Interesting Questions for High School Students

Prime Numbers

How many prime numbers do we have?  Infinite!  Can you prove there are infinite prime numbers?

Mathematical Induction

When we have to prove a mathematics theorem, Mathematical Induction is one of the useful proofs.

It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

Let S(n) denote the statement involving the integer variable n. The Principle of Mathematical Induction states:

If S(1) is true and, if for each integer k ≥ 1, S(k+1) is true whenever S(k) is true; then S(n) is true for all n ≥ 1.

However, the induction itself is also a 'theorem'.  Do you know how to prove the Mathematical Induction?

Lines and Dots under Euclidean geometries

We know that line is made up of dots.  The question is: How many dots are there in a segment of line?  The answer is: Infinity.  If it is infinity, then does it mean that all lines are made up of the "same number" of dots - infinite numbers of dots?  Can you prove it?