Hello! In the next 2 minutes I am going to introduce you to the concept of mathematical induction. Why is it important? Because mathematical induction provides an easy and short demonstration for equations, divisibility problems and other difficult problems that require a general proof.
Used in different ways by mathematicians like Platon, Euclid or al-Karaji, this method of proof was not recognized as a valid one until 1575 when Francesco Maurolico first attempts to illustrate the concepts that make the mathematical induction an reliable and strong method of proof.
This method of proof is done in 2 steps. The first step is also known as “the base case”, and it attempts to demonstrate the initial statement. When we want to prove that an equation is true for all the natural numbers, we first have to find out if the initial case is valid, so we check if the first number satisfies the equation.
The second step is called “the inductive step”, and it makes the transition from one number to the other. In this step, we have to prove that if the statement is true for one number, it is also true for it’s successor. We assume that k satisfies the equation and demonstrate that k+1 also satisfies the equation, usually using P(k).
If we combine the two steps, we have that the first initial case satisfies the equation and that for each number that satisfies the equation, the next one does it too. If P(1) is true, then P(2) is true, then P(3) is true, P(4) is true and so on. Inductively, any natural number satisfies the equation.
Mathematical Induction looks very similar to the “domino effect”. If we tear down the first domino, it will tear down the second one and so on, until all the dominos are down. With the first element and the inductive step, we can prove that the statement is true for all the cases.
In conclusion, it is obvious that Mathematical Induction has a lot of application in algebra and geometry, but it is also a very important logical concept, essential for any mathematician.