To prove that \( \sqrt{2} \) is irrational, we use a classical proof by contradiction.

### Steps of the proof:

1. **Assume the opposite**: Suppose \( \sqrt{2} \) is a rational number. By definition, a rational number can be expressed as a fraction of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers, \( q \neq 0 \), and the fraction is in its lowest terms (i.e., \( p \) and \( q \) have no common factors other than 1, they are coprime).

   So, let's assume:
   \[
   \sqrt{2} = \frac{p}{q}
   \]
   Squaring both sides, we get:
   \[
   2 = \frac{p^2}{q^2}
   \]
   This implies:
   \[
   p^2 = 2q^2
   \]

2. **Examine the implications**: From \( p^2 = 2q^2 \), we see that \( p^2 \) is an even number because it equals \( 2q^2 \) (which is obviously even). If \( p^2 \) is even, then \( p \) itself must also be even (because the square of an odd number is odd). So, let \( p = 2k \) for some integer \( k \).

3. **Substitute and simplify**: Now, substitute \( p = 2k \) into the equation \( p^2 = 2q^2 \):
   \[
   (2k)^2 = 2q^2
   \]
   This simplifies to:
   \[
   4k^2 = 2q^2
   \]
   Dividing both sides by 2, we get:
   \[
   2k^2 = q^2
   \]
   This implies that \( q^2 \) is also even, which means \( q \) must be even as well.

4. **Contradiction**: If both \( p \) and \( q \) are even, then they have a common factor of 2. But this contradicts our original assumption that \( p \) and \( q \) are coprime (they should not have any common factors other than 1).

### Conclusion:
Since our assumption that \( \sqrt{2} \) is rational leads to a contradiction, it follows that \( \sqrt{2} \) must be irrational.

---

Would you like more details on any step or have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the definition of an irrational number?
2. Can the same method be used to prove that \( \sqrt{3} \) is irrational?
3. Why is it necessary for the numbers \( p \) and \( q \) to be coprime?
4. How does this proof relate to the Fundamental Theorem of Arithmetic?
5. Can the method of contradiction be used for other types of mathematical proofs?

**Tip**: In proofs by contradiction, it's important to assume the exact opposite of what you're trying to prove and carefully follow the logical implications of that assumption until a contradiction arises.

Explore a step-by-step proof by contradiction to demonstrate that √2 is an irrational number. This proof covers key concepts in number theory, including the properties of rational numbers, coprime integers, and the application of the Fundamental Theorem of Arithmetic. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation