An Intuitive Understanding : Mathematical Induction

Jesse Racicot
Mar 13 · 6 min read

The following article will describe mathematical induction with a straightforward analogy and gloss over the technical details. As strongly as I do believe in the necessity of rigor when approaching mathematics, it seems to be too much of an investment for the reader who is not unaccustomed to mathematics. With that said, I would like to enlighten readers and spread knowledge of mathematics, not provide them with another bed-time story.

As a side note, I would like to address why the language of mathematics seems “incomprehensible”. It really is a shame that people shy away from mathematics in large part because of this. A common perception is that mathematicians conjure up a complicated language so they can torture children at a young age and the students that do get through it may grow up and play an elitist game with cool symbols, big words and little meaning. This is an especially hard claim to dispute because a large portion of mathematics seems to lack “applications” in the real world, whatever “applicable” means. The least I could say to defend the seemingly obscure notation in mathematics is that it inhibits the “intellectual sleight of hand” that we see in many other debates. That is, when discussions get a little “wordy”, it can be very hard to spot the holes in an argument. Mathematics doesn’t seem to have problems declaring a clear winner in a “debate”, and mathematical notation might be the reason why.

Now, most of mathematics is modeled after an intuitive understanding of something in the physical world. From there, it takes years of work to develop a concise and abstract way of presenting it. By the time a mathematical theory reaches this point, it can become difficult to decipher what this mathematical object was intended to represent and the encompassing idea seems to have faded away. That being said, I would encourage a beginner to take a less rigorous approach to learning mathematics, such as relating the topic to something easily understood and hard to forget. From there, they can take steps towards dissecting the definition and drawing parallels with their intuition.

To motivate the idea behind mathematical induction, I will ask a simple question and work through an answer that most people would agree with. From there, I will reveal the principle of mathematical induction and highlight the similarities. By then, you’ll either have a firm, intuitive grasp on induction or you will be fast asleep. At any rate, there is undeniable utility in this article going forward.

Can you climb this ladder ?

Being that the ladder seems unending, I shouldn’t ask whether you can reach the end, but rather, “Can you eventually reach any step on this ladder?”

It seems like a bit of an odd question to ask, without giving a few conditions. Obviously, I would need the reader to assume that they had all the time in the world and actually wanted to climb the ladder. The brave readers who decided to try overcoming their fear of heights would also have to pack a change of underpants.

All jokes aside, I will allow the reader to assume the following two conditions :

  1. You know how to get onto the first step of the ladder.
  2. If you find yourself at any step on the ladder, you know how to get onto the next step of the ladder.

Now, equipped with those two assumptions, do you think you would eventually get to any step on the ladder ? Let’s try a scenario.

Suppose I asked you to get to the 3rd step.

Well, you can use the first assumption to start your journey. You know how to get onto the first step of the ladder. In reality, this is quite easy for most people. Take any one of your feet and lift them onto the first step.

Great, you have found your way onto the first step of the ladder. Now, if you aren’t sure what to do next you can refer back to the assumptions. You have used your first assumption and it got us this far but it seems to be of no use at this point. Let’s have another look at the second assumption :

If you find yourself at any step on the ladder, you know how to get onto the next step of the ladder.

Well, the first step is in some sense, any step on the ladder. You claimed to know how to get onto the next step (I think the “how” might be something similar to how you got onto the first step). So, go ahead and use this fact and you will find yourself on the second step of the ladder.

You’re still not quite there. The second step is nice but I asked if you could get to the third step (besides, those pickles on the top shelf are still out of reach). Fortunately, it’s starting to get pretty obvious how you should maneuver up this ladder. You just need to use that crucial second assumption. The second step is just an arbitrary (any) step on the ladder and that second assumption tells you that you can get to the third.

Low and behold, you made it to the third step and without much trouble. Hopefully it’s becoming clear what the path to ladder-climbing success is. If I were to ask you to get to the 50th step, you would first have to get onto the ladder and once settled on that first step, you would climb another 49 steps to reach the 50th. That is, you would use the first assumption to climb onto that first step and then use that second assumption 49 times to continue climbing steps.

A small word of caution. It isn’t enough to have only one of these assumptions. If you said you could get onto the first step but didn’t know how to move from one step to the next, then you wouldn’t get past the first step. Moreover, suppose you could freely move from one step to the next (you are equipped with assumption 2) but couldn’t get onto that first step (did not have assumption 1). In that case, you would have a hard time reaching any step on the ladder because you couldn’t get onto it in the first place. (We don’t consider the ground floor “any” step on the ladder, sorry!)

The Principle of Mathematical Induction

Mathematical induction is commonly taught as a proof technique in undergraduate math courses. There are occasions where one would like to prove a statement that holds for all positive integers. While there are many methods to prove statements of that variety, there are times where mathematical induction seems to be the appropriate route.

It can be stated as follows :

Suppose the following conditions hold for any statement involving a positive integer, n:

  1. The statement holds for n=1.
  2. Whenever the statement holds for some n ≥ 1, it also holds for n + 1.

then the statement holds for any positive integer.

Recall our first assumption :

You know how to get onto the first step of the ladder.

This is analogous to, “The statement holds for n=1”.

Second assumption :

If you find yourself at any step on the ladder, you know how to get onto the next step of the ladder.

This corresponds with, “Whenever the statement holds for some n ≥ 1, it also holds for n + 1.”

Consider the statement, “I can get to the n-th step on this infinite ladder”. That is certainly a statement about a positive integer, n. Moreover, you know how to get onto the first step and you know how to move from one step to the next once you’re on the ladder. That is, you could say :

  1. I can get to the n-th step for n=1.
  2. Whenever I am on the n-th step for some n ≥ 1, I can also get to the (n+1)-th step.

According to the principle of mathematical induction, we can say this :

I can get to the n-th step for any positive integer n.

So you can get to any step on the ladder. You knew this already. Mathematics just captured this intuitive idea of climbing the ladder and made it a principle. When confronted with an assertion about all positive integers, it suffices to show the assertion holds for the number one, and that when it holds for some number, it holds for the next number. Showing that second condition isn’t always as obvious as climbing a step on the ladder, but the idea that you could climb as far as you like, once you know the “general” way to climb from step to step, is taken as self-evident truth in mathematics. It is captured in the principle of mathematical induction.

Jesse Racicot

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Pure mathematics enthusiast