Properties Of Rational Number

Number system of Class 9

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 PROPERTIES OF RATIONAL NUMBER

1 is called multiplicative identity and PROPERTIES OF RATIONAL NUMBER is called multiplicative inverse of a or reciprocal of a.

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

∴ L.H.S. PROPERTIES OF RATIONAL NUMBERR.H.S.

question Prove that PROPERTIES OF RATIONAL NUMBER is an irrational number.

Solution: Let PROPERTIES OF RATIONAL NUMBER where r be a rational number

Squaring both sides

PROPERTIES OF RATIONAL NUMBER

⇒ 3 + 2 – 2PROPERTIES OF RATIONAL NUMBER =r2

⇒ 5 – 2PROPERTIES OF RATIONAL NUMBER =r2

Here, 5 –2 PROPERTIES OF RATIONAL NUMBER is an irrational number but r2 is a rational number.

∴ L.H.S. PROPERTIES OF RATIONAL NUMBER R.H.S.

Hence it contradicts our assumption that PROPERTIES OF RATIONAL NUMBER is a rational number.

(b) Irrational Number in Decimal Form :

PROPERTIES OF RATIONAL NUMBER= 1.414213 ...... i.e. it is not-recurring as well as non-terminating.

PROPERTIES OF RATIONAL NUMBER = 1.732050807 ...... i.e. it is non-recurring as well as non-terminating.

question Insert an irrational number between 2 and 3.

Solution: PROPERTIES OF RATIONAL NUMBER

question Find two irrational number between 2 and 2.5.

Solution: 1st Method : PROPERTIES OF RATIONAL NUMBER

Since there is no rational number whose square is 5. So PROPERTIES OF RATIONAL NUMBER is irrational..

Also PROPERTIES OF RATIONAL NUMBER is a irrational number.

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

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

question Find two irrational number between PROPERTIES OF RATIONAL NUMBERand PROPERTIES OF RATIONAL NUMBER.

Solution: 1st Method : PROPERTIES OF RATIONAL NUMBER

Irrational number betweenPROPERTIES OF RATIONAL NUMBERand PROPERTIES OF RATIONAL NUMBER

PROPERTIES OF RATIONAL NUMBER

2nd Method : As PROPERTIES OF RATIONAL NUMBER = 1.414213562 ...... andPROPERTIES OF RATIONAL NUMBER = 1.732050808......

As , PROPERTIES OF RATIONAL NUMBERand PROPERTIES OF RATIONAL NUMBER has 4 in the 1st place of decimal while PROPERTIES OF RATIONAL NUMBER has 7 is the 1st place of decimal.

∴ 1.501001000100001......., 1.601001000100001...... etc. are in between PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER

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 PROPERTIES OF RATIONAL NUMBER.

  • Square root of a prime number is also irrational number. For example,PROPERTIES OF RATIONAL NUMBER etc.

PROPERTIES OF IRRATIONAL NUMBERS:

S. No.

Properties

Example

1.

Sum of two irrationals need not be an irrational.

PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER

2.

Difference of two irrationals need not be an irrational.

PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER

3.

Product of two irrationals need not be an irrational.

PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER

4.

Quotient of two irrationals need not be an irrational.

PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER

5.

Sum of a rational and an irrational is irrational.

5 and PROPERTIES OF RATIONAL NUMBER

6.

Difference of a rational and an irrational is irrational.

5 and PROPERTIES OF RATIONAL NUMBER

7.

Product of a rational and an irrational is irrational.

5 and PROPERTIES OF RATIONAL NUMBER

8.

Quotient of a rational and an irrational is irrational.

2 and PROPERTIES OF RATIONAL NUMBER

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) PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER are RF of each other, as PROPERTIES OF RATIONAL NUMBER which is rational.

(ii) PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER are RF of each other, as PROPERTIES OF RATIONAL NUMBER which is rational.

(iii) PROPERTIES OF RATIONAL NUMBER and PROPERTIES OF RATIONAL NUMBER are RF of each other, as PROPERTIES OF RATIONAL NUMBER which is rational.

question Rationalise the denominator of the following:

(i) PROPERTIES OF RATIONAL NUMBER (ii) PROPERTIES OF RATIONAL NUMBER (iii) PROPERTIES OF RATIONAL NUMBER

Solution: (i) PROPERTIES OF RATIONAL NUMBER

(ii) PROPERTIES OF RATIONAL NUMBER

(iii) PROPERTIES OF RATIONAL NUMBER

LAWS OF RADICALS:

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

S. No.

Law of exponents

1.

PROPERTIES OF RATIONAL NUMBER

2.

PROPERTIES OF RATIONAL NUMBER

3.

PROPERTIES OF RATIONAL NUMBER

4.

PROPERTIES OF RATIONAL NUMBER

question Simplify (i) PROPERTIES OF RATIONAL NUMBER (ii) PROPERTIES OF RATIONAL NUMBER (iii) PROPERTIES OF RATIONAL NUMBER (iv) PROPERTIES OF RATIONAL NUMBER

Solution: (i) PROPERTIES OF RATIONAL NUMBER [PROPERTIES OF RATIONAL NUMBER ]

(ii) PROPERTIES OF RATIONAL NUMBER [PROPERTIES OF RATIONAL NUMBER ]

(iii) PROPERTIES OF RATIONAL NUMBER PROPERTIES OF RATIONAL NUMBER

(iv) PROPERTIES OF RATIONAL NUMBER PROPERTIES OF RATIONAL NUMBER

REPRESENTATION OF IRRATIONAL NUMBER ON NUMBER LINE:

e.g. Represent PROPERTIES OF RATIONAL NUMBERon the number line. 

First take PROPERTIES OF RATIONAL NUMBER(OQ) on the number line.

Steps of Construction for PROPERTIES OF RATIONAL NUMBER

PROPERTIES OF RATIONAL NUMBER

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

PROPERTIES OF RATIONAL NUMBER

or PROPERTIES OF RATIONAL NUMBER.

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

Hence, it is clear, that PROPERTIES OF RATIONAL NUMBER

Thus, the point P represents the number PROPERTIES OF RATIONAL NUMBERon the number line l.

EXISTENCE OF SQUARE ROOT OF A POSITIVE REAL NUMBER:

For any positive real number x, we have

PROPERTIES OF RATIONAL NUMBER

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 PROPERTIES OF RATIONAL NUMBER

For example:

Represent PROPERTIES OF RATIONAL NUMBER on the number line.

In order to represent PROPERTIES OF RATIONAL NUMBER on number line, we follow the following steps:

1. Draw a line and mark a point A on it.

2. Mark a point B on the line drawn in step 1 such that AB = 9.3 cm.

3. Mark a point P on AB produced such that BP = 1 unit.

4. Find mid-point of AP. Let the mid-point be O.

PROPERTIES OF RATIONAL NUMBER

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 PROPERTIES OF RATIONAL NUMBER

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