infinity - how can x-2 = x+2?

Question

If a hotel has an infinite number of rooms, but two are removed, does that make it less than infinite? What if two are added? I think it's still infinite but I don't get it. Suppose infinity is x. Then prove that x+2 =x-2. It's not possible. What is wrong with mathemeticians.

But algebraically it doesn't make sense!!!

x-2=x+2

It follows that

x-x=2+2.

Hence,

0=4.

How can this be?

Answer 1

Yes you are correct.

Infinity -2 and infinity + 2 are both still considered infinity.

In fact, infinity is so great that it is equal to infinity + or - any real number.

Infinity is a concept, not a number. It is absurd, and many people say it is "the greatest number" just to be able to think about it. I think you should ask whoever asked you this question how a hotel could have an infinite number of rooms, since, once again, infinity is not a number.

To answer your other question: I still haven't figured out what's wrong with mathematicians, but I'm working on it.

Answer 2

David Hilbert came up with the hotel room analogy about 100 years ago, and it's still one of the best.

Suppose a hotel with infinitely many rooms (numbered 1,2,3, and continuing infinitely) is full, and then a new guest arrives. Can the manager make room for the extra guest? Sure, put the new guest in room 1, move the occupants of room 1 to room 2, move the occupants of room 2 to room 3, and so on up to infinity. In this way, the manager has added one to infinity and he still has infinity.

Now suppose infinitely many new guests arrive at the already-full hotel. Can the manager find space for them? Sure. For each occupant already in the hotel, move them to the room which is 2 times their current room number. Then the infinitely-many occupants of the hotel now occupy only the even numbered rooms, so the manager can put the infinitely-many new guests in the odd numbered rooms. So the manager has multiplied infinity by 2 and he still has infinity.

Now here's the punchline: the hotel manager was already earning an infinite amount of money from his original guests (as long as each guest paid more than $0), so going through all that trouble of adding infinitely more guests was a waste of his time. He will still profit the same infinite amount of money as he would have profited with half as many guests.

And that's Hilbert's Hotel in a nutshell.

Answer 3

If you know anything about mathematics, you know that infinity is not a value. You can't add anything to it or subtract anything from it. I learned that when I was 11 or 12. Infinite is unbounded. Infinity is that which is boundless. To attempt to add anything to something that is boundless and therefore increasing its size indicates a lack of understanding. The problem therefore,m doesn't seem to be with the matheaticians.

Here's something to think about....
Two hotels with infinite rooms... all occupied.

One burns down.. but all the people escape.

So they go to the other hotel.

The management of that hotel decide to put all the existing in rooms numbered 2n-1 where n is the room they occupied before the move. The odd integers are infinite, so there's room.

Now that leaves every even numbed room vacant. So the put the refugees into even rooms 2n where n is the number of the room they occupied in the burned down hotel. The even integers are also infinite.

Infinity is a concept. It doesn't have a finite value. Got it?

Answer 4

The concept of an unending growth of anything (generally of a variable) is mathematically expressed by saying that it has become "infinite" or "infinitely large" symbolically denoted by "∞".

In some sense, one may think of infinity as if it is an ocean. If you take out two buckets (or any finite number of buckets) of water from the ocean the remainder will still be an ocean. We will not call it "ocean minus two buckets of ocean".

Similarly, if you add two buckets (or any finite number of buckets) of water to the ocean it will still be an ocean. We will not call it "ocean plus two buckets of ocean".

ADDENDUM:

Under Additional Details, x - 2 = x + 2, you have rightly written
"It follows that x - x = 2 + 2. Hence, 0 = 4". Actually you
missed to write 0*x = 4 or x = 4/0 approaching ∞.

Remember whenever you continue dividing any finite fixed
number by a decreasing number the result will grow with no end e.g.

10/5 = 2, 10/4 = 2.5, 10/3 ≈ 3.3, 10/2 = 5, 10/1.5 ≈ 6.67.

This fact we exhibit by saying that: (any finite fixed #)/x approaches to ∞, where x is a decreasing number.

Answer 5

There is an algebra that governs operations on infinities, but it's not the same as the algebra for finite numbers.

"What is wrong with mathemeticians."

What is wrong with people who can't spell "mathematician"?

Answer 6

Things get complicated with infinite. Infinite is not a determined quantity, thus adding and substracting from it might look like paradoxes but they are not.

infinite + 2 is not equal to infinite - 2

when you say x+2 you indicate an operation, that can be performed, with inifinite the operation can not be performed, only indicated or reduced (kinda tricky).

Answer 7

The thing is, you can't treat infinity like a number because it's not; it's a concept. Subtracting 2 from infinity is treating it like a number.

Infinity falls outside of the realm of real numbers.

lim (x + 2) = infinity, and
x -> infinity

lim (x - 2) = infinity
x -> infinity

That doesn't tell us x + 2 = x - 2. It only tells us their limits give the same result.

Answer 8

Basically, as x approaches infinity the (-2) and (+2) terms become irrelevant. Thus, in the limiting case x=x, which is what you were trying to prove.

Answer 9

Infinity is not a number and you certainly can't add or subtract anything from it. Simple algebra doesn't allow simple addintion, subtraction etc rules to apply in case of infinty.

Answer 10

because algebraically infinity isn't considered a real number, the same way there is no square root of a negative number therefore not all equations apply to it