What are surds?

Consider the number 2 .
We can not find any number in the form m/n where m and n are integers and n#0, whose square is 2.
That is to say there is no rational number r where rxr is 2.
Hence Ö2 is said to be a surd.

All surds are irrational numbers

Square roots of all numbers between perfect squares like are all surds.

e.g. Ö2,Ö3,Ö5,Ö6,Ö7,Ö8


Surds are irrational that is they have decimal which is non-recurring and unending

E.g.
Ö2 is 1.4142135623730.....
Ö3 is 1.7320508075688…

Ö4 is not a surd. Why?

Because Ö4 is 2 which is a rational number.

3Ö2 is also a surd. Why?


There is no rational number whose cube is 2. Hence cube root of 2 is a surd.

In general nÖa is a surd if there is no rational number r whose nth power is a

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Now try these examples

Which of these are surds?
Ö18 Ö45 Ö36 Ö72
Ö22 Ö24 Ö128

 

Remember these rules

(nÖa )n = a
nÖa nÖb = nÖ(ab)
Öa + Öb is not equal to Ö(a+b)
(Öa)2 is a
Ö(ab) = ÖaÖb
Ö(a/b) = Öa/Öb

Which of these are surds?

Ö15 Ö60
3Ö9 3Ö24
2Ö10 ÷ 4Ö40

Answer : None of these are surds. First one has a value 30, second has a value 6 and third has a value 1/4

Rationalisation

Multiplying a surd by another to get a rational number is called Rationalisation

1/(Ö2+Ö3)

(Ö2-Ö3)
= -------------------
( (Ö2+Ö3) ) (Ö2-Ö3) ))

(Ö2-Ö3)
------
2-3

= -(Ö2-Ö3)=Ö3-Ö2

 

Question:Is Ö(4/9) rational ?

Yes. Because 2/3 X 2/3 is 4/9
Root of 4/9 is 2/3 which is a rational number

 

 


1>Rationalise the following fractions


1/(2+Ö3)
4/(Ö2-Ö5+1)

Multiply and divide byÖ2+Ö5+1) which will again give a surd 4x(Ö2+Ö5+1)/(-4-2Ö5). Next Multiply and divide by (-4+2Ö5).


2>Insert an irrational numbers between 4.5 and 4.7

4.52 = 20.25 and 4.7 2 is 22.09 . Two integers between them are 21 and 22 . Hence two irrational numbers between these are Ö21 and Ö22. There are other answers also.(like Ö21.5 etc.)




Recurring decimals

Any fraction which has prime factors of denominator as 2 and 5 only will have terminating decimal value

e.g. ½, 3/8 , 7/5, 1/80 etc

All other fractions will have recurring decimals.

e.g. 1/3 = 0.3333333
4/7 = 0.5714285714285714

Example : Convert the following recurring decimal to vulgar fraction
0.090909090909


Now bring the first recurring pattern before decimal digit
If m is 0.09090909 then 100 m is 9.09090909..
100m – m is 9
i.e. 99 m is 9
So m is 9/99 = 1/11

i.e. 0.09090909.. = 1/11