Rational and Irrational Numbers Introduction

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The sets of numbers which every student must remember are:

• The set of natural numbers

N = {1, 2, 3, 4, 5, …}

• The set of whole numbers

W = {0, 1, 2, 3, 4, 5,…}

• The set of integers

Z = I = {…, -3, -2, -1, 0, 1, 2, 3,…}

• The set of rational numbers

Q

• The set of irrational numbers

is a set of numbers that cannot be expressed as rational numbers.

(i)

(ii) forms the set of real numbers

(iii)

Set of Real Numbers

Some Definitions

Consecutive integers

They are integers that differ from each other by one,

e.g., 8, 9, 10 are consecutive integers, also 180, 179, 178, 177 are consecutive integers.

Factor

A factor of a number divides it exactly without leaving any remainder,

e.g., Factors of 18 are 1, 2, 3, 6, 9, 18.

Common factor

A number which is a factor of two or more numbers is called a common factor of these numbers.

e.g., 4 is a common factor of 16 and 20.

Prime numbers

It can only be divided by one or itself i.e., a prime number has no factors other than 1 and itself.

e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, … are examples of prime numbers.

Composite number

It has factors other than one and itself,

e.g., 21 is a composite number.

Prime factors

They are prime numbers which are factors of a given number.

e.g., Prime factors of 42 = 2 x 3 x 7.

Numbers prime to each other

Such numbers have only 1 as common factor even though they may be composite numbers

e.g., 21 and 32 are prime to each other.

We generally write (21, 32) = 1, which means 21 and 32 are prime to each other i.e., they have only one factor in common viz., 1.

Rule of Divisibility

Fraction

A fraction is a part of the whole.

Types of fractions

• A Proper fraction has its numerator less than its denominator.

• An Improper fraction has its numerator greater than its denominator.

• A Mixed fraction consists of two parts, one is an integer and the other a proper fraction.

• Vulgar fraction is a fraction in which the denominator is not 10, 100, 1000, etc.

Simplification of fractions

Remember 'BODMAS', Do the various operations in the order given below:

Open brackets from within and go outwards in the order:

-, (), {} and lastly [].

B 1 Brackets

O 2 Of

D 3 Division

M 4 Multiplication

A 5 Addition

S 6 Subtraction