1, 2, 3, 4, 5...............

All numbers which are...well... "Natural"!!!

**Real Numbers**

Whole Numbers

0

+

1, 2, 3, 4, 5....

Natural Numbers

Just add the "hole"!!!

0, 1, 2, 3, 4, 5......

Whole Numbers

......-5, -4, -3, -2, -1

+

Integers

Rational Numbers

Any number which can be written in the form of

p/q {p,q = integers; q = 0}

Ex. 2/3; 4; 0; 7/2; 6547/2143......

Irrational Numbers

A number whose exact value is a:

Non-Terminating & Non-Repeating Decimal

Ex. 0.001000100001.... ; √2, 3, π

Types of Irrational Numbers

1. Non-terminating, non-repeating

Ex. 0.313113111311111...; 5.252252225...; 0.554445554444

2. √m; where m is not a perfect square

Ex. √2, √3, √5,√53......

3. m; where m is not a perfect cube

Ex. 2, 3, 5, 53......

4. The Number π (pie). Its approx. value is 22/7. π has a value which is non-terminating, non-repeating.

/

**Properties of Real Numbers**

Addition Properties

Multiplication Properties

Addition Properties

1. Closure Property

Real Number 1 + Real Number 2 = Real Number 3

2. Associative Law

(a + b) + c = a + (b + c)

3. Commutative Law

a + b = b + a

4. Existence of Additive Identity

a + 0 = 0 + a = a

5. Existence of Additive Inverse

a + (-a) = (-a) + a = 0

1. Closure Property

Real Number 1 x Real Number 2 = Real Number 3

2. Associative Law

(ab)c = a(bc)

3. Commutative Law

ab = ba

4. Existence of Multiplicative Identity

a . 1 = 1 . a = a

5. Existence of Multiplicative Inverse

a . (1/a) = (1/a) . a = 1

Multiplication Properties

**Rationalisation**

Rationalising Factor (RF)

If product of two irrational number is rational, then they are called the rationalising factor of each other

Ex. (a + b) and (a - b) are RF of each other.

Q. Find a rational number between 1/4 and 1/3.

Exercise

Q. Find five rational number between 3/5 and 4/5.

Exercise

Exercise

1. Add (3√2 + 7 √3) and (√2 - 5√3)

3. Simplify: (√13 - √6)(√13 + √6)

2. Divide 15√15 by 3√5

Exercise

1. Rationalise the denominator of:

a. 1 / (3 + √5)

b. 4 / (√7 + √2)

c. (√4 + √3) / (√4 - √3)

Exercise

2. If a and b are rational numbers and

(4 + 3√5) / (4 - 3√5) = a + b√5,

Find the value of a and b.

**Class IX Refresher**

Numbers whose square is non-negative

**Laws of Radicals**

For a real no. a > 0 and rational numbers

p

&

q

i) a x a = a

p

q

(p + q)

**ii) (a ) = a**

p

q

pq

iii) a / a = a

p

q

(p - q)

iv) a x b = (ab)

p

p

p

Exercise