How to prove that a number is irrational?
Understand the Problem
The question is asking for methods or strategies to demonstrate that a particular number cannot be expressed as a fraction of two integers, which is the definition of an irrational number.
Answer
Assume it is rational, then derive a contradiction.
To prove a number is irrational, assume it is rational, then show that this assumption leads to a contradiction.
Answer for screen readers
To prove a number is irrational, assume it is rational, then show that this assumption leads to a contradiction.
More Information
Proof by contradiction is a very powerful technique and is often used because assuming the opposite of what you want to prove and arriving at a contradiction provides a clear and logical conclusion.
Tips
A common mistake is not rigorously demonstrating the contradiction or misunderstanding the definition of rational numbers.
Sources
- Proving Irrationality - number theory - Mathematics Stack Exchange - math.stackexchange.com
- A proof that the square root of 2 is irrational - Homeschool Math - homeschoolmath.net
- irrationality (proofs of) - Platonic Realms - platonicrealms.com
