As has been discussed, the concept of infinity can be difficult to grasp, let alone comprehend! However, infinity has several extremely practical and useful applications in much of the mathematics we do from day to day; from the simplest computations to some of the most complicated formulas. Some examples are listed below:

Mathematicians have been able to prove and justify many relationships that would have been impossible prior to our current understanding of the potential and actual infinity. Here, I will focus on only a few: namely Riemann Sums, Calculus related relationships, and infinite geometric series.

Riemann Sums

Named after the 19th century German mathematician Bernhard Riemann, the Riemann Sum is a way of approximating the area under a curve using a summation. Riemann sums have incredible relation to integrals in Calculus, which we will discuss later. Explore the following applets to understand what a Riemann Sum is and why they are helpful:

When it comes to finding area, there are objects that most of us are relatively familiar with where we can easily find the area. Some of these objects may include rectangles, triangles, trapezoids, etc. However, some object areas may be significantly more difficult to compute. Consider the following graph:

How would you find the area of the blue shaded region? Do you know a 'formula' for that? One strategy would be to use a Riemann Sum. Consider constructing a series of rectangles beneath the curve in question, as shown below:

If we calculated the area under this curve by adding up the areas of these eight rectangles, we might get close to the actual area, but our calculated area would be a little too small. However, in order to more closely calculate the area, we could split the interval into even more rectangles: because the more the rectangles, the closer our area approximation will be.

Now, what if the number of rectangles approached infinity. How close would our approximation be? It would be extremely close! The difference would almost be inconsequential! This is an example of the 'potential infinity' we discussed in the History section of this page: infinity is never actually 'achieved,' but the number of rectangles seems to increase without end. This method of summation can be used with almost any function we can graph on a coordinate plane.

With that in mind, it would be extremely difficult and impractical to compute the area of infinitely many rectangles! Rather, using Calculus, we can use the idea of Riemann Sums to compute the area under a curve.

Calculus Relationships

One of the most important relationships in Calculus is the integral. In short, If you were to consider any curve graphed on a coordinate plane, we could use the idea of a Riemann Sum with infinitely many rectangles of infinitesimally small width. This is basically the idea of an integral, except an integral calculates this area exactly! We won't go into extensive detail on integration here, but for further information on this topic consider watching the following video by Khan Academy:

Basically all of Calculus, and especially the integral, rely extensively on the idea of infinity. Before we were able to conceptualize infinity, we could approximate these areas with a relative amount of accuracy, but not nearly as accurately as we can now!

Let's explore this just a little further. Consider as an example the formula for the area of a circle: $$A=\pi r^2$$ How did mathematicians derive this formula? Where does it come from? There is a detailed Calculus derivation that we won't go into here, but there is another geometric derivation for this formula. Engage in the following applet to discover where this area formula comes from:

The formula for the area of a circle comes from the idea of dissecting a circle into infinitely many sections. The applet helps the user to discover the formula for the area of a circle by using the formula for a triangle.

The relevance of infinity in Calculus doesn't end with the integral, it has importance in almost every other Calculus application you can think of! Calculus wouldn't exist without infinity, and there are multiple fields that require the use of applied Calculus! So basically, without infinity, much of the complex mathematics we use in the real world would be null and void.

Infinite Geometric Series

A geometric sequence is a sequence of numbers where each successive number is found by multiplying the previous number in the sequence by a common ratio. Consider the following sequence as an example: $$2, 4, 8, 16, 32, 64, 128$$ Each term in this sequence is found by multiplying the previous term by 2. An infinite geometric series is the sum of an infinite geometric sequence. The sum of an infinite geometric series can be equated using the following formula: $$S=\frac{a_1}{1-r}$$ where \(a_1\) is the first term in the series and \(r\) is the common ratio. However, an infinite geometric series will only have a sum if \(r\) is less than 1.

Consider the following infinite geometric series as an example: $$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, ...$$ Using the formula above, we get the following: \(\frac{\frac{1}{2}}{1-\frac{1}{2}}=1\). So the sum of this infinite series is equal to 1! Take a look at the actual series, add up as many of the numbers as you feel necessary. Are you convinced? The following applet gives a visual, geometric representation to help convince you that it's true (this applet will also allow you to change the common ratio to see many different sums):

In this infinite geometric series, the sum won't just approach 1, or get really really close to 1, the sum will be exactly 1! This kind of precision is only possible with the ability to sum infinitely many terms.

The following applet provides another interesting and helpful geometric visualization of the sum of an infinite geometric series (In the applet, notice that \(\frac{1}{8}\) of the square is shaded in red and blue respectively, and notice how much of the final square is shaded:

The unique nature of these infinite sums is attributed to the fact that they are indeed infinite! The fact that we can get a finite sum from adding up infinitely many numbers is simply mind-blowing! The mathematical relationships associated with the infinite seem to 'never end,' and they will continue to perplex mathematicians for years and years to come.