by Jason Marshall
Pop quiz time: Do you think it’s possible to add up an infinite number of anything and still get a finite answer? In other words, if I hand you an infinite number of measuring sticks, is it possible that laying them all out end-to-end could produce anything other than an infinitely long line?
You might be tempted to jump to the conclusion that the answer is “No, an infinite number of anything must add up to an infinite number!” And maybe you’re right. But instead of guessing, today we’re going to take a look and see what math has to say. And to do that we’re going to use something called a mathematical series.
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What’s the Difference Between a Mathematical Series and a Sequence
As you may recall, we talked about something called a mathematical sequence in earlier articles. To refresh your memory, a sequence in math is simply a list of numbers that are arranged in a particular order. The first five positive integers 1, 2, 3, 4, 5 and the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, … which are related to the golden ratio are two examples of mathematical sequences. The first is what’s called a finite sequence since it’s made up of a non-infinite number of things (more technically known as elements). Since the Fibonacci numbers go on and on forever, they’re what’s called an infinite sequence.
But we’re not talking about sequences today, so why am I bringing them up? Well, as it turns out, there’s something in math called a series that’s very closely related to the idea of a sequence. And it’s precisely this idea of a series that we need to understand in order to answer our question about the length of an infinite number of measuring sticks. So, let’s get to it.
What Is a Mathematical Series?
What exactly is a series? Well, a series in math is simply the sum of the various numbers, or elements of a sequence. For example, to make a series from the sequence of the first five positive integers 1, 2, 3, 4, 5, just add them up. So, 1 + 2 + 3 + 4 + 5 = 15 is a series. Pretty simple right? Since this series is made from a finite sequence—and therefore contains a finite number of terms—it’s what’s called a finite series.
A mathematical series is simply the sum of the elements of a mathematical sequence.
On the other hand, since the Fibonacci sequence is an infinitely long sequence of numbers, the series formed by adding together all the Fibonacci numbers is what’s called an infinite series. And, as a couple of seconds of thought will prove to you, since each number in the Fibonacci sequence keeps getting larger and larger, the sum of all the numbers in the series must be infinite. So the sum of an infinitely long sequence of numbers—an infinite series—sometimes has an infinite value. But, going back to the question that we started with, is it always infinite? Or can it sometimes be finite?
How Can You Picture the Meaning of a Series?
To answer this question, allow me to paint a picture in your mind. Imagine drawing a perfect square on a perfect piece of paper. Let’s say that the area of this square is 1. Now imagine drawing a line down the middle of this square to make two rectangles. What’s the area of each? Well, since they’re each half the size of the square, they must each have an area of 1/2. Next, divide one of these two new rectangles in half. Now do this again, and again, and again, and so on…literally forever!
What do you end up with? Well, if you think about it, you’ll see that the rectangles you’ve drawn inside your original square have areas that follow the sequence: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, and so on. Do you see the pattern? Each element of this sequence is just the fraction 1/2 raised to a higher and higher integer power. For example, 1/2 = (1/2)^1, 1/4 = (1/2)^2, 1/8 = (1/2)^3, and so on. When you add all of the elements of this sequence together, you get the series:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + …
Though this series has an infinite number of terms, the fact that the original square we divided into smaller and smaller rectangles is not infinitely large means that the series must actually be finite. In fact, it must be equal to 1. Why? Because that’s the area of the original square. Which means that some infinite series can indeed have finite values!
Can an Infinite Number of Things Have a Finite Sum?
So, what about the question that kicked this whole thing off? Can an infinite number of measuring sticks stacked end-to-end have a finite length? The answer depends on whether we’re talking about real world measuring sticks or idealized mathematical ones. In the real world, there are limits on how small things can be. For example, it’d be pretty tough to make a measuring stick that’s smaller than a single atom. But in the mathematical world, you can imagine making measuring sticks that get smaller and smaller and infinitely smaller. Then, in exactly the same way that the total area of the infinite number of ever shrinking rectangles is finite, the total length of the stacked measuring sticks could be finite too.
What do you think? Pretty fascinating stuff, right? Indeed, mathematical series are very cool. And, as we’ll see next week, they’re very useful in the real world too!
Number of the Week
Before we finish up, it’s time for this week’s featured number selected from the various numbers of the day posted to the Math Dude’s Facebook page over the past week. This week’s number is actually a collection of three related numbers: 2.7, 1.9, and 1.5. How are they related? They’re three answers to the question: How long would it take you to travel around the Earth? Your options are:
So which would you choose? These numbers really give you some perspective on just how big the Earth is, and just how fast the space shuttle flies!
If you liked those numbers, be sure to become a fan of the Math Dude on Facebook where you’ll learn a new number that’s just as interesting as these every single weekday. If that’s not enough to convince you to check it out, I’ve also started to post daily math puzzles that have proven to be quite popular. Head over to the Math Dude’s Facebook page and see for yourself. Of course, if you’re on Twitter, please follow me there too and get updates about the podcast, the numbers of the day, the daily math puzzles, and all the latest math and science news. Finally, if you have any math questions, please feel free to ask me via Facebook, Twitter, or by email at email@example.com.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!