Any number subtracted from itself is equal to zero. Therefore, you might think that infinity minus infinity is also equal to zero; however, this is not true. Let’s find out the answer together. 

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First let’s assume that infinity minus infinity is equal to zero. 

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∞ - ∞ =  0 

 

Adding one to both sides of the equation: 

 

∞ - ∞ + 1 =  0 + 1

 

Simplified we get the following equality: 

 

∞ - ∞ = 1 

 

This would mean that ∞ - ∞ is equal to both one and zero, which simply can’t be true. Substituting one with any other number results in ∞ - ∞ being equal to any real number. For this reason, ∞ - ∞ is undefined. Taking a different approach, we’ll prove that it is undefined. 

 

Let’s assume the previous equation to be true again: 

∞ - ∞ =  0 

 

We know ∞ + ∞ =  ∞ to be true, so let’s substitute it in the equation. 

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∞ + ∞ - ∞ = 0

 

Because we already assumed ∞ - ∞ = 0, we can substitute it in the equation: 

 

∞ + 0 =  0 

 

Which is then simplified to: 

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∞  =  0 

 

This simply cannot be true, hence:

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∞ - ∞ = undefined

 

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References

Phil for Humanity. “What Does Infinity Minus Infinity Equal?” Phil for Humanity, Phil for Humanity, 1 Jan. 2010, https://www.philforhumanity.com/Infinity_Minus_Infinity.html.

“What Is the Result of ∞ - ∞?” GeeksforGeeks, 28 June 2021, https://www.geeksforgeeks.org/what-is-the-result-of-%E2%88%9E-%E2%88%9E/. 

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