To Infinity and Beyond!

1. Introduction

Let’s start off with something that puzzled mathematicians for over 300 years: infinity.

There are two definitions that we can give to infinity. One is “in the limit,” and the other “infinite set.” I will explain what each means and how things can go wrong at infinity. Lastly, I will draw pictures to prove a mathematical fact involving infinity. You will be surprised by just how obvious it is.

2. Infinity = “in the limit”

Roughly between 1650 and 1850, mathematicians wondered, if we take steps that are infinitely many, what happens at the end (“in the limit”)?

The pictures below show a circle whose diameter is equal to 1.

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By definition, the circumference of this circle is equal to \pi, which is about 3.14. If we consider two triangles that fit the circle just right, one from the inside and the other from the outside, we see that their perimeters “sandwich” the value of \pi.

Now, a triangle is a polygon that has 3 sides. So if we consider a polygon with 5 sides, do the two perimeters still sandwich the value of \pi? What if we consider a polygon with 10 sides? 20, 40, or even 100 sides?

We expect that whatever happens after finitely many steps happens after infinitely many steps. And indeed, the two perimeters of a polygon will continue to sandwich \pi as we increase the number of sides.

However, it turns out that this is not always the case.

a. What can possibly go wrong?

A famous example involves the graph of 1 over x. We take the graph and rotate it about the horizontal axis to get a three-dimensional object. It looks like a horn, doesn’t it?

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Using calculus, we can show that when the length L goes to infinity, the surface area of the horn goes to infinity but the volume stays finite. In other words, if you want to paint the outside of the horn as the horn grows, you are going to need an infinite amount of paint. On the other hand, if you pour the paint inside, you just need a finite amount!

We are baffled. Our experience with and intuition of “finite-step” objects do not necessarily translate to “infinite-step” objects.

3. Infinity = “infinite set”

It took mathematicians about 200 years to understand what was going on. Just when the dust seemed to have settled down, in the late 19th century, a man named Cantor brought up the notion of an infinite set.

Now, a set is a collection of objects, and an infinite set just means there are infinitely many objects. Examples of infinite set include:

  • Natural numbers (0, 1, 2, 3, \cdots)
  • Integers (\cdots, -3, -2, -1, 0, 1, 2, 3, \cdots)
  • Rational numbers (all integers and fractions)
  • Real numbers (all numbers)

a. What can possibly go wrong?

Pop quiz. Which of the following infinite sets contains the most numbers? Is it,

  1. Natural numbers (0, 1, 2, 3, \cdots)
  2. Integers (\cdots, -3, -2, -1, 0, 1, 2, 3, \cdots)
  3. Even numbers (\cdots, -4, -2, 0, 2, 4, \cdots)
  4. Rational numbers (all integers and fractions)

We expect that the rational numbers have the most numbers. After all, the integers seem to have twice as many numbers as the natural numbers. The even numbers are missing half of the integers, so they can’t be it. And rational numbers? They include both integers and fractions.

It turns out all four have equally many numbers. How can that be?

b. Brief explanation

With infinite sets, size is an issue. A part of an infinite set can have equally many objects as the whole. Note that this does not happen with a finite set. Take a deck, which has 52 cards, and consider just the diamonds. For sure, there are fewer cards that are diamonds than there are the cards themselves, right? Again, what happens in the finite doesn’t necessarily translate to the infinite.

An infinite set can have equally many objects as another infinite set. Does this mean all infinite sets have equally many objects? It turns out that this isn’t the case either. Some infinite sets can have more objects than others. They are, in some sense, more infinite! For example, there are far more real numbers than there are rational numbers. And the reason? Irrational (and transcendental) numbers.

4. A Proof Without Words

Finally, I want to show you that the number of real numbers between 0 and 1 equals the number of all real numbers. In other words, if you consider all numbers that are larger than 0 and smaller than 1, there are as many of them as if you had considered all numbers.

The classic approach in a real analysis class is to map the two open intervals using the hyperbolic tangent and show that this map is 1-1 and onto. Bleh.

Here is a much simpler proof. See if you understand what each picture is trying to say:

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5. The Beyond

In summary, infinity can mean two things: “in the limit” and “infinite set.” Extending a result from the finite case to the infinite case can work sometimes. But sometimes, it does not. By carefully studying what happens at infinity, we can understand math better.


This blog post is the transcript of a speech that I gave at my local Toastmasters club last November. It is still my favorite among the five that I have given as of now.

To entertain the audience and stay on the point, I deliberately did not pursue why things go wrong at infinity. If you are interested, I recommend Eli Maor’s “To Infinity and Beyond.” I learned that amazing proof-by-picture from this book.

Had I had more than 7 minutes for a Toastmasters speech, I would like to have considered two more examples.

First is the Koch snowflake. This is a 2D analog of the 3D horn (which is named Gabriel’s horn). We take an equilateral triangle and replace the middle third from each side with a smaller equilateral triangle. Repeat this process many times. The perimeter of the snowflake will go to infinity but the area will stay finite (bounded, to be precise).

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So if you want to build a fence for your dog, do not go for the shape of a Koch snowflake!

The other example is Hilbert’s hotel. This hotel has infinitely many rooms, and infinitely many guests have checked in already. When a new guest arrives to check in, it seems like there are no more rooms for this guest. But we actually can make a room, by moving the guest in room \,n\, to room \,(n + 1)\, for each n = 1,\,2,\,3,\,\cdots. Then, the new guest checks into room 1, which is now empty.

For a great explanation and animation, please see TED’s video: The Infinite Hotel Paradox.


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