1.3 Set of Real Numbers

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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 ?





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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.

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1.3 Set of Real Numbers

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