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What is the result of ∞ – ∞?

  • Difficulty Level : Hard
  • Last Updated : 28 Jun, 2021
Geek Week

It is known that a number subtracted from itself will result in the value 0, but there is the confusion that subtracting infinity from infinity is zero or not. But it’s not so. In because infinity is not a Real Number.

Assumptions:

  • Firstly, assume that infinity subtracted from infinity is zero i.e., ∞ – ∞ = 0.
  • Now add the number one to both sides of the equation as ∞ – ∞ + 1 = 0 + 1.
  • As ∞ + 1 = ∞ and 0 + 1 = 1, then to simplify both parts of the equation as ∞ – ∞ = 1.

It is impossible for infinity subtracted from infinity to be equal to one and zero. Using this type of math, it would be easier to get infinity minus infinity to equal any real number. Therefore, infinity subtracted from infinity is undefined.

Now subtract ∞ from ∞ to get an exact pie by using our famous mathematician (Riemann’s Paradox) concept.

  • 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + … + ∞.
  • Separating the positive and negative terms from this series:
    • 1 + 1/3 + 1/5 + 1/7 + ……
    • -1/2 – 1/4 – 1/6 – 1/8 – …….
  • Now, if one adds only positive terms, it will get ∞ and if one adds negative terms, it will get -∞.
  • Riemann’s rearrangement theorem says that if one has got a convergent series whose positive terms add up to ∞, and whose negative terms add up to -∞, then it can rearrange the series into a series that has any sum one wants. So, perform this operation for the same for π(pi) with this particular series.
  • The value of π(pi) is positive(3.14359). So, the first term of our new series will be 1 and have positive terms up until it comes close to π. So we will add it by 1/151 and make it 3.1471.
  • Now users will use negative terms to get just under.
  • So use -1/2 . Now π becomes 2.6471, which is not exact π.
  • So adding some positive terms again like this, adding and subtracting, and surely will get it exactly π.
  • This is so because at any stage of this process, the positive terms that are left over will add up to , and the negative terms that are left over will add up to ∞. Therefore, one can always be sure no matter how far users are under or over. We can take enough terms to get under or over.
  • So,  π = ∞ – ∞ That’s why mathematicians have decided to let this be undefined because it does not exist, and probably it doesn’t have any worthy meaning associated with it.

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