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# What is infinity minus infinity?

Me and my buddies were reasoning in physics class and the question arose: what is infinity minus infinity. Now to me, the answer is quite simple, zero, but i haven't thought it out deeply. Care to differ? No answers promoting individual destructive agendas please.

Relevance

depends on what type of infinity you are talking about.

For example:

If you are talking about the difference in the infinite number in the set of rational numbers and the infinite number in the set of integers then that infinite - infinite = infinite

I guess what I am trying to say is that it depends on the size of the infinite. If the inifinities are both the same size then it is 0. If one is bigger than others than it's a whole other ballgame.

If you are working with limits, it depends on the expression. lim (n->4) 1/(x-4) - 1/(x-4) = 0, for instance, but lim (n->1) 2/(x-1)-1/(x-1) = infinity.

Note that an infinite set has the same number of elements as a part of itself. So therefore the infinity of all positive integers minus the infinity of all positive integers greater than 10 is a 10-element set, but the infinity of all integers minus the infinity of all odd integers is infinite: the set of all even integers.

In Conway's surreal numbers, w - w = 0, where w is omega, the first transfinite number after the natural numbers.

So the answer can be 0, but it can be any other number, or even infinity.

Infinity minus infinity really is irrelavent and unknowable.

Think of it this way:

Infinity times two is still infinity. So, subtract the first infinity from the second infinity and you still have infinity left.

Infinity plus one is still infinity. So, subtract the first infinity from the second infinity and you only have one left..

I can do this an infinite number of different ways an get an infinite number of different answers.

Therefore, the question is moot.

Welll, it depends on your definitions.

If your definition of infinity is absolute (ie a number which is beyond anyone's ability to count), then you are right.

But if infinities are variable, different, or fluid, then you're wrong.

We know that there are an infinity of fractions between the whole numbers 1 and 2, and between 2 and 3.

Is the inifinity of fractions between whole numbers 1 and 3 twice as much as that between 1 and 2?

If your answer = no, then for you, 'infinity' is an absolute, so infinity minus infinity = 0.

If you answer = yes, then for you there are varying intensities of infinity, and so the answer could be infinity, or 0, or negative infinity, depending on the intensities of the infinities involved.

• Anonymous

Depends on how each infinity is defined and determined.

For example, two times infinity is still infinity. But then twice infinity minus infinity would (by definition) equal infinity. Make sense?

What if you take the expression (x^2)/x as x -> infinity.

x^2 is infinity also.

But x^2 / x = x so infinity divided by infinity would still be infinity.

As I said, it all depends on how infinity is defined. Tricky since it's not a definite number.

• Julia
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It could be considered undefined like dividing by zero, because infinity is immeasurable, so you could never subtract an immeasurable amount.

On the other hand, infinity could be considered like a varible x, so if you subtracted the same infinity, also variable x, you would get zero.

Unfortunately, it is all conceptual making it hard to put on paper. It seems impossible to take an an amount which cannot be measured away from an amount that cannot be measured.

Also, to subtract is to withdraw or take away, as a part from a whole. Infinity is not a whole because it is boundless, so you can't really subtract from it.

OVERALL, my answer is that you can't subtract infinity because it goes on forever, therefore you would be subtracting forever, resulting in INFINITY - INFINITY = INFINITY.

On a practical level - and this does occur in physics, engineering,.. - if one expression gets arbitrarily large, and so does another what can be said about their difference?

Put more formally:

If f(x) -> ∞ as x -> ∞ and g(x) -> ∞ as x -> ∞

then what happens to f(x) - g(x) as x -> ∞ ?

The correct answer, as given by several people here, is that it can be anything. Here are some specific examples:

0: when f(x) = x and g(x) = x

88: when f(x) = x+88 and g(x) = x

∞: when f(x) = 2x and g(x) = x

-∞: when f(x) = x and g(x) = 2x

undefined: when f(x) = x+1 if x is an integer, else x and g(x) = x

• Anonymous

Infinity minus infinity is the infinitesimal time between...

the perfection of a rose in full bloom and it's decay.

a bullet touching the skin and penetration.

a shared breath and lips touching for the first kiss.

the last heartbeat and the soul's departure.

seeing your death unfolding before you and then dying.

not knowing and suddenly feeling at one with the world.

an act of love and the beginning of a new life.

living a busy life and looking up at the night sky and realizing you are less than a dust mote in the universe.

Infinity minus infinity is an indeterminant form which means it could be any real number.

• Dr. J.
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