Q1. Which method is primarily used to prove a number is irrational?
✔ We assume the number is rational and derive a contradiction to prove it is irrational.
Q2. What condition must integers p and q satisfy in the proof?
✔ p and q must have no common factors other than one to ensure the fraction is in simplest form.
Q3. After squaring root 5 equals p/q, what equation do we get?
✔ Squaring both sides of root 5 equals p/q results in 5 equals p squared over q squared.
Show transcript
[Slide 1] Understanding Irrational Numbers
Welcome students. Today we learn how to prove numbers like root 5 are irrational. We assume they are rational and find a contradiction.
[Slide 2] Proof by Contradiction Rule
We start by assuming the opposite. Let the number equal p divided by q where p and q are coprime integers.
[Slide 3] Proving Root Five is Irrational
For root 5, squaring gives 5 equals p squared over q squared. This implies 5 divides p squared and thus p itself.
[Slide 4] Watch Out for Errors
Many students forget p and q must be coprime. You must show they share a common factor to prove contradiction.
[Slide 5] Key Takeaways Summary
Remember the steps clearly for your board exams. Always conclude that the assumption was wrong.