Throughout this blog, I’ll refer many times to getting my answers through inductive reasoning, and it will probably benefit you to know what that is. Inductive Reasoning is, along with deductive reasoning, one of the two methods of figuring things out with logic. While deduction makes a more airtight case than induction, it is still an extremely useful tool that would be foolish to ignore. It is frequently used in mathematics, and may perhaps be necessary for any problem involving infinite numbers.
A common example of inductive reasoning is used in figuring out whether the sun will rise tomorrow. It rose this morning, and yesterday morning, and every morning of my life, and every recorded instance of mornings in history. Because the pattern fits so many intervals uninterrupted and without a single pattern break, it is said to be a strong inductive argument.
Mathematicians also use inductive reasoning to figure out, for example, what you get when you divide 1 by 3. We all know it’s (roughly) .333… , which is read as “point 3 repeating”. How does one reach this conclusion? We can’t keep dividing 10 by 3, dropping the remainder of 1, adding a 0, and dividing again by 3 over and over again. There isn’t enough time to do the whole problem, and it would be pointless to do it infinitely anyway because we all spot the pattern rather quickly. Because we know that this pattern will repeat every time without fail, we inductively reason that, no matter how many digits we add to the problem, we’ll get nothing but threes.
It’s important to note that this pattern doesn’t have a reason to break or interrupt — what is true about it at the beginning is true at the end, even if we keep it up into infinity. Strangely, some mathematicians would prefer to believe that the pattern does suddenly change when a number becomes infinitely long. Why? We’ll explore that in further blog posts, but the important thing to note here is that inductive reasoning will tell us what a pattern will do for any length of math problem, even an infinitely long one, and they are all strong arguments because there are an infinite number of examples in which each one works.
One final note: While I welcome criticism, please do not claim that I am wrong unless you can explain how I could have been right. If you don’t believe that inductive reasoning is useful for figuring out a problem like 1/3, please tell me how one does figure out such a problem. If you don’t have an alternative, there’s a good chance that there isn’t one.