To prove that \( \sqrt{5} \) is irrational, we will use a **proof by contradiction**. Here's the step-by-step process:

### Assumption
Assume, for the sake of contradiction, that \( \sqrt{5} \) is rational. By definition of a rational number, this means that \( \sqrt{5} \) can be expressed as a fraction of two integers, say:

\[
\sqrt{5} = \frac{p}{q}
\]

where \( p \) and \( q \) are integers with no common factors (i.e., \( p \) and \( q \) are coprime) and \( q \neq 0 \).

### Squaring both sides
If \( \sqrt{5} = \frac{p}{q} \), then squaring both sides gives:

\[
5 = \frac{p^2}{q^2}
\]

Multiplying both sides by \( q^2 \) to eliminate the fraction:

\[
5q^2 = p^2
\]

### Implication for divisibility
This equation, \( 5q^2 = p^2 \), implies that \( p^2 \) is divisible by 5. Since 5 is a prime number, this means that \( p \) itself must be divisible by 5 (because if a prime divides the square of a number, it must divide the number itself). Therefore, we can write:

\[
p = 5k
\]

for some integer \( k \).

### Substituting into the equation
Now, substitute \( p = 5k \) into the equation \( 5q^2 = p^2 \):

\[
5q^2 = (5k)^2
\]
\[
5q^2 = 25k^2
\]
\[
q^2 = 5k^2
\]

### Contradiction
This equation, \( q^2 = 5k^2 \), implies that \( q^2 \) is divisible by 5, which means that \( q \) is also divisible by 5 (for the same reason as before: if a prime divides the square of a number, it must divide the number itself).

Now we have both \( p \) and \( q \) divisible by 5, which contradicts our initial assumption that \( p \) and \( q \) are coprime (i.e., they have no common factors).

### Conclusion
This contradiction shows that our original assumption—that \( \sqrt{5} \) is rational—is false. Therefore, \( \sqrt{5} \) must be **irrational**.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How can you prove that \( \sqrt{2} \) is irrational using a similar method?
2. What are the implications of irrational numbers in number theory?
3. Can all square roots of non-perfect squares be irrational?
4. How do we categorize numbers as rational or irrational?
5. What are some real-world applications of irrational numbers?

**Tip:** Proof by contradiction is a powerful technique in mathematics, often used when direct proof seems difficult.

Discover how to prove that √5 is irrational with a detailed proof by contradiction. This proof involves key concepts from number theory, including the properties of rational and irrational numbers. Suitable for students in Grades 10-12.

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