When you're just starting out in mathematics, infinity is little more than a neat concept.

Infinity, at that point, is simply the idea that numbers go on forever, and once you accept that, you seem to be fine with the concept of infinity. As you learn more about math, though, you start running into more and more problems relating to infinity, and the concept starts to get weird.

David Hilbert came up with a wonderful example that helps people begin to grasp unusual concepts concerning infinity. His example is known as Hotel Infinity, and is explained here by Martin Gardner, with some clear and amusing illustrations.

Imagine a hotel with an infinite number of rooms (especially easy if, like me, you live in Las Vegas). Also, imagine that an infinite number of people are staying there, so every room is occupied. What happens when 1 person, a UFO pilot in Martin Gardner's version, wants a room? Everybody can be moved to a room number that's 1 higher than their current room, so the first room is now available for the UFO pilot.

Similarly, if 5 couples show up, everyone can be moved to a room number that's 5 higher than their current room, so five rooms are now available for the new couples.

Let's make this more challenging: What happens if an infinite number of people now want a room? You can't simply have everybody move to a room number that's infinitely higher than their current room, so how would you solve this problem?

The answer is surprisingly simple. Have everybody move to a room number that's twice as big as their current room number! Now, the infinite number of previous guests are all staying in even numbered rooms, and the infinite number of new guests can now move into the odd-numbered rooms! Since there are an infinite number of even numbers and odd numbers, this works.

In the late 19th and early 20th centuries, Georg Cantor started talking about different sizes of infinities in a manner similar to this, and even the great mathematical minds of the time scoffed. Eventually, however, mathematicians did come to accept this idea. How exactly can there be different sizes of infinities? You can learn more about this unusual concept in a basic way via Martin Gardner's Ladder of Alephs article. Videos from TED-Ed and Numberphile examine this concept in more detail.

Even though such discoveries about infinity are relatively new, even the ancient Greeks understood the importance of analyzing infinity. Zeno of Elea developed several paradoxes involving infinity which still challenge mathematicians today. TED-Ed's video below explains the Dichotomy paradox:



A video from Numberphile discusses both the Dichotomy paradox and the Achilles and the Tortoise paradox, and how they relate to infinity:



Not that Hilbert's Hotel Infinity thought experiment even makes this clear. It's even a little startling to realize that it can help you reduce this to a simple algebra problem.

In BetterExplained.com's newest post, An Intuitive Introduction to Limits, these odd ideas about infinity help you understand the concept of limits in calculus. The introduction sums up the challenge perfectly: “Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh. Here’s how I learned to enjoy them:” Concrete examples, including a buffering soccer video, make even this odd concept clear.

If you grasped limits from that article, you're probably ready for the concept of an infinite series, explained in detail in this 15-minute video from WhyU. It's amazing how a little knowledge of infinity can quickly take you through such advanced concepts.

If you're confused by the infinite series video, take some time and go back through the earlier concepts of infinity to make sure you understand them. Start by reading the first half of this post, followed by the next quarter of this post, then the next eighth of the post, then the next sixteenth...