90.1.3 Irrational Numbers Part 1.3 Surd

The collection of all rational numbers and irrational numbers together make up the collection of real numbers, which is denoted by R.

there are three types of the irrational number

a) Non Terminating Decimals

b) π

c) Pure Root or Pure Surd

a) Non Terminating Decimals

We have already learn about the non terminating decimals. Non Terminating Decimals has no fixed value so that these are called irrational numbers.

b) π

The value of π

The ratio of the circumference of a circle to its diameter is a constant, denoted by л. Its value is given by Aryabhata I in the following stanza:

चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम् । अयुतविष्कम्भस्यासनो वृत्तपरिणाह ॥

"Add 4 to 100, multiply by 8 and add to 62,000; this is approximately the circumference of a circle whose diameter is 20,000." - ABh. II, 10 

This means a circle of diameter 20,000 units has its circumference approximately equal to

(100+ 4) x 8+ 62,000 i.e., 62,832 so that we get

        Circumference.           62,832

π =  _________________ =  ___________ = 3.1416

             Diameter                 20,000

It is remarkable that Aryabhata I is the first Indian mathematician to have given the value of π. Which is correct to four decimal places. Even then, he mentions that this value of x is approximate ("asanna").

c) Root or Surd

Definition of a Surd

A root of a positive real number is called a surd if we cannot remove the root symbol after simplification.

All surds are irrational numbers, but the converse is not true. We know that both $\pi$ and $\mathrm{<br>}$e are irrational numbers; but they are not surds.

 

Pyar Root or Pure Surd is an irrational number.

For √1 

√1 is not a Pure Root or Pure Surd because it's root value is one. So it is a rational number.

√1 = 1

√2 = 1.4142.....

√3 = 1.7321.....

√4 = 2

√5 = 2.2361.....

√6 = 2.4494.....

√7 = 2.6457.....

√8 = 2.8284.....

√9 = 3

√10 = 3.1622.....

√11 = 3.3166......

√12 = 3.4641.....

√13 = 3.6055.....

√14 = 3.7416.....

√15 = 3.8729.....

√16 = 4

√17 = 4.1231.....

√18 = 4.2426......

√19 = 4.3588.....

√20 = 4.4721.....

√21 = 4.5826.....

√22 = 4.6904.....

√23 = 4.7958.....

√24 = 4.8990.....

√25 = 5

We find that √1, √4, √9, √16 and √25 are not Pure Root or Pure Surd so they are rational number and others are Pure Root or Pure Surd.

Pure Root or Pure Surd

The numbers return in the form of root is called Pure Surd or pure root. We can say that root or surd of prime numbers are Pure Surd or pure root.

Like that √2, √3, √5, √7 and √11 etc 

Rational Numbers

But √1, √4, √9, √16 and √25 are not Surd because these surds have their Rational Value so this are rational numbers.

Different Types of Surds

Let’s understand these different types of surds.

Types Of Surds.

1. Simple surd

2. Pure surd

3. Similar surds

4. Mixed surds

5. Compound surds

6. Binomial surds

7. Trinomial surds

8. Conjugate surds

Simple Surd: 

When there is only a number present in the root symbol, then it is known as a simple surd. For example √2 or √5

Pure Surd: Surds that are irrational are called pure surds. For example √3

Similar Surd: When surds have the same common factors, they are known as similar surds.

√3, 7√3, 10√3, –2√3

Two or more surds are said to be similar or like surds if they can be so reduced as to have the same surd-factor.

√5, 7√125, √20, –5√45

Mixed Surds: When numbers can be expressed as a product of rational and irrational numbers, it is known as a mixed surd.

5√2, 7√3

Compound Surds: 

The addition or subtraction of two or more surds is known as a complex surd.

√2 + √5, √7 – √11 + √3, √2 – √3, √5 – √7

Binomial Surd: 

When two surds give rise to one single surd, the resultant surd is known as binomial surds.

3√2 + √3, √2 + √5, √2 – √3, √5 – √7

Trinomial Surds

A compound surd which contains exactly three surds is called a trinomial surd.

√7 – √11 + √3

Conjugate Surds

Two binomial surds which are differ only in signs (+/–) between them are called conjugate surds.

For example, √7 + √3 and √7 – √3

★★★★★

Order of Surds

The order of a surd is the index of the root / redical .

√2 = order is 2

√3 = order is 2

√5 = order is 2

√17 = order is 2

Same as

$\text{exception 25:}$

$\sqrt[3]{3}$ =Order is 3

$\sqrt[4]{3}$ = Order is 4

$\sqrt[5]{3}$ =Order is 5

We can add subtract multiplied or divide only the same kind of the orders and like terms numbers.$\text{exception 25:}$

Exercise 1

a) Reduce into the lowest form of the following Surds :

Or

Write into standard form of surds:

(1) √8

(2) √9

(3) √12

(4) √18

(5) √24

(6) √27

(7) √48

(8) √50

(9) √9

(10) 2√9

(11) 2√27

(12) 3√32

(13) √64

(14) √128

(15) √36

(16) √25

(17) 2√125

(18) 5√50

(19) 6√32 

(20) 3√144 

(21) 2√49

Sol. (1) 

$\mathrm{= }\sqrt[\mathrm{<br>}]{8}$

$\mathrm{= }\sqrt[2]{\mathrm{2\; \times \; 2\times 2}}$

= 2

Sol. (10) 

$\mathrm{=\; 2}\sqrt[\mathrm{<br>}]{9}$

$\mathrm{=\; 2}\sqrt[\mathrm{<br>}]{\mathrm{3\; \times \; 3\; \times \; 3}}$

= 2 × 3

= 6

Addition Subtraction Multiplication And Division Of Same Order

Addition 

We add or subtract only numbers of the same order and the same surd. Numbers with unlike orders and unlike surd are never added or subtracted and sign of Addition or subtraction is placed between them.

हम केवल सामान ऑर्डर या समान करणी की संख्याओं को ही आपस में जोड़ने या घटे घटाते । असामान ऑर्डर या असमान करणी वाली संख्याएं कभी भी जोड़ा या घटाया नहीं जाता है। और उनके बीच जोड़ या घटाव का चिह्न लगाया जाता है।

Exercise 2

a) Add the following Surds 

(1) √2 and √2

(2) √2 and 2√2

(3) √2 and 3√2

(4) 2√3 and √3

(5) 2√2 and 2√2

(6) 3√2 and 2√2

(7) 3√2 and 7√2

(8) √3 and √3

(9) √3 and 2√3

(10) √2 and 3√2

(11) 2√3 and √3

(12) 2√5 and 5√5

(13) 3√3 and 2√3

(14) 3√7 and 7√7

(15) √5 and √5

Sol. (1)  Add √2 and √2

= (√2 ) + (√2)

= 1√2  + 1√2

= 2√2 

Sol. (2)  Add √2 and 2√2

= (√2 ) + (2√2)

= 1√2  + 2√2

= 3√2 

Sol. (4)  Add 2√3 and √3

= (2√3 ) + (√3)

= 2√3  + 1√3

= 3√3 

Sol. (12)  Add 2√5 and 5√5

= (2√5 ) + (5√5)

= 2√5  + 5√5

= 7√5 

$\mathrm{<br>}$

b) Add the following Surds 

(1) √2, √2 and √2

(2)  2√2, √2 and 2√2

(3) √2,  2√2 and 3√2

(4) 2√3,   2√3 and √3

(5) 2√2,  3√2 and 2√2

(6) 3√2 , 5√2 and 2√2

(7) 3√5 , 2√5 and 7√5

(8) √3 , 2√3 and √3

(9) √3 3√3 and 2√3

(10) √2,  2√2 and 3√2

Sol. (1)  Add √2, √2 and √2

= (√2 ) + (√2) + (√2)

= 1√2  + 1√2 + 1√2

= 3√2 

Sol. (2)  Add 2√2, √2 and 2√2

= (2√2 ) + (√2) + (2√2)

= 2√2  + 1√2 + 2√2

= 5√2 

Sol. (7)  3√5 , 2√5 and 7√5

= (3√5 ) + (2√5) + (7√5)

= 3√5 + 2√5 + 7√5

= 12√5 

c) Add the following Surds 

(1) √2,  2√2, √2 and √2

(2)  2√2,   5√2, √2 and 2√2

(3) √2,  7√2, 2√2 and 3√2

(4) 2√3,   2√3,  2√3 and √3

(5) 2√2,   3√2,  4√2 and 5√2

(6) 3√2 ,   2√2, 5√2 and 2√2

(7) 3√5 , 2√5   2√5, and 7√5

(8) √3 ,   3√3, 2√3 and √3

(9) √3,   2√3,  3√3 and 2√3

(10)   2√2, √2,  2√2 and 3√2

Sol. (1)  Add  √2,  2√2, √2 and √2

= (√2 ) + (2√2) + (√2) + (√2)

= 1√2  + 2√2 + 1√2 + 1√2

= 5√2 

Sol. (4)  Add 2√3,   3√3,  4√3 and √3

= (2√3) + (3√3) + (4√3) + (√3)

= 2√3 + 3√3 + 4√3 + √3

= 10√3 

d) Add the following Surds 

(1) √2 and √3

(2) √3 and 2√2

(3) √3 and 3√5

(4) 2√3 and √3

(5) 2√2 and 2√4

(6) 3√2 , 5√3 and 2√2

(7) 3√5 , 2√3 and 7√5

(8) √3 , 2√3 and √7

(9) √3 3√2 and 2√3

(10) √2, 2√3 and 3√2

(11) √2,  2√3, √2 and √3

(12)  2√2,   5√5, √2 and 2√2

(13) √4,  7√2, 2√3 and 3√2

(14) 2√9,   2√3,  2√5 and √3

(15) 2√2,   3√3,  4√4 and 5√7

(16) 3√2 ,   2√3, 5√2 and 2√3

(17) 3√5 , 2√25   2√9, and 7√5

(18) √3 ,  3√2, 2√3 and √3

(19) √3,  2√3,  3√2 and 2√3

(20)   2√2, √4,  2√3 and 3√2

Add

Q.1 Add 2√3 + 4√5  and  3√3 –7√5 

Sol. (2√3 + 4√5) + (3√3 –7√5)       

= 2√3 + 4√5 + 3√3 –7√5    

= 2√3 +3√3 –7√5 + 4√5)

=     5√3 –3√5

Q.2 Add 2√2 + 5√3  and  √2 + 3√3 

Sol. (2√2 + 5√3) + (√2 + 3√3 )       

= 2√2 + 5√3 + √2 + 3√3

= 2√2 + √2 + 5√3 + 3√3 

 = 3√2 + 8√3 +

Exercise 4

a) Subtract the following Surds 

(1) √2 from √2

(2) √2 from 2√2

(3) √2 from 3√2

(4) 2√2 from √2

(5) 2√2 from 2√2

(6) 3√2 from 2√2

(7) 3√2 from 7√2

(8) √3 from √3

(9) √3 from 2√3

(10) √2 from 3√2

(11) 2√3 from √3

(12) 2√3 from 2√3

(13) 3√3 from 2√3

(14) 3√3 from 7√3

(15) √5 from √5

b) Subtract the following Surds 

(1) √2 from √3

(2) √2 from 2√5

(3) √3 from 3√2

(4) 2√3 from √2

(5) 2√4 from 2√9

(6) 3√2 from 4√8

(7) 3√3 from 7√27

(8) √3 from 5√3

(9) √5 from 2√6

(10) √2 from 3√7

(11) 2√7 from √8

(12) 2√3 from 2√9

(13) 3√3 from 2√4

(14) 3√4 from 7√3

(15) √5 from 5√25

Multiply the following Surds together 

d) Multiply the following Surds 

(4) 2√3 and √3

(5) 2√2 and 2√4

(6) 3√2 , 5√3 and 2√2

(7) 3√5 , 2√3 and 7√5

(8) √3 , 2√3 and √7

(9) √3 3√2 and 2√3

(10) √2, 2√3 and 3√2

(1) √2,  2√3, √2 and √3

(2)  2√2,   5√5, √2 and 2√2

(3) √4,  7√2, 2√3 and 3√2

(4) 2√9,   2√3,  2√5 and √3

(5) 2√2,   3√3,  4√4 and 5√7

(6) 3√2 ,   2√3, 5√2 and 2√3

(7) 3√5 , 2√25   2√9, and 7√5

(8) √3 ,  3√2, 2√3 and √3

(9) √3,  2√3,  3√2 and 2√3

(10)   2√2, √4,  2√3 and 3√2

Divide the followings

Exercise 5

a) Divide the following Surds 

(1) √2 from √2

(2) √2 from 2√2

(3) √2 from 3√2

(4) 2√2 from √2

(5) 2√2 from 2√2

(6) 3√2 from 2√2

(7) 3√2 from 7√2

(8) √3 from √3

(9) √3 from 2√3

(10) √2 from 3√2

(11) 2√3 from √3

(12) 2√3 from 2√3

(13) 3√3 from 2√3

(14) 3√3 from 7√3

(15) √5 from √5

b) Divide the following Surds 

(1) √2 from √3

(2) √2 from 2√5

(3) √3 from 3√2

(4) 2√3 from √2

(5) 2√4 from 2√9

(6) 3√2 from 4√8

(7) 3√3 from 7√27

(8) √3 from 5√3

(9) √5 from 2√6

(10) √2 from 3√7

(11) 2√7 from √8

(12) 2√3 from 2√9

(13) 3√3 from 2√4

(14) 3√4 from 7√3

(15) √5 from 5√25

Q. Multiply 2√3 by 3√5

Sol. 2√3 × 3√5         

= 2 × 3 × √3 ×√5

= 6√15

Learn These Identities

Identity 1

(a + b) (a + b) 

(a + b)² = a² + b² + 2ab 

                  Or

a² + b² + 2ab = (a + b)²

Identity 2

(a – b) (a – b) 

(a – b)² = a² + b^{2 –} 2ab 

                  Or

a² + b^{2 –} 2ab = (a – b)²

Identity 3

(a +b)(a –b)  = a² – b²  

                  Or

a² – b²  = (a +b)(a –b)

निम्न को साधारण गुणा तथा सर्वसमिका के प्रयोग द्वारा सरल करो । 

Simplify the following by using multiplication and Identity method:

1. (6 – 4√2)²

2. (6 + 4√2)²

3. (2 + 2√3)²

4. (2 – 2√3)²

5. (√2 – √3)²

6. (√3 + √5)²

7. (√2 – 2√7)²

8. (√11 + 2√11)²

9. (√5 – 2√2)²

10. (6 + √3)²

 11. ( √7 – √3)²      

निम्न को साधारण गुणा तथा सर्वसमिका के प्रयोग द्वारा सरल करो । 

Simplify the following by using multiplication and Identity method:

1. (6 + 4√2) (6 + 4√2)

2. (6 – 4√2) (6 – 4√2)

3. (6 + 4√2) (6 + 4√2)

4. (6 – 4√2) (6 + 4√2)

5. (6 + 4√2) (6 – 4√2)

6. (2 + 2√3) (2 + 2√3)

7. (2 – 2√3) (2 – 2√3)

8. (√11 – √7)(√11 – √7)

1l9. (√11 + √7)(√11 + √7)

10. (√11 + √7)(√11 – √7)

11. (√11 – √7)(√11 + √7)

12. (5 + √7)(2 + √7)

13. (3√5 + √3) (3√5 - √3)          

14. (3√5 + √2) (√5 - 2√3)      

15. (√11 + √2) (√11 – 2√2)

1. Rationalging / Rationalzing the following:

              1

(1)        __.               

             √2

              1

(2)        __.               

             √3

              1

(3)        __.               

             √5

              1

(4)        __.               

             √7

              1

(5)        __.               

             √11

              2

(6)        __.               

             √2

              3

(7)        __.               

             √3

              2

(8)        __.               

             √6

              7

(9)        __.               

             √7

              5

(10)        __.               

             √5

               7

(11)        __.               

             3√3

               6

(12)        __.               

             3√2

               6

(13)        __.               

             2√3

               5

(14)        __.               

             2√5

               7

(15)        __.               

             2√7

1. Rationalging / Rationalzing the following:

                  1

(1)        _______               

             2 + √2

                 1

(2)        ______               

             3 + √3

                  5

(3)        _________              

                2 + √5

                  1

(4)        ________               

             √2 + √3

                   1

(5)        ________               

             √3 + √5

                   2

(6)        ________              

             √7 – √3

                   5

(7)        ________              

             √5 – √3

                   2

(8)        ________              

             7 + 3√3

                   7

(9)        ________              

             3√3 + 2√2

                   1

(10)        ________              

               7 – 3√2

                 √3 – 1

(11)        ________              

                 √3 + 1

                √3 + 1

(12)        ________              

                √3 – 1

                5 + √6

(13)        ________              

                5 – √6

                5 – √6

(14)        ________              

                5 + √6

                7 + 3√5

(15)        ________              

                7 – 3√5

                √5 + √6

(16)        ________              

                √5 – √6

                √7 – √5

(17)        ________              

                √7 + √5

                √32 + √48

(18)        ___________              

                  √8 + √2

                √48 + √50

(19)        ___________              

                √48 + √18

                 √24 – √5

(20)        ___________              

                √45 – √24

Now find the value of a and b

                7 + 3√5

(16)        ________    = a + b√5           

                7 – 3√5

                3 + 4√5

(17)        ________    = a – b√5           

                4 – 3√5

                5 + 2√3

(18)        ________    = a + b√3           

                7 + 4√3

                5 + 3√3

(19)        ________    = a – b√5           

                7 + 4√3

                5 + √12

(16)        ________    = a + b√3           

                7 + 3√48

Ans.

18. [2].                              19. [(9+4√6)/15]

20.  [(9+4√30)/21]

Simplify the following:

               √3 – 1             √3 + 1

(1)        ________    +     ________    

               √3 + 1              √3 – 1

               √3 – 1             √3 + 1

(2)        ________    –     ________    

               √3 + 1              √3 – 1

               5 – √6             5 + √6

(3)        ________    +     ________    

               5 + √6              5 – √6

               5 – √6             5 + √6

(4)        ________    –     ________    

               5 + √6              5 – √6

               5 + √6             5 – √6

(5)        ________    +     ________    

               5 – √6              5 + √6

               7 + 3√5              7 – 3√5

(6)        ________    +        ________    

               7 – 3√5             7 + 3√5

               7 + 3√5              7 – 3√5

(7)        ________     –      ________    

               7 – 3√5             7 + 3√5

               √7 + √5              √7 – √5

(8)        ________      +      ________    

               √7 – √5             √7 + √5

Now find the value of a and b

               √3 – 1             √3 + 1

(9)        ________    +     ________    = a + b √3

               √3 + 1              √3 – 1

               √7 – 1             √7 + 1

(10)      ________    +     ________    = a + b √7

               √7 + 1              √7 – 1

               7 + √5              7 – √5

(11)       ________    +    ________    = a + b √5

               7 – √5             7 + √5

               √2 + √3          √2 – √3

(12)        ________    +   __________    = a + b √3

               3√2 – 2√3      3√2 + 2√3

Simplify the following:

                 1                  2                   1

(1)       _______  +   ________   +  ________

             2 + √3        √5 – √3         2 – √5

                 1                  1                   1

(2)       _______  +   ________   +  ________

             1 + √2        √2 + √3         √3 + √4

                 1                  1                   1

(3)       _______  +   ________   +  ________

           √5 + √3        √3 + √2         √5 – √2

                 1                  2                   1

(4)       _______  +   ________   +  ________

             2 + √3        √5 – √3         2 – √5

              7√3              2√5                3√2

(5)       _______  –   ________   –  ___________

          √10 – √3       √6 – √5        √15 – 3√2

              2√6              6√2                8√3

(6)       _______  +   ________   –   _________

           √2 + √3        √6 + √3          √6 + √2

Simplify the following:

                       1

(1)         _____________             

               2 + √3  + √7

                       1

(2)         _____________             

               1 + √5  + √6

                       1

(3)         _____________             

               1 + √2  – √3

                       4

(4)         _____________             

               2 + √3  + √7

                       4

(5)         _____________             

               1 + √2  + √3

                       3

(6)         _____________             

               5 + √3  –2 √7

                       1

(7)         _____________             

               √3 + √2 – √5

                       1

(8)         ______________             

               √3 + √5 – 2√2

                       1

(9)         _____________             

               √5 + √3 – √8

2. Pure surd

3. Similar surds

4. Mixed surds

5. Compound surds

6. Binomial surds

7. Trinomial surds

8. Conjugate surds

Q. Divide 8√20 by 2√5

Sol. 8√20 ÷ 2√5                            

= 4 × √4 

= 8

Q. Simplify the following expressions:

(√2 + √3) (√5 –√7)  

= √2(√5 –√7) + √3) (√5 –√7)  

= √10  – √14   + √15  – √21

Q.1 Add (3√5 + √3) , (-3√5 + 5√3)

Q.2 Subtract  (√5 + 7√3)  from ( 2√5 - √3)

Q.3 Multiply  3√5 and  5√3

Q.4 Divide  6√10 by  3√2

Q.5 Simplify the following expressions:

(i) (3√5 + √3) (3√5 - √3)          

(ii)  (3√5 + √2) (√5 - 2√3)

 (iii)  ( √7 - √3)2                                   

(iv) (√11 + √2) (√11 - 2√2)

6. Explore

Simplify

$$\sqrt{75}+\sqrt{50}$$

Check whether the terms are ‘like surds’.

★★★★★

TYPES OF SURDS

Following are the different types of surds.

1. Simple surd

2. Pure surd

3. Similar surds

4. Mixed surds

5. Compound surds

6. Binomial surds

7. Trinomial surds

8. Conjugate  surds

Simple Surd

A surd having only one term is called a simple surd.

For example, 

√2, √3

Pure Surd

A surd which is completely irrational is called pure surd.

In other words, a surd which has unity only as rational factor the other factor being irrational is called pure surd

For example,

√3, √5

Similar Surds

Two or more surds are said to be similar or like surds if they have the same surd-factor.

√3, 7√3, 10√3, -2√3

All the above surds are similar or like surds. Because, we have the same surd factor √3.

And also, two or more surds are said to be similar or like surds if they can be so reduced as to have the same surd-factor.

√5, 7√125, √20, -5√45

Each of the above surds can be expressed with the same surd-factor √5.

7√125 = 7√(25 x 5)

= 7(5√5)

= 35√5

√20 = √(4 x 5)

= 2√5

-5√45 = -5√(9 x 5)

= -5(3√5)

= -15√5

The given surds can be expressed as

√5, 35√5, 2√5, -15√5

All the above surds are similar or like surds. Because, we have the same surd factor √5.

Mixed Surds

Surds which are not completely irrational and they can be expressed as a product of a rational number and an irrational number.

For example,

5√2, 7√3

Compound Surds

An expression which contains addition or subtraction of two or more surds is called compound surd.

For example,

√2 + √5, √7 - √11 + √3

Binomial Surd

A compound surd which contains exactly two surds is called a binomial surd.

3√2 + √3

Trinomial Surds

A compound surd which contains exactly three surds is called a trinomial surd.

√7 - √11 + √3

Conjugate Surds

Two binomial surds which are differ only in signs (+/–) between them are called conjugate surds.

For example,

√7 + √3 and √7 - √3

In Addition or subtraction 

√2 = 3

$\sqrt[2]{2}$

2

$\text{exception 25:}$$\text{exception 25:}$$3\sqrt[2]{2}$

5×5×32$3\sqrt[2]{2}$

2

$\sqrt[2]{\mathrm{2 }}$

−−√2$\sqrt[2]{\mathrm{5\times 5}\times 3}$

$\sqrt[2]{\mathrm{5\times 5}\times 3}$

result. 

Now 8–√2$\sqrt[2]{8}$ + 18−−√2$\sqrt[2]{18}$ = $\mathrm{<br>}\sqrt[2]{\mathrm{<br>}}$ + 32–√2$3\sqrt[2]{2}$ = 52–√2$5\sqrt[2]{2}$.

Similarly we will find out subtraction of 75−−√2$\sqrt[2]{75}$, 48−−√2$\sqrt[2]{48}$.

75√2$\sqrt[2]{75}$= 25×3−−−−−√2$\sqrt[2]{\mathrm{5\times 5}\times 3}$= 52×3−−−−−√2$\sqrt[2]{5^2\times 3}$= 53–√2$5\sqrt[2]{3}$

48−−√2$\sqrt[2]{48}$ = 16×3−−−−−√2$\sqrt[2]{16\times 3}$ = 42×3−−−−−√2$\sqrt[2]{4^2\times 3}$= 43–√2$4\sqrt[2]{3}$

So 75−−√2$\sqrt[2]{75}$ - 48−−√2$\sqrt[2]{48}$ = 53–√2$5\sqrt[2]{3}$ - 43–√2$4\sqrt[2]{3}$ = 3–√2$\sqrt[2]{3}$.