Representing Real Numbers on Number Line and Law of Exponents - Class 9, Mathematics | Mathematics for NDA PDF Download

REPRESENTATION OF IRRATIONAL NUMBERS ON A NUMBER LINE

Method-Ist:- Plot  on a number line Constructing the 'square root spiral':- Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point O and draw a line segment OP₁ of unit length. Draw a line segment P₁P₂ perpendicular to OP₁ of unit length [see figure]. Now draw a line segment P₂P₃ perpendicular to OP₂. Then draw a line segment P₃P₄ perpendicular to OP₃.

Continuing in this manner, we can get the line segment P_{n – 1}P_n by drawing a line segment of unit length perpendicular to OP_{n – 1}. In this manner, we will have created the points : P₁, P₂, P₃, ....., P_n, ...., and joined them to create a beautiful spiral depicting

OB = √2 = OF

OC = √3 = OG

OD = √4 = OH

OE = √5 = OI

Method : II
To represent √2 on the real number line :
Let l be a real number line and O be a point representing 0 (zero) . Take OA = 1 unit. Draw AB ⊥ OA such that AB = 1 unit.

with O as a centre and OB radius, draw an arc, meeting line 7 at P.
Then, OB = OP = √2 unit    

Thus, the point P represent √2 on the number line .

To represent √3 on the real number line :
 

Then, OC = OQ = √3 unit

Thus, the point Q represent √3 on the number line.

To represent √5 on the real nunber line :

[By Pythagorus theorem]

Then, OB = OP = √5
Thus, the point P represents √5 on the number line.

To represent √6 on the real number line :

Then, OC = OQ = √6 unit
Thus, the point Q represent √6 on the number line.    D

Ex. Insert a rational and an irrational number between 2 and 3.
Sol.

If a, b are rational numbers, then  is a rational number between them.
 A rational number between 2 and 3 is 

Ans.
If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then  is an irrational number lying between a and b.
Hence, the irrational number between 2 and 3 is Ans.

Ex. Find two irrational numbers between 2 and 2.5.
Sol. If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then  is an irrational number lying between a and b.

Irration number between 2 and 2.5 is 

Similarly, irrational number between 2 and √5 is 

So, required numbers are 

EXPONENTS OF REAL NUMBERS
exponents OR index or index number or power

or 

If a number is multiplied by itself a number of times, then it can be written in the exponential form

3 × 3 = 3²      OR       x × x × x × .... n times = xⁿ

      x = any rational number

              n = Positive Integer

In    –7, x, are called bases and 2, 4, 3, n are called exponents or index.

Zero Exponent : For any non zero rational number x we define a⁰ = 1

Positive integral Power :- Let a be a real number and n be a positive integer. Then we define aⁿ as

aⁿ = a × a × a × a ×.... × a (n times)

Where aⁿ is called the nth power of a. The real number a is called the base and n is called the exponent of the n^th power of a.

Example : (i) 3⁴ = 3 × 3 × 3 × 3 = 81

Negative integral Power :- Let a be any non-zero real number and n be a positive integer. Then, we define

a^-n = 

Example :

LAWS OF INTEGRAL EXPONENTS
First law (Product Law):- Let a be any real number and m, n are positive integers, then

a^m . aⁿ = a^{m + n}

Generalisation a^x × a^y × a^z × ...... = a^{x + y + z + ........}

Example :-
(i) 7⁴ × 7⁵ = 7^{4 + 5} = 7⁹

Second Law (Quotient Law) :- Let a be a non zero real number and m, n are positive integers, then

Example :-

Third Law (Power Law) :- Let a be a positive real number and m, n are positive integers, then
(a^m)ⁿ = a^mn = (aⁿ)^m

Example :-
(i) (2³)⁴ = 2³ × ⁴ = 2¹²

(ii) (5^–2)^–3 = 5(^–2) × (^–3) = 5⁶

Fourth Law :- Let a, b be two real numbers and n is a positive integer, then
(i) (ab)ⁿ = aⁿ . bⁿ

Example :-

(i) (10)⁴ = (2 × 5)⁴ = 2⁴ × 5⁴

RATIONAL EXPONENTS OF A REAL NUMBER
Principal n^th root of a Positive real number:- Let a be a positive real number and n be a positive integer. Then, the principal n^th root of a is the unique positive real number x such that xⁿ = a.
The principal n^th root of a positive real number a is denoted by

Principal nth Root of a Negative Real Number :- Let a be a negative real number and n be an odd positive integer. Then, the principal nth root of a is defined as

i.e. the principal nth root of a is minus of the principal nth root of |a|.

Examples :-

Remark :- n^th root of a is not defined. If a is negative real number and n is an even positive integer because an
even power of a real numbr is always positive.
Example :-
  is meaningless quantity.
Justification :- Let  which is not possible as x² should always be positive.

LAWS OF EXPONENTS
Let a, b > 0 be a real number, and let m and n be rational numbers.
Then, we have

(i) a^m . aⁿ = a^{m + n}

(iii) (a^m)ⁿ = a^mn

(iv) (ab)^m = a^mb^m

We can also generalise the laws of exponent for the nth root of a number.
These are given below :-

  or 

 or 

 or    

Ex. Evaluate each of the following:-

(i) 2⁵ × 5²                      (ii) (2³)²

Sol. (i) We have 2⁵ × 5² = (2 × 2 × 2 × 2 × 2) × (5 × 5) = 32 × 25 = 800

(ii) We have (2³)² = (2)^{3 × 2} = 2⁶ = 64

(v) We have