Notesale: Turn your study into money

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NUMBER SYSTEM AND SETS
Mathematics is a basic tool in our daily life
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Storekeepers need mathematical computation in their bookkeeping
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Electronics principles are frequently stated by means of
mathematical formulas
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In such a way
Defense warfare becomes more and more complex, mathematics achieves ever
increasing importance as an essential tool
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Mathematics better equips us to do his present works
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Statistically it has been
found that one of the best indicators of a man's potential success as a learner is his
understanding of mathematics
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An attempt is made throughout to give an understanding of why the rules of
mathematics are true
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Many of us have areas in our mathematics background that are hazy, barely
understood, or troublesome
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These chapters attempt to treat the subject on an adult
level that will be interesting and informative
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The
process is based on the idea of ONE-TO-ONE COR-RESPONDENCE, which is easily
demonstrated by using the fingers
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Having outgrown finger counting, we use numerals
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One of the simplest numeral systems is the Roman
numeral system, in which tally marks are used to represent the objects being
counted
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By this method, one makes short vertical marks until a total of four is reached;
when the fifth tally is counted, a diagonal mark is drawn through the first four
marks
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A number may have many "names
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The important thing to remember is that a number is an idea; various symbols used to
indicate a number are merely different ways of expressing the same idea
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They are the positive whole numbers, or to use the more precise
mathematical term, positive INTEGERS
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For example, 5, 32, and 7,049 are all integers
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The idea of ranking numbers in terms of tens, hundreds, thousands, etc
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PLACE-VALUE
Although a system such as the Roman numeral system is adequate for recording
the results of counting, it is too cumbersome for purposes of calculation
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The idea of 0 as a number
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Positional notation (Place Value)
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The
convention used in our number system is that each digit has a higher place value
than those digits to the right of it
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The base which is most commonly used is ten, and
the system with ten as a base is called the decimal system ( decem is the Latin word
for ten)
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One exception to this rule occurs when the subject of an entire discussion
is some base other than ten
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DECIMAL SYSTEM
In the decimal system, each digit position in a number has ten times the value of the
position adjacent to it on the right
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The 1 on the right is said to be in the "units place," with the understanding that
the term "unit" in our system refers to the numeral 1
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" Since ten plus one is eleven, the
symbol 11 represents the number eleven
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Figure A-1 shows the names of several digit positions in the decimal system
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”
The number may be expressed in mathematical symbols as follows:
2 * 10 *10 + 3 * 10 + 5 * 1
Notice that this bears out our earlier statement: each digit position has 10 times the
value of the position adjacent to it on the right
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" Expressed in mathematical symbols, this
number is as follows:
4 * 1000 + 3 * 100 + 7 * 10 + 2 * 1
This presentation may be broken down further, in order to show that each digit
position as 10 times the place value of the position on its right, as follows:
4 * 10 * 100 + 3 * 10 * 10 + 7 * 10 * 1 + 2 * 1
The comma which appears in a number symbol such as 4,372 is used for "pointing off"
the digits into groups of three beginning at the right-hand side
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Some of these groups are shown in table A-2
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Place values and grouping
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Notice
that the word "and" is not
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***Practice problems:***
1
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Answer: 7,281
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Write the meaning, in words, of the symbol 23,469
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3
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4
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) does
it belong?
Answers: Billions
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How-ever, since the base in this system is two, only two digit symbols are needed for
writing numbers
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In order to understand why only two
digit symbols are needed in the binary system, we may make some observations
about the decimal system and then generalize from these
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For
example, in the decimal system the symbol for ten, the base, is 10
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"
Notice the implication of this where other bases are concerned: Every system uses
the same symbol for the base, namely 10
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Suppose that a
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Then the
only digit symbols needed would be 0, 1, 2, 3, and 4
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" In general, in a number system using base N, the largest number for
which a single-digit symbol is needed is N minus 1
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An example of a binary number is the symbol 101
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Figure A-2 shows that the place
value of each digit position in the binary system is two times the place value of the
position adjacent to it on the right
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0 1 0 1
EIGHTS

FOURS

TWOS

UNITS

Figure A-2 – Digit Positions in the binary system
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" Thus 101 is the binary
equivalent of decimal 5
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Since 7 is
equal to 4 plus 2 plus 1, we say that it "contains" one 4, one 2, and one unit
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The most common use of the binary number system is in electronic digital computers
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One of the reasons for this is the fact that electrical and electronic equipment utilizes
many switching circuits in which there are only two operating conditions
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Details concerning binary arithmetic are be-yond the
scope of this volume, but are available in Mathematics
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Write the decimal equivalents of the binary numbers 1101, 1010, 1001 and 1111
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2
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Answer: 1100, 111, 1110 and 11
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At the time of writing of
this topic, much emphasis is being placed on so-called modern math in the public
schools
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In the following paragraphs, a very brief introduction to some of the set theory of
modern math is presented
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DEFINITIONS AND SYMBOLS
The word "set" implies a collection or grouping of similar objects or symbols
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A set of tools would be an example of a group of objects not
necessarily similar in appearance but similar in purpose
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The elements of a mathematical set are usually symbols, such as numerals, lines, or
points
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This is often done where the elements
of the set are not too numerous
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In other words, equality between sets has nothing
to do with the order in which the elements are arranged
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That is, the elements of {2, 2, 3, 4} are simply 2, 3, and 4
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***Practice problems:***
1
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Answer: {1,3,5,7,9}
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Use the correct symbols to designate the set of names of days of the week which
do not contain the letter “s”
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List the elements of the set of natural numbers greater than 15 and less than 20
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Suppose that we have sets as follows:
A = {1,2,3} , B= {1,2,2,3} , C= {1,2,3,4} , D= {1,1,2,3}
Which of these sets are equal?
Answer: A = B= D

SUBSETS
Since it is inconvenient to enumerate all of the elements of a set each time the set is
mentioned, sets are often designated by a letter
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In symbols, this relationship could be Stated as Follow:
S={1,2,3,4,5,6,7,8,9}
Now suppose that we have another set, T, which comprises all positive even integers
less than 10
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This establishes the SUBJECT
relationship; T is said to be a subset of S
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This set
comprises the counting numbers (natural numbers) and includes, as subsets, all of
the sets of numbers which we have discussed
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This means that the successive elements of
the set continue to increase in size without limit, each number being larger by 1 than
the number preceding it
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One way to represent the set of natural numbers symbolically would be as follows:
{ 1 , 2 , 3 , 4 , 5 , 6 , …}
The three dots, called ellipsis, indicate that the pattern established by the numbers
shown continues without limit
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POINTS AND LINES
In addition to the many sets which can be formed with number symbols, we
frequently find it necessary in mathematics to work with sets composed of points or
lines
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The mark which
is made on a piece of paper is merely a symbol representing the point
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Thus a pencil dot
is only a rough picture of a point, useful for indicating the location of the point but
certainly not to be confused with the idea
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" Picturing this arrangement by drawing dots on paper, we would have a
"dotted line
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Once again, it is important to emphasize that the picture is
only a symbol which represents an ideal line
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The foregoing discussion leads to the conclusion that a line is actually a set of points
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The idea of arranging points together to form a line may be extended to the
formation planes (flat surfaces)
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It is also determined by two intersecting lines
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A line segment is a sub-set of the set of points comprising a
line
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A ray is not a line
segment, because it does not terminate at both ends;
It may be appropriate to refer to a ray as a "half-line
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All three—lines, line segments, and rays—are subsets of the set of points comprising a
plane
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To illustrate the construction of a number line, let us place
the elements of the set of natural numbers in one-to-one correspondence with points
on a line
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The starting point is labeled 0, the next
point is labeled 1, the next 2, etc
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See fig
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) Such an arrangement is often referred to as a scale, a familiar
example being the scale on a thermometer
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A number Line
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The number line is an ideal device for picturing the relationship between integers and
other numbers such as fractions and decimals
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Examples are the points representing the numbers 1/2 (located hallway between 0
and 1) and 2
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An interesting question arises , concerning the "in-between" points on the number
line: How many points (numbers) exist between any two integers? To answer this
question, suppose that we first locate the point halfway between 0 and 1, which
corresponds to the number 1/2
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The result of the next such halving
operation would be 1/8, the next 1/16, etc

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