How to prove an irrational number?
Understand the Problem
The question is asking for methods to demonstrate that a particular number is irrational. This includes understanding the properties of irrational numbers and possibly providing a proof, such as the classic proof that √2 is irrational.
Answer
Assume the number is rational, derive a contradiction, conclude it's irrational.
To prove a number is irrational, assume it can be written as a ratio of two integers, derive a contradiction, and conclude that the number cannot be rational.
Answer for screen readers
To prove a number is irrational, assume it can be written as a ratio of two integers, derive a contradiction, and conclude that the number cannot be rational.
More Information
One of the most famous proofs of irrationality is the proof that the square root of 2 cannot be expressed as a fraction.
Tips
A common mistake is not simplifying the fraction to its simplest form (ensuring p and q are coprime) at the start of the proof.
Sources
- Proving Irrationality - number theory - Mathematics Stack Exchange - math.stackexchange.com
- Irrational Numbers - Definition, List, Properties, Examples, Symbol - byjus.com
- A proof that the square root of 2 is irrational - Homeschool Math - homeschoolmath.net
