Prove that root 5 is irrational number?

Question

Answer

Answer

Answer

We know that the rational numbers are in the form of p/q form where p,q are intezers. hence our assumption is wrong and √5 is an irrational number. '√5 is an “irrational number

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0
⇒√5=p/q

On squaring both the sides we get,
⇒5=p²/q²
⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p
p is a multiple of 5
⇒p=5m
⇒p²=25m² ————-(ii)
From equations (i) and (ii), we get,
5q²=25m²
⇒q²=5m²
⇒q² is a multiple of 5
⇒q is a multiple of 5
Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number

Hence proved

Answer

Answer

Answer

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0
⇒ √5 = p/q

On squaring both the sides we get,
⇒5 = p²/q²
⇒5q² = p² —————–(i)

p²/5 = q²

So 5 divides p
p is a multiple of 5
⇒ p = 5m
⇒ p² = 25m² ————-(ii)
From equations (i) and (ii), we get,
5q² = 25m²
⇒ q² = 5m²
⇒ q² is a multiple of 5
⇒ q is a multiple of 5
Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number.

Hence proved

Answer

Answer

Answer

Assume 5 as a rational no. then we need to find two coprime integers that are a and b where b≠0          root 5=a/b5b²=a²5|a²5|a so 5 divides a² and a.  (1)equationa=5c for some positive Integer ca²=25c²      5b²=25c²   [a²=5b²]b²=5c²5|b²5|b       eq. 2from 1 and 2 we find that a and b have atleast 5 as a common factor. This contradicts that a and b are co-prime root 5 is irrational.

Answer

Refer the text book lesson number 1.

Answer

Introduction:
To prove that root 5 is an irrational number, we need to show that it cannot be expressed as a ratio of two integers. In this response, we will use the proof by contradiction method to show that root 5 is an irrational number.

Proof by Contradiction:
Assume that root 5 is a rational number, which can be expressed as a ratio of two integers, p and q, where q is not equal to zero and p and q have no common factors.

Expressing root 5 as a fraction:
We can express root 5 as a fraction by squaring both sides of the equation root 5 = p/q.

root 5 = p/q
5 = p^2/q^2
p^2 = 5q^2

Parity of p and q:
Since p^2 is a multiple of 5, p must be a multiple of 5. Therefore, p is an odd number because if p were even, then p^2 would be a multiple of 4, but not of 5. Similarly, q must be odd because if q were even, then q^2 would be a multiple of 4, but not of 5.

Expressing p and q as 5m + 1 and 5n + 1:
Since both p and q are odd, we can express them as 5m + 1 and 5n + 1, respectively, where m and n are integers.

Substituting p and q in p^2 = 5q^2:
Substituting p = 5m + 1 and q = 5n + 1 in the equation p^2 = 5q^2, we get:

(5m + 1)^2 = 5(5n + 1)^2
25m^2 + 10m + 1 = 25n^2 + 10n + 1
5m^2 + 2m = 5n^2 + 2n

Deducing that both m and n are even:
The left-hand side of the equation is odd, while the right-hand side is even. This is a contradiction, so our assumption that root 5 is rational must be false. Therefore, root 5 is an irrational number.

Conclusion:
In conclusion, we used the proof by contradiction method to show that root 5 cannot be expressed as a ratio of two integers, and therefore, it is an irrational number.

Answer

Similar Class 10 Doubts

Prove that root 5 is irrational?

Prove that 1/root 2 is a irrational number?

prove that root 6 is an irrational number

Prove that 5 2 root 7 is an irrational no.?

Prove that for any prime positive integer p, root p is an irrational number.?

Top Courses for Class 10View all

Science Class 10

Mathematics (Maths) Class 10

Social Studies (SST) Class 10

English Class 10

Past Year Papers for Class 10

Top Courses for Class 10

Top Courses for Class 10

Related Content

About Class 10

Class 10 Study Material

Class 10 All Subjects

NCERT Textbook Solutions

Other Classes

Important Links

Company