Math Problem Statement
prove that root 2 is not rational
Solution
To prove that is not a rational number, we use a proof by contradiction. Here's the step-by-step process:
Step 1: Assume the Opposite
Assume that is rational. If is rational, then it can be expressed as a fraction of two integers and in simplest form (i.e., the fraction is fully reduced with no common factors other than 1 between and ):
where and are integers, , and .
Step 2: Square Both Sides
Square both sides of the equation to eliminate the square root:
Step 3: Multiply by
To remove the fraction, multiply both sides by :
This equation implies that is an even number because it is equal to , which is clearly even (since it's two times an integer).
Step 4: Analyze the Parity of
If is even, then itself must also be even (since the square of an odd number is odd). Therefore, we can write as for some integer .
Step 5: Substitute into the Equation
Substitute into the equation :
Step 6: Simplify the Equation
Divide both sides of the equation by 2:
This equation implies that is also even, so must also be even.
Step 7: Reach a Contradiction
Now we have found that both and are even. This means that and have at least 2 as a common factor. But this contradicts our original assumption that is in the simplest form (i.e., ).
Conclusion
The contradiction arises from our initial assumption that is rational. Therefore, we must conclude that is not a rational number.
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Related Questions:
- Can you prove that is not rational?
- What is the difference between rational and irrational numbers?
- How do we prove that is irrational?
- Can the square root of any non-perfect square be a rational number?
- How does the proof for being irrational differ from proving is rational?
Tip:
When proving by contradiction, always ensure that your assumption contradicts a known fact or leads to an impossible situation. This method is powerful for proving irrationality and other mathematical properties.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Proof by Contradiction
Irrational Numbers
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12