Math Problem Statement

Prove square root of 2 is irrational

Solution

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

  1. Assumption: Assume that 2\sqrt{2} is rational. This means it can be expressed as a fraction of two integers aa and bb in simplest form, such that: 2=ab\sqrt{2} = \frac{a}{b} where aa and bb have no common factors (other than 1) and b0b \neq 0.

  2. Square Both Sides: Squaring both sides gives: 2=a2b22 = \frac{a^2}{b^2} Multiplying both sides by b2b^2 yields: a2=2b2a^2 = 2b^2

  3. Analyze a2a^2: This equation shows that a2a^2 is even because it is equal to 2b22b^2 (which is clearly even).

  4. Conclusion About aa: Since a2a^2 is even, aa must also be even (the square of an odd number is odd). Thus, we can express aa as: a=2ka = 2k for some integer kk.

  5. Substitute aa: Substituting back into the equation a2=2b2a^2 = 2b^2: (2k)2=2b24k2=2b22k2=b2(2k)^2 = 2b^2 \quad \Rightarrow \quad 4k^2 = 2b^2 \quad \Rightarrow \quad 2k^2 = b^2

  6. Conclusion About bb: This shows that b2b^2 is also even, which means bb is even.

  7. Contradiction: If both aa and bb are even, they share at least the common factor of 2. This contradicts our initial assumption that aa and bb have no common factors.

  8. Conclusion: Therefore, our assumption that 2\sqrt{2} is rational must be false. Thus, 2\sqrt{2} is irrational.

If you have any questions or need further clarification, feel free to ask! Here are some related questions you might consider:

  1. What are some other examples of irrational numbers?
  2. How can we prove the irrationality of other square roots, like 3\sqrt{3}?
  3. What is the significance of rational vs. irrational numbers in mathematics?
  4. Can you provide a visual representation of this proof?
  5. What are some applications of irrational numbers in real life?

Tip: Familiarize yourself with properties of even and odd numbers, as they are often used in proofs involving integers.

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Math Problem Analysis

Mathematical Concepts

Number Theory
Irrational Numbers
Proof by Contradiction

Formulas

\(\sqrt{2} = \frac{a}{b}\)
\(a^2 = 2b^2\)
\(a = 2k\)
\(b^2 = 2k^2\)

Theorems

Proof by Contradiction
Properties of Even and Odd Numbers

Suitable Grade Level

Grades 9-12