group as shown:

Remember that the correction factor is added only to groups that exceed 910  (1001BCD). Convert the following numbers to BCD and add:

Q107.

Q108.

Q109.

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Q110.

SUMMARY

Now that you've completed this chapter, you should have a basic understanding of number systems. The number systems that were dealt with are used extensively in the microprocessor and computer fields. The following is a summary of the emphasized terms and points found in the "Number Systems" chapter.

The UNIT represents a single object.

A NUMBER is a symbol used to represent one or more units.

The RADIX is the base of a positional number system. It is equal to the number of symbols used in that number system.

A POSITIONAL NOTATION is a system in which the value or magnitude of a number is defined not only by its digits or symbol value, but also by its position. Each position represents a power of the radix, or base, and is ranked in ascending or descending order.

The MOST SIGNIFICANT DIGIT (MSD) is a digit within a number (whole or fractional) that has the largest effect (weighing power) on that number.

The LEAST SIGNIFICANT DIGIT (LSD) is a digit within a number (whole or fractional) that has the least effect (weighting power) on that number.

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The BINARY NUMBER SYSTEM is a base 2 system. The symbols 1 and 0 can be used to represent the state of electrical/electronic devices. A binary 1 may indicate the device is active; a 0 may indicate the device is inactive.

The OCTAL NUMBER SYSTEM is a base 8 system and is quite useful as a tool in the conversion of binary numbers. This system works because 8 is an integral power of 2; that is, 23  = 8. The use of octal numbers reduces the number of digits required to represent the binary equivalent of a decimal number.

The HEX NUMBER SYSTEM is a base 16 system and is sometimes used in computer systems. A binary number can be converted directly to a base 16 number if the binary number is first broken into groups of four digits.

The basic rules of ADDITION apply to each of the number systems. Each system becomes unique when carries are produced.

SUBTRACTION in each system is based on certain rules of that number system. The borrow varies in magnitude according to the number system in use. In most computers, subtraction is accomplished by using the complement (R's or R's-1) of the subtrahend and adding it to the minuend.

To CONVERT A WHOLE BASE 10 NUMBER to another system, divide the decimal number by the base of the number system to which you are converting. Continue dividing the quotient of the previous division until it can no longer be done. Extract the remainders - the remainder from the first computation will yield the LSD; the last will provide the MSD.

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To CONVERT DECIMAL FRACTIONS, multiply the fraction by the base of the desired number system. Extract those digits that move to the left of the radix point. Continue to multiply the fractional product for as many places as needed. The first digit left of the radix point will be the MSD, and the last will be the LSD. The example to the right shows the process of converting 248.3210  to the octal equivalent (370.2438).

BINARY numbers are converted to OCTAL and HEX by the grouping method. Three binary digits equal one octal digit; four binary digits equal one hex digit.

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To CONVERT binary, octal, and hex numbers to DECIMAL use the POWERS of the base being converted.

BINARY-CODED DECIMAL (BCD) is a coding system used with some microprocessors. A correction factor is needed to correct invalid numbers

ANSWERS TO QUESTIONS Q1. THROUGH Q110.

A1.     Unit

A2.     Number

A3.     Arabic

A4.     The number of symbols used in the system

A5.     17310

A6.     103, 102, 101, 100

A8.

(a)  MSD - 4, LSD - 0

(b)  MSD - 1, LSD - 6

(c)  MSD - 2, LSD - 4

(d)  MSD - 2, LSD - 1

A9.     111112

A10.    111012

A11.    1000012

A12.    1011112

A13.    10002

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A14.    110111102

A15.    100002

A16.    10112

A17.    111012

A18.    112

A19.    11102

A20.    111112

A21.    22110

A22.    011000112

A23.    -00012

A24.    108

A25.    608

A26.    10158

A27.    223068

A28.    1518

A29.    248

A30.    3218

A31.    368

A32.    3368

A33.    3778

A34.    1048

A35.    77678

A36.    DD8D16

A37.    11FDB16

A38.    125F16

A39.    1202016

A40.    191AB16

A41.    1AA816

A42.    33516

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A43.    93516

A44.    953116

A45.    36B316

A46.    10ABC16

A47.    42F0F16

A48.    10010002

A49.    11000012

A50.    111100112

A51.    0.11102

A52.    0.01012

A53.    10001.011012

A54.     78

A55.    538

A56.    7638

A57.    0.74678

A58.    0.002038

A59.    374.1278

A60.    2A16

A61.    5316

A62.    B016

A63.    1EB16

A64.    0.B89316

A65.    28

A66.    128

A67.    578

A68.    0.148

A69.    0.638

A70.    67.258

A71.    216

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A72.    B16

A73.    2F16

A74.    0.316

A75.    0.CC16

A76.    37.5416

A77.    1110112

A78.    1010010102

A79.    1000000112

A80.    0.1001011102

A81.    0.1110112

A82.    11110.1012

A83.    3C16

A84.    14A16

A85.    0.0C16

A86.    C.8816

A87.    1000112; 438

A88.    110112; 338

A89.    0.1110012; 0.718

A90.    1000101.1012; 105.58

A91.    1810

A92.    12410

A93.    8510

A94.    0.312510

A95.    0.62510

A96.    109.937510

A97.    1510

A98.    5210

A99.    25310

A100.    0.510

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A101.    0.76562510

A102.    8.2812510

A103.    3610

A104.    16510

A105.    21910

A106.    998.312510

A107.    1000BCD

A108.    1001BCD

A109.    0001 0001BCD

A110.    0010 0010BCD

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