# What's infinity minus infinity?

I was wondering earlier today, what's infinity minus infinity? Would it be positive infinity, zero, negative infinity, or is it impossible to solve? While reasoning for about 10 minutes, here's the useful things I came up with on the subject that could be used to reason the answer with. 1: There are different size infinities, some being infinitely bigger than others. For example: there are and infinite amount more decimals than whole numbers, because for every whole number there is an infinite amount of decimals (1.1, 1.01, 1.001, etc.) 2: Infinity is an "idea", not a number, so you may not be able to do this problem at all. 3: I think that we could all agree that infinity + infinity would equal infinity. So if adding infinity together don't change it's state, why would subtracting it change it's state? (i.e. 0+0=0, and 0-0=0). I personally don't think this is solvable, but I would like to hear what you think :P

### 13 Answers

- Anonymous7 years agoFavorite Answer
At first, you may think that infinity subtracted from infinity is equal to zero. After all, any number subtracted by itself is equal to zero, however infinity is not a real (rational) number. I am going to prove what infinity minus infinity really equals, and I think you will be surprised by the answer.

First, I am going to define this axiom (assumption) that infinity subtracted from infinity is equal to zero:

∞ - ∞ = 0

Next, I am going to add the number one to both sides of the equation.

∞ - ∞ + 1 = 0 + 1

Since ∞ + 1 = ∞ and 0 + 1 = 1, then we are going to simplify both parts of the equation:

∞ - ∞ = 1

Woops! It is impossible for infinity subtracted from infinity to be equal to one and zero. Using this type of math, we can get infinity minus infinity to equal any real number. Therefore, infinity subtracted from infinity is undefined.

- JoanLv 44 years ago
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.

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- Anonymous7 years ago
Infinity is a trickier concept than you imagine.

First, infinity - infinity doesn't make sense since infinity is not a number. It's like asking what's apple - orange? Subtraction is defined for finite number, not for infinity or fruity.

Second, your statement about decimals and whole numbers isn't quite right.

If you look at ALL decimals, including infinite, nonrepeating ones, that is correct. But if you only look at finite ones, then it is incorrect.

The constructs you are describing here are known as transfinite numbers, and there is a body of mathematics surrounding them. The smallest transfinite number has the name aleph-null. That would be a Hebrew letter with a 0 subscript, but I don't know how to type that. This is called "countably infinite," because it applies to the cardinality of the counting numbers. "Cardinality" means the number of elements of a set. The next largest (probably--it's really an unresolved question) is aleph 1, the cardinality of the real numbers (or the real numbers in the unit interval). These are uncountable. There are more real numbers than whole numbers.

Note that there are as many positive integers as there are integers, because we can put them into a one to one correspondence:

0 to 1

1 to 2

-1 to 3

2 to 4

-2 to 5

n > 0 to 2n

n < 0 to 2n + 1

etc.

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- BenLv 57 years ago
∞-∞=∞

we can't be sure which number the idea of infinity is and infinity can't be claimed as the same as another infinity therefore it is still infinity or indeterminate depending on the mathematician you ask.

1 Comment: if there are infinite numbers, you can't say there are infinite more decimals because if non decimal numbers never end in progressive counting, how could there ever be more of some thing than that? there are infinite decimals and non decimal numbers but it is impossible to say that one of them has more than the other

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- 7 years ago
It's considered an indeterminate form. Just like infinity divided by infinity is indeterminate.

Addition and multiplication keep it infinity but their opposites are impossible to solve. Because exactly what you said--the different sizes. There's no way to know which one is "bigger".

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- Anonymous7 years ago
Then it's to infinity... and beyond!

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- 6 years ago
The answer is INDETERMINATE, which is what Ashley said. All the other answers are wrong. indeterminate means it is equal to everything. This is NOT the same thing as UNDEFINED, which means there is no solution.

0/0, ∞/∞, 0 × ∞, 0^0, 1∞ and ∞^0 are also indeterminate.

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- 7 years ago
Since infinity is not defined as a positive number, my answer would be infinity. but you could argue it is zero since we are taking of infinity as a number?

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- 6 years ago
I referred to upanishad (Indian Vedic philosophy) and found that "if you remove infinity from infinity, it remains still the same. ie no change.

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Order of operations is from left to right u ate doing it wrong buddy