Properties Of Rational Number

If a,b,c are three rational numbers.

(i) Commutative property of addition. a + b = b + a

(ii) Associative property of addition (a+b)+c = a+(b+c)

(iii) Additive inverse a + (-a) = 0

0 is identity element, -a is called additive inverse of a.

(iv) Commutative property of multiplications a.b. = b.a.

(v) Associative property of multiplication (a.b).c = a.(b.c)

(vi) Multiplicative inverse

1 is called multiplicative identity and is called multiplicative inverse of a or reciprocal of a.

(vii) Distributive property a.(b+c) = a.b + a.c

∴ L.H.S. R.H.S.

question Prove that is an irrational number.

Solution: Let where r be a rational number

Squaring both sides

⇒

⇒ 3 + 2 – 2 =r²

⇒ 5 – 2 =r²

Here, 5 –2 is an irrational number but r² is a rational number.

∴ L.H.S. R.H.S.

Hence it contradicts our assumption that is a rational number.

(b) Irrational Number in Decimal Form :

= 1.414213 ...... i.e. it is not-recurring as well as non-terminating.

= 1.732050807 ...... i.e. it is non-recurring as well as non-terminating.

question Insert an irrational number between 2 and 3.

Solution:

question Find two irrational number between 2 and 2.5.

Solution: 1st Method :

Since there is no rational number whose square is 5. So is irrational..

Also is a irrational number.

2nd Method: 2.101001000100001.... is between 2 and and it is non-recurring as well as non-terminating.

Also, 2.201001000100001......... and so on.

question Find two irrational number between and .

Solution: 1st Method :

Irrational number betweenand

2nd Method : As = 1.414213562 ...... and = 1.732050808......

As , and has 4 in the 1st place of decimal while has 7 is the 1st place of decimal.

∴ 1.501001000100001......., 1.601001000100001...... etc. are in between and

question Find two irrational number between 0.12 and 0.13.

Solution: 0.1201001000100001......., 0.12101001000100001 .......etc.

question Find two irrational number between 0.3030030003..... and 0.3010010001 .......

Solution: 0.302020020002...... 0.302030030003.... etc.

question Find two rational number between 0.2323323332..... and 0.252552555255552.......

Solution: 1st place is same 2.

2nd place is 3 & 5.

3rd place is 2 in both.

4th place is 3 & 5.

Let a number = 0.25, it falls between the two irrational number.

Also a number = 0.2525 an so on.

IRRATIONAL NUMBERS: 

An irrational number is a non-terminating and non-recurring decimal and cannot be put in the form .

• Square root of a prime number is also irrational number. For example, etc.

PROPERTIES OF IRRATIONAL NUMBERS:

RATIONALIZATION FACTOR (RF):

If the product of the irrational numbers is rational then each one is called the rationalizing factor of the other.

If a and b are integers and x, y are natural numbers, then 

(i) and are RF of each other, as which is rational.

(ii) and are RF of each other, as which is rational.

(iii) and are RF of each other, as which is rational.

question Rationalise the denominator of the following:

(i) (ii) (iii)

Solution: (i)

(ii)

(iii)

LAWS OF RADICALS:

Let a > 0 be a real number, and let p and q be rational numbers, then we have:

question Simplify (i) (ii) (iii) (iv)

Solution: (i) [ ]

(ii) [ ]

(iii)

(iv)

REPRESENTATION OF IRRATIONAL NUMBER ON NUMBER LINE:

e.g. Represent on the number line. 

First take (OQ) on the number line.

Steps of Construction for

1. On the hypotenuse OB of right angled ΔAOB, draw BC perpendicular to OB such that BC = OA = 1 unit.

2. Join OC.

In ΔOCB, by Pythagoras Theorem, we have

or .

3. With O as centre and OC as radius, draw an arc to intersect number line
at P.

Hence, it is clear, that

Thus, the point P represents the number on the number line l.

EXISTENCE OF SQUARE ROOT OF A POSITIVE REAL NUMBER:

For any positive real number x, we have

To find the positive square root of a positive real number, we follow the following steps.

•  Obtain the positive real number x (say).
• Draw a line and mark a point A on it.
• Mark a point B on the line such that AB = x units.
• From point B mark a distance of 1 unit and mark this new point as P.
• Find the mid-point of AP and mark the point as O.
• Draw a circle with centre O and radius OP.
• Draw a line perpendicular to AP passing through B and intersecting the semicircle at D. Length BD is equal to

For example:

Represent on the number line.

In order to represent on number line, we follow the following steps:

5. Taking O as the centre and OP = OA as radius draw a semi-circle. Draw a line passing through B perpendicular to OB cutting the semi-circle at D. 

6. Taking B as the centre and BD as radius draw an arc cutting OP produced at E. Distance BE represents