Make You Think Thursdays: Infinity

written by Emma Bartley

Think of the biggest number you can think of. Now go one number after that. And one hundred after that. Numbers are infinite; they are endless. Ready to get your mind blown? Let’s talk about infinity.

Counting

Within an interval (let’s say 0 to 1), you can have an infinite amount of numbers. And if you choose any two numbers within that infinite set, there are infinite more numbers between them.

Achilles and the Tortoise

“Achilles, the fleet-footed hero of the Trojan War, is engaged in a race with a lowly tortoise, which has been granted a head start. Achilles’ task initially seems easy, but he has a problem. Before he can overtake the tortoise, he must first catch up with it. While Achilles is covering the gap between himself and the tortoise that existed at the start of the race, the tortoise creates a new gap. The new gap is smaller than the first, but it is still a finite distance that Achilles must cover to catch up with the animal. Achilles then races across the new gap. To Achilles’ frustration, while he was scampering across the second gap, the tortoise was establishing a third. The upshot is that Achilles can never overtake the tortoise. No matter how quickly Achilles closes each gap, the slow-but-steady tortoise will always open new, smaller ones and remain just ahead of the Greek hero.”

Source: http://www.slate.com/articles/health_and_science/science/2014/03/zeno_s_paradox_how_to_explain_the_solution_to_achilles_and_the_tortoise.html

.999=1

Let’s start with one divided by 3. The answer is 0.333 with an infinite amount of decimals. Then, if you multiply that number by 3, you will get .999 with an infinite amount of decimals. However, a number multiplied and divided by the same number remains unchanged. So although you end up with .999 after dividing and then multiplying by 3, the answer must also be 1.

To put it mathematically:

1÷3 = .33333……

.33333 x 3 = .999999….

BUT

1÷3×3 = 1

So 1 = .9999….

Still confusing? Let’s think about this way. No matter what number you add to .9999, the number will not be small enough to equal exactly 1. For example, if you try adding .11111 to .99999 with an infinite amount of decimals, the answer will be 1.00000999 and so on. There will always be an infinite amount of numbers that gets closer to 1 when you add it but is slightly bigger than 1. The number that you would add to get to 1 simply does not exist. Thus, .9999 must equal 1.

Deck of Cards

When you shuffle a deck of cards, there’s a good chance that it has never been shuffled in that way before. We think that 52 cards is a pretty small number, but it’s ridiculous how many different configurations you can put 52 cards in. In fact, the number of possibly combinations for a deck of cards is 80658175170943878571660636856403766975289505440883277824000000000000. The way you shuffle your cards is probably the first that it has ever been shuffled in history.

Zeno’s Paradox

You do an infinite amount of things in finite time. Even just walking to your room is covering an infinite amount of distances: first you have to cover half the distance, then half the remaining distance, and half the remaining distance, and so on forever. Fortunately, you can cover those infinitely many distances in finite time. This is known as Zeno’s Paradox.