root 5 is irrational proove it please in simpler and lenghthy way that i understand

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It is only with the help of  assumption method.you know that √5 is an irrational no. but we have to prove this so,let us consider √5 as  a rational number.(then at ending point prove that it is wrong)-rational no. can be written in the form of a/b , where a and b are coprime (no other common factor than 1), and b not = 0.√5=a/b  (Square both side)5b^2 =a^2a=5c; so, 5b^2=25a^2b^2=5a^2here  we can see that  5 divides a^2 and b^2 both it means 5 divides both a and b , but if √5 is rational then a and b are co-prime this show us that our assumption is wrong.Hence √5 is irrational!

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check ur ncert

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Understanding Rational and Irrational Numbers
Rational numbers are numbers that can be expressed as a fraction where both the numerator and denominator are integers, and the denominator is not zero. Examples include 1/2, 3, and -4. Conversely, irrational numbers cannot be expressed as such a fraction. Examples include π and √2.
Proof that √5 is Irrational
To prove that √5 is irrational, we will use a contradiction approach.
Step 1: Assume the Opposite
- Assume √5 is rational.
- This means that it can be expressed as a fraction a/b, where a and b are integers with no common factors (i.e., they are in simplest form).
Step 2: Set Up the Equation
- If √5 = a/b, then squaring both sides gives us 5 = a²/b².
- Rearranging yields a² = 5b².
Step 3: Analyze the Equation
- Since a² = 5b², this implies that a² is divisible by 5.
- If a² is divisible by 5, then a must also be divisible by 5 (because the square of a number is only divisible by a prime if the number itself is).
Step 4: Substitute Back
- Let a = 5k for some integer k.
- Substituting back, we have (5k)² = 5b², which simplifies to 25k² = 5b².
- Dividing both sides by 5 gives 5k² = b².
Step 5: Conclusion of Contradiction
- This shows that b² is also divisible by 5, meaning b must be divisible by 5 as well.
- Since both a and b are divisible by 5, they share a common factor, contradicting our assumption that a/b is in simplest form.
Final Conclusion
- Therefore, our original assumption that √5 is rational must be false.
- This means that √5 is irrational.

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