Конспект: попередження: "Нескінченність" не число

Warning: "Infinity" is Not a Number

When you were a kid you probably played the following game with one of your friends:

You: I bet I have the most Halloween candy!

Your Friend: No way, I have like a hundred candies

You: Well, I have, like, a million candies

Your Friend: Oh, I miscounted I have a quadrillion candies

You: Liar! And it doesn't matter because I have INFINITY candies

Your Friend: Well you are stupid and I have infinity plus one candies

...

And that went on for a while until the inevitable short fight, tears, and of course- candy.

But, to the point, you CANNOT HAVE INFINITY CANDIES. Infinity is NOT A NUMBER! Infinity simply is an idea- the idea that a number or quantity increases forever. (If you've ever heard the Latin phrase ad infinitum it literally means "to go on forever.") Having infinity plus one is the same as having infinity candies.

So when you see a limit like this one:

$\lim_{x \rightarrow 0^+} \frac{1}{x}$

And determine that it equals $+\infty$ this means that the limit just gets super big- endlessly big even. Whenever you pick an x value and get a big y value from the function, you could always just pick a smaller x value and get an even bigger y value. In this way, the limit doesn't really equal infinity- we just say it does to represent the idea that the limit gets infinitely big.

Think about it! You could never pick a function whose limit is exactly one more than the limit in the previous example- $\infty + 1$ is basically meaningless.

If you only remember one thing: INFINITY IS NOT A NUMBER IT IS AN IDEA!

So the next time your friend says they have infinity candies you can confidently prove that they do not, and win the mathematical Halloween candy hoarding contest (Personal Advice: don't actually do this)

As a brief addition to the previous discussion, as not to sow confusion, I have to include a small caveat. Some infinities are bigger than others. This can become very relevant in more difficult limits if you end up in a situation where you have infinity over infinity or the like. The idea here is that some infinities get really big markedly faster than other infinities and thus behave differently. Nonetheless, the idea that infinity is not a number still applies

(Video) Why is Infinity Not a Real Number?

by Jim Fowler

Here's a good video by Jim Fowler explaining why you can't think of infinity as a number.

(Video) Infinity is Bigger Than You Think

by Numberphile

Here's a fun video by Numberphile talking about infinity.

The Hotel Paradoxe

Let's take that we have a Hotel that has infinite number of rooms which are all occupied.

To let those people get a room, we will move each of our habitant to the next room, so who is in room 1 will move to 2, and so on, and since we have infinite number of rooms the process will never end, and so we created more rooms than we already have

The fact that infinity isn't a number but it's a concept an Idea every number you can think of there someone bigger, so basical each number that exist is a part of the infinity.

Infinity plus a finite number is infinity and how come an infinite number can be bigger than another infinite number therefore ∞+n=∞ and ∞ +∞ =∞

**So we didn't create new rooms we already had them, but we created new space for new quests **

OK PEACE