FORMULABANK

When proving something, you often need to prove it with algebra or provide a counterexample using a calculation.

A counterexample is when you can conclude through a calculation and/or algebra that the hypothesis does not hold.

Example of a counterexample:

Prove that when you double either the radius or the height of a cylinder, the volume remains the same.

We assume the height is 10 and the radius is 2.

$\pi \cdot 2^2\cdot 10\approx 125.6637$

Doubling the height: $\pi \cdot 2^2\cdot 20\approx 251.3274$

Doubling the radius: $\pi \cdot 4^2\cdot 10\approx 502.6548$

The two results are not the same as the original calculation (125.6637). Therefore, we have proven with a counterexample that the hypothesis is incorrect.