Module 13 - Introduction to Number Systems and Logic
Navy Electricity and Electronics Training Series (NEETS)
Chapter 1:  Pages 1-1 through 1-10

Chapter 1

NUMBER Systems

Learning Objectives

Learning objectives are stated at the beginning of each chapter. These learning objectives serve as a preview of the information you are expected to learn in the chapter. The comprehensive check questions are based on the objectives. By successfully completing the NRTC, you indicate that you have met the objectives and have learned the information. The learning objectives are listed below.

Upon completion of this chapter, you should be able to do the following:

1.    Recognize different types of number systems as they relate to computers.

2.    Identify and define unit, number, base/radix, positional notation, and most and least significant digits as they relate to decimal, binary, octal, and hexadecimal number systems.

3.    Add and subtract in binary, octal, and hexadecimal number systems.

4.    Convert values from decimal, binary, octal, hexadecimal, and binary-coded decimal number systems to each other and back to the other systems.

5.    Add in binary-coded decimal.

Introduction

How many days' leave do you have on the books? How much money do you have to last until payday? It doesn't matter what the question is - if the answer is in dollars or days or cows, it will be represented by numbers.

Just try to imagine going through one day without using numbers. Some things can be easily described without using numbers, but others prove to be difficult. Look at the following examples:

I am stationed on the aircraft carrier Nimitz. He owns a green Chevrolet. The use of numbers wasn't necessary in the preceding statements, but the following examples depend on the use of numbers:

I have $25 to last until payday. I want to take 14 days' leave. You can see by these statements that numbers play an important part in our lives.

BACKGround and HIsTORY

Man's earliest number or counting system was probably developed to help determine how many possessions a person had. As daily activities became more complex, numbers became more important in trade, time, distance, and all other phases of human life.

As you have seen already, numbers are extremely important in your military and personal life. You realize that you need more than your fingers and toes to keep track of the numbers in your daily routine.

Ever since people discovered that it was necessary to count objects, they have been looking for easier ways to count them. The abacus, developed by the Chinese, is one of the earliest known calculators. It is still in use in some parts of the world.

Blaise Pascal (French) invented the first adding machine in 1642. Twenty years later, an Englishman, Sir Samuel Moreland, developed a more compact device that could multiply, add, and subtract. About 1672, Gottfried Wilhelm von Leibniz (German) perfected a machine that could perform all the basic operations (add, subtract, multiply, divide), as well as extract the square root. Modern electronic digital computers still use von Liebniz's principles.

ModeRN use

Computers are now employed wherever repeated calculations or the processing of huge amounts of data is needed. The greatest applications are found in the military, scientific, and commercial fields. They have applications that range from mail sorting, through engineering design, to the identification and destruction of enemy targets. The advantages of digital computers include speed, accuracy, and man- power savings. Often computers are able to take over routine jobs and release personnel for more important work¾work that cannot be handled by a computer.

People and computers do not normally speak the same language. Methods of translating information into forms that are understandable and usable to both are necessary. Humans generally speak in words and numbers expressed in the decimal number system, while computers only understand coded electronic pulses that represent digital information.

In this chapter you will learn about number systems in general and about binary, octal, and hexadecimal (which we will refer to as hex) number systems specifically. Methods for converting numbers in the binary, octal, and hex systems to equivalent numbers in the decimal system (and vice versa) will also be described. You will see that these number systems can be easily converted to the electronic signals necessary for digital equipment.

TYPES of NUMBER Systems

Until now, you have probably used only one number system, the decimal system. You may also be familiar with the Roman numeral system, even though you seldom use it.

The DECIMAL NUMBER System

In this module you will be studying modern number systems. You should realize that these systems have certain things in common. These common terms will be defined using the decimal system as our base. Each term will be related to each number system as that number system is introduced.

Each of the number systems you will study is built around the following components: the UNIT, NUMBER, and Base (RADIX).

Unit and Number

The terms unit and number when used with the decimal system are almost self-explanatory. By definition the unit is a single object; that is, an apple, a dollar, a day. a number is a symbol representing a unit or a quantity. The figures 0, 1, 2, and 3 through 9 are the symbols used in the decimal system. These symbols are called Arabic numerals or figures. Other symbols may be used for different number systems. For example, the symbols used with the Roman numeral system are letters - V is the symbol for 5, X for 10, M for 1,000, and so forth. We will use Arabic numerals and letters in the number system discussions in this chapter.

Base (Radix)

The base, or radix, of a number system tells you the number of symbols used in that system. The base of any system is always expressed in decimal numbers. The base, or radix, of the decimal system is 10. This means there are 10 symbols - 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 - used in the system. A number system using three symbols - 0, 1, and 2 - would be base 3; four symbols would be base 4; and so forth. Remember to count the zero or the symbol used for zero when determining the number of symbols used in a number system.

The base of a number system is indicated by a subscript (decimal number) following the value of the number. The following are examples of numerical values in different bases with the subscript to indicate the base:

7592₁₀             214₅         123₄        656₇

You should notice the highest value symbol used in a number system is always one less than the base of the system. In base 10 the largest value symbol possible is 9; in base 5 it is 4; in base 3 it is 2.

Positional Notation and Zero

You must observe two principles when counting or writing quantities or numerical values. They are the POSITIONAL NOTATION and the ZERO principles.

Positional notation is a system where the value of a number is defined not only by the symbol but by the symbol's position. Let's examine the decimal (base 10) value of 427.5. You know from experience that this value is four hundred twenty-seven and one-half. Now examine the position of each number:

If 427.5 is the quantity you wish to express, then each number must be in the position shown. If you exchange the positions of the 2 and the 7, then you change the value.

Each position in the positional notation system represents a power of the base, or radix. a Power is the number of times a base is multiplied by itself. The power is written above and to the right of the base and is called an EXPONENT. Examine the following base 10 line graph:

Now let's look at the value of the base 10 number 427.5 with the positional notation line graph:

You can see that the power of the base is multiplied by the number in that position to determine the value for that position.

The following graph illustrates the progression of powers of 10:

All numbers to the left of the decimal point are whole numbers, and all numbers to the right of the decimal point are fractional numbers. a whole number is a symbol that represents one, or more, complete objects, such as one apple or $5. a fractional number is a symbol that represents a portion of an object, such as half of an apple (.5 apples) or a quarter of a dollar ($0.25). a mixed number represents one, or more, complete objects, and some portion of an object, such as one and a half apples (1.5 apples). When you use any base other than the decimal system, the division between whole numbers and fractional numbers is referred to as the RADIX Point. The decimal point is actually the radix point of the decimal system, but the term radix point is normally not used with the base 10 number system.

Just as important as positional notation is the use of the zero. The placement of the zero in a number can have quite an effect on the value being represented. Sometimes a position in a number does not have a value between 1 and 9. Consider how this would affect your next paycheck. If you were expecting a check for $605.47, you wouldn't want it to be $65.47. Leaving out the zero in this case means a difference of $540.00. In the number 605.47, the zero indicates that there are no tens. If you place this value on a bar graph, you will see that there are no multiples of 10¹.

Most Significant Digit and Least Significant Digit (MSD and LSD)

Other important factors of number systems that you should recognize are the MOST SIGNIFICANT DIGIT (MSD) and the LEAST SIGNIFICANT DIGIT (LSD). The MSD in a number is the digit that has the greatest effect on that number. The LSD in a number is the digit that has the least effect on that number. Look at the following examples:

You can easily see that a change in the MSD will increase or decrease the value of the number the greatest amount. Changes in the LSD will have the smallest effect on the value. The nonzero digit of a number that is the farthest LEFT is the MSD, and the nonzero digit farthest RIGHT is the LSD, as in the following example:

In a whole number the LSD will always be the digit immediately to the left of the radix point.

Q1.    What term describes a single object?

Q2.    A symbol that represents one or more objects is called a .......... .

Q3.        The symbols 0, 1, 2, and 3 through 9 are what type of numerals?

Q4.    What does the base, or radix, of a number system tell you about the system?

Q5.   How would you write one hundred seventy-three base 10?

Q6.    What power of 10 is equal to 1,000? 100? 10? 1?

Q7.    The decimal point of the base 10 number system is also known as the .......... .

Q8.    What is the MSD and LSD of the following numbers

Carry and Borrow Principles

Soon after you learned how to count, you were taught how to add and subtract. At that time, you learned some concepts that you use almost everyday. Those concepts will be reviewed using the decimal system. They will also be applied to the other number systems you will study.

ADDITION - Addition is a form of counting in which one quantity is added to another. The following definitions identify the basic terms of addition:

AUGEND - The quantity to which an addend is added

ADDEND - a number to be added to a preceding number

SUM - The result of an addition (the sum of 5 and 7 is 12)

CARRY - a carry is produced when the sum of two or more digits in a vertical column equals or exceeds the base of the number system in use

How do we handle the carry; that is, the two-digit number generated when a carry is produced? The lower order digit becomes the sum of the column being added; the higher order digit (the carry) is added to the next higher order column. For example, let's add 15 and 7 in the decimal system:

Starting with the first column, we find the sum of 5 and 7 is 12. The 2 becomes the sum of the lower order column and the 1 (the carry) is added to the upper order column. The sum of the upper order column is 2. The sum of 15 and 7 is, therefore, 22.

The rules for addition are basically the same regardless of the number system being used. Each number system, because it has a different number of digits, will have a unique digit addition table. These addition tables will be described during the discussion of the adding process for each number system.

A decimal addition table is shown in table 1-1. The numbers in row X and column Y may represent either the addend or the augend. If the numbers in X represent the augend, then the numbers in Y must represent the addend and vice versa. The sum of X + Y is located at the point in array Z where the selected X row and Y column intersect.

Table 1-1. - Decimal Addition Table

To add 5 and 7 using the table, first locate one number in the X row and the other in the Y column. The point in field Z where the row and column intersect is the sum. In this case the sum is 12.

SUBTRACTION. - The following definitions identify the basic terms you will need to know to understand subtraction operations:

 
·    SUBTRACT - To take away, as a part from the whole or one number from another

 
·    MINUEND - The number from which another number is to be subtracted

·    SUBTRAHEND - The quantity to be subtracted

 
·    REMAINDER, or Difference - That which is left after subtraction

 
·    BORROW - To transfer a digit (equal to the base number) from the next higher order column for the purpose of subtraction.

Use the rules of subtraction and subtract 8 from 25. The form of this problem is probably familiar to you:

It requires the use of the borrow; that is, you cannot subtract 8 from 5 and have a positive difference. You must borrow a 1, which is really one group of 10. Then, one group of 10 plus five groups of 1 equal 15, and 15 minus 8 leaves a difference of 7. The 2 was reduced by 1 by the borrow; and since nothing is to be subtracted from it, it is brought down to the difference.

Since the process of subtraction is the opposite of addition, the addition table 1-1 may be used to illustrate subtraction facts for any number system we may discuss.

In addition,

X + Y = Z

In subtraction, the reverse is true; that is,

Z – Y = X

OR

Z – X = Y

Thus, in subtraction the minuend is always found in array Z and the subtrahend in either row X or column Y. If the subtrahend is in row X, then the remainder will be in column Y. Conversely, if the subtrahend is in column Y, then the difference will be in row X. For example, to subtract 8 from 15, find 8 in either the X row or Y column. Find where this row or column intersects with a value of 15 for Z; then move to the remaining row or column to find the difference.

The BINARY NUMBER System

The simplest possible number system is the BINARY, or base 2, system. You will be able to use the information just covered about the decimal system to easily relate the same terms to the binary system.

Unit and Number

The base, or radix - you should remember from our decimal section¾is the number of symbols used in the number system. Since this is the base 2 system, only two symbols, 0 and 1, are used. The base is indicated by a subscript, as shown in the following example:

1₂

When you are working with the decimal system, you normally don't use the subscript. Now that you will be working with number systems other than the decimal system, it is important that you use the subscript so that you are sure of the system being referred to. Consider the following two numbers:

11     11

With no subscript you would assume both values were the same. If you add subscripts to indicate their base system, as shown below, then their values are quite different:

11₁₀       11₂

The base ten number 11₁₀  is eleven, but the base two number 11₂  is only equal to three in base ten. There will be occasions when more than one number system will be discussed at the same time, so you MUST use the proper Subscript.

Positional Notation

As in the decimal number system, the principle of positional notation applies to the binary number system. You should recall that the decimal system uses powers of 10 to determine the value of a position. The binary system uses powers of 2 to determine the value of a position. a bar graph showing the positions and the powers of the base is shown below:

All numbers or values to the left of the radix point are whole numbers, and all numbers to the right of the radix point are fractional numbers.

Let's look at the binary number 101.1 on a bar graph:

Working from the radix point to the right and left, you can determine the decimal equivalent:

Table 1-2 provides a comparison of decimal and binary numbers. Notice that each time the total number of binary symbol positions increase, the binary number indicates the next higher power of 2. By this example, you can also see that more symbol positions are needed in the binary system to represent the equivalent value in the decimal system.