CBSE / NCERT Maths Solutions Class 10 – Real Numbers – Irrational Theorems
In this article, we will discuss Irrational Numbers from Class 10 Maths Chapter 1 Real Numbers.
Irrational numbers which when expressed in decimal forms are expressible as non-terminating and nonrepeating decimals are known as irrational numbers.
0.1010010001…is a non-terminating and non-repeating decimal.
0.12112111211112 is irrational.
0.232322223 is irrational.
22/7 is irrational.
Some Results on Irrationals
Let P be a prime number and a be a positive integer. If p divides then show that p divides a.
Proof: – Let p be a prime number and a be a positive integer such that p divides
We all are aware that every positive integer can be expressed as the product of primes.
Let a = p1 , p2 ……. pn where , ….. , are primes p1, p2 ……. pn
a2 = (p1, p2 ……. pn) (p1, p2 ……. pn)
a2 = (p12, p22 ……. pn2)
P divides a2
P is one of p1, p2 ……. pn
P divides a
Learn how to use Theorem 1 in solving questions, click CBSE Maths Class 10th for demos.
Prove that √2 is irrational.
Proof: – Let √2 be rational and let its simplest form be a/b
Then a an b are integers having no common factors other than 1 and b is not equal 0
2 divides a
2 is prime and divides a²
Let a=2c for some integer c
Putting a=2c we get
2 divide b²
2 divide b
2 is prime and divides b²
2 divide b
2 is a common factor of a and b
But this contradicts the fact that a and b have no common factor other than 1
The contradiction arises by assuming that √2 is rational
Hence √2 is irrational
Grab Maths Class 10 tutorials on our website.
Prove that √3 is irrational
If possible √3 be rational and let its simplest form be a/b
Then a and b are integers having no common factors other than 1 and b is not equal 0
3=a²/b² on squaring both sides
3 divides a
3 is prime and 3 divides a²
3 divides a
Let a=3c for some integer c
Putting a =3c we get
3 divide b²
3 divide b
3 is prime and 3 divide b². Thus, 3 is a common factor of a and b
But this contradicts the fact that a and b have no common factors other than 1.
The contradiction arises by assuming √3 is rational.
Hence √3 is irrational.
In the same manner, we can prove that each of the numbers √5, √7 is irrational.
Special packages are available for Class 10th Maths, Science, and English.
For Sample papers for Class 10th, subject wise test series, CBSE Guide, CBSE exam pattern, offline animated video classes for all subjects from Class 1 to 12. Professional courses and Skill development courses are also available, kindly visit www.takshilalearning.com for details.