Math Problem Statement
Prove square root of 2 is irrational
Solution
To prove that is irrational, we can use a proof by contradiction.
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Assumption: Assume that is rational. This means it can be expressed as a fraction of two integers and in simplest form, such that: where and have no common factors (other than 1) and .
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Square Both Sides: Squaring both sides gives: Multiplying both sides by yields:
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Analyze : This equation shows that is even because it is equal to (which is clearly even).
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Conclusion About : Since is even, must also be even (the square of an odd number is odd). Thus, we can express as: for some integer .
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Substitute : Substituting back into the equation :
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Conclusion About : This shows that is also even, which means is even.
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Contradiction: If both and are even, they share at least the common factor of 2. This contradicts our initial assumption that and have no common factors.
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Conclusion: Therefore, our assumption that is rational must be false. Thus, is irrational.
If you have any questions or need further clarification, feel free to ask! Here are some related questions you might consider:
- What are some other examples of irrational numbers?
- How can we prove the irrationality of other square roots, like ?
- What is the significance of rational vs. irrational numbers in mathematics?
- Can you provide a visual representation of this proof?
- 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