Proving √2 is Irrational: A Brief Explanation

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In summary, the conversation discusses a proof of the irrationality of √2. It defines rational, even, and odd numbers, and then presents a proof by contradiction. The proof shows that if √2 were rational, it would lead to a contradiction, thus proving that it is not rational. The summary also confirms that the proof presented in the conversation is correct.
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Jamin2112
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Homework Statement



As the title says.

Homework Equations



Rational number: a/b for some integers a, b
Even number: 2k for some integer k
Odd number: 2j+1 for some integer j

The Attempt at a Solution



Assume √2 is a rational number. Then it can be expressed a/b for some integers a and b. Reduced to it’s lowest form, a and b cannot both be even numbers.

√2 = a/b ----> √2b=a ----> 2b2=a2 ---->a2 is even ----> a is even ----> a=2k for some integer k ----> 2b2=(2k)2 ----> b2=2k2 ----> b is even: a contradiction because both a and b cannot be even.

Hence √2 is not a rational number.
 
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Your proof is correct!
 

Related to Proving √2 is Irrational: A Brief Explanation

1. What does it mean for a number to be irrational?

A number is considered irrational if it cannot be expressed as a ratio of two integers (whole numbers). This means that it cannot be written as a fraction in the form of a/b, where a and b are integers.

2. How is irrationality proven for a specific number, such as √2?

To prove that a number is irrational, we must show that it cannot be written as a ratio of two integers. This can be done through a proof by contradiction, where we assume that the number can be written as a fraction and then show that this leads to a contradiction.

3. What is the proof by contradiction used to prove the irrationality of √2?

The proof by contradiction for √2 involves assuming that √2 can be written as a fraction a/b, where a and b are integers. We then manipulate this assumption to show that it leads to a contradiction, proving that √2 cannot be expressed as a ratio of two integers.

4. Why is proving the irrationality of √2 important in mathematics?

Proving the irrationality of √2 is important because it helps us understand the properties of irrational numbers and the concept of irrationality. It also has important applications in fields such as geometry, as √2 is the length of the diagonal of a square with side length 1.

5. Can the proof of √2's irrationality be applied to other numbers?

Yes, the proof by contradiction used to prove the irrationality of √2 can be applied to other numbers as well. This method can be used to show that other square roots, such as √3 and √5, are also irrational. It can also be extended to prove the irrationality of other types of numbers, such as cube roots or logarithms.

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