Math Problem Statement
prove that is irrational number. Use that result to prove 2 is an irrational number
Solution
Proof that is an Irrational Number:
We will use proof by contradiction.
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Assume is rational. This means we can express as a fraction of two integers , where and are coprime integers (i.e., and ).
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Square both sides to eliminate the square root.
Multiply both sides by to clear the denominator:
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Analyze the equation.
This equation implies that is divisible by 7, meaning that must also be divisible by 7 (since 7 is a prime number and a property of primes is that if a prime divides a square, it must divide the original number).
So, let for some integer .
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Substitute back into the equation.
Simplifying:
Divide both sides by 7:
This implies that is divisible by 7, so must also be divisible by 7.
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Contradiction:
Now we have shown that both and are divisible by 7, which contradicts our assumption that . Therefore, the assumption that is rational must be false.
Thus, is irrational.
Proof that is Irrational:
Next, we use the fact that is irrational to prove that is also irrational.
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Assume is rational. This means , where is a rational number.
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Isolate :
Since is rational and is rational, is rational. Thus, is rational.
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Divide both sides by 3:
Since is rational and 3 is rational, is rational. This implies that is rational.
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Contradiction:
This contradicts the fact that is irrational (as proven earlier). Therefore, the assumption that is rational must be false.
Hence, is irrational.
Do you need further clarification or more details on any part of the proof?
Here are 5 related questions to further explore this topic:
- How can we extend this proof technique to other square roots of non-perfect squares?
- Why do we require the assumption that in the proof by contradiction?
- Can you generalize this result for numbers of the form where is irrational?
- How can you prove the irrationality of other numbers like or ?
- Is it possible for the sum of a rational and an irrational number to be rational?
Tip: Always begin a proof by contradiction with a clear assumption that is opposite to what you want to prove, and then work logically until you reach a contradiction.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Irrational Numbers
Proof by Contradiction
Formulas
√7 is irrational
2 + 3√7 is irrational
Theorems
Properties of Prime Numbers
Rational Number Definition
Contradiction Method
Suitable Grade Level
Grades 9-12