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

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:

Convert this total to decimal to check your answer:

Remember that the correction factor is added only to groups that exceed 9₁₀  (1001_BCD). 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

A7.           Radix point

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.    1000_BCD

A108.    1001_BCD

A109.    0001 0001_BCD

A110.    0010 0010_BCD

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