# What does Infinity Minus Infinity Equal?

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

∞ - ∞ + 1 = 0 + 1 |
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Since

**∞ + 1 = ∞**and

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

∞ - ∞ = 1 |
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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**.

Let's prove this another way. Again, assume this is true:

∞ - ∞ = 0 |
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Since we know that ∞ = ∞ + ∞, then if we substitute this equation into the first infinity in the equation above, we get:

(∞ + ∞) - ∞ = 0 |
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Which is equal to:

∞ + ∞ - ∞ = 0 |
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Since we already assumed ∞ - ∞ = 0, then we can substitute to this:

∞ + 0 = 0 |
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This simplifies to:

∞ = 0 |
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This is obviously incorrect, so we can conclude that:

∞ - ∞ ≠ 0 |
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Specifically, we can conclude that:

∞ - ∞ = undefined |
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Let's try proving that infinity minus infinity is not a number (n). Assume this is true:

∞ - ∞ = n |
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Since we know that ∞ = ∞ + ∞, then if we substitute this equation into the first infinity in the equation above, we get:

(∞ + ∞) - ∞ = n |
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Which is equal to:

∞ + ∞ - ∞ = n |
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Since we already assumed ∞ - ∞ = n, then we can substitute to this:

∞ + n = n |
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Subtract n from both sides of the equation, then this simplifies to:

∞ = 0 |
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This is obviously incorrect, so we can again conclude that:

∞ - ∞ ≠ n |
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Specifically, we can again conclude that:

∞ - ∞ = undefined |
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by Phil B.