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The Set of Real
Numbers
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Learning Objectives:
Upon completion, you should be able to:
Identify subsets of the set of real numbers;
Recognize various forms of rational
numbers;
Distinguish rational numbers from
irrational numbers;
Locate numbers on the real number line.
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What is a number?
Numbers rule the universe Pythagoras
All is number - Pythogoreans
A number is a mathematical object used in
counting and measuring.
A notational symbol which represents a
number is called a numeral.
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Natural Numbers
All natural numbers are truly natural. We
find them in nature.
Also called as counting numbers
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Some Subsets of
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Time to Think!
What is the largest prime number that you
know?
Is ? Is ?
Are and disjoint?
Are and disjoint?
Are and disjoint?
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Whole Numbers
Is zero a number?
How can a number of nothing be a
number?
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Integers
Negative numbers are natural numbers
with negative (minus) sign.
A positive number added to its negativeresults to 0.
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Some Subsets of
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Rational Numbers
Numbers that can be expressed as a ratio
or quotient of two integers
In form of fractions, terminating decimals
and non-terminating but repeating
decimals.
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Illustration
All integers are rational numbers.
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Irrational Numbers
Numbers that can not be expressed as the
ratio of two integers
Decimal numbers that neither repeat norterminate
Note that irrational numbers cannot be a
rational number, i.e.,
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Illustration
Some of the irrational numbers..
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Real Numbers
The union of the set of rational numbers
and the set of irrational numbers
Relating the numbered sets we have,
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Time to Think!
Perform the following operations:
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Real Number Line
One-dimensional coordinate system
There is a one-to-one correspondence
between the set of points on a line and the
set of all real numbers.
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Real Number Line
Locate the following numbers in the number
line:
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Properties of
Real Numbers
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Learning Objectives:
At the end of the lesson, you should be able
to:
Enumerate field axioms;
Illustrate the closure property for real
numbers;
Identify the identity and inverse elements
for addition and multiplication;
Explain the density property.
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Operations on
Addition
Multiplication
Subtraction
Division
Exponentiation
Root Extraction
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Closure Property
A set is closed (under an operation) if and
only if the operation on two elements of
the set produces another element of theset.
If an element outside the set is produced,
then the operation is not closed.
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Closure Property
For any , and
That is, is closed under and .
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Time to Think!
Is closed under and ?
Is closed under and ?
Is closed under and ?
Is closed under and ?
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Commutative Property
For any
That is, we can add and multiply real
numbers in any order.
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Associative Property
For any
That is, we can group numbers in asum/product in any way we want and still
get the same answer.
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Distributive Property
of multiplication over addition
For any
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Existence of Identity Elements
Zero added to any number results to the
number itself.
Any number multiplied to one gives the
number itself.
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Existence of Identity Elements
For any , there exist two unequal
numbers (read zero and one, resp.)
such that
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Existence of Inverse Element
Given a certain number, what will you add
to it so that the result will be the additive
identity element zero?
What will you multiply to it so that the
result is the multiplicative identity one?
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Existence of Inverse Element
For any , there exists another
element of , denoted by such that
For any , , there exists another
element of , denoted by such that
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Subtraction & Division
Subtracting a number means adding the
negative of that number
Dividing by a number means multiplying
the reciprocal of that number
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Density Property
We can always find a real number that lies
between any two real numbers.
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Illustration
State the property that justifies the truth of
the following statements:
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Summary:
A number is either rational or irrational.
There is a one-to-one correspondence
between the set of real numbers and theset of points on the line.
Real numbers satisfy the field axioms;
We need to be familiar with the propertiesof real numbers in order to solve algebra
problems.