## Leonhard Euler

Infinity Minus Infinity

Seden Nalbant

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.

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

∞ - ∞ = 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.

∞ + ∞ - ∞ = 0

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

∞ + 0 = 0

Which is then simplified to:

∞ = 0

This simply cannot be true, hence:

∞ - ∞ = undefined

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/.