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Ryuu
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Recently, an AT&T commercial has been running on TV where a moderator asks some children about the largest number they could think of.
At the end, one kid replies “∞ times ∞”, which of course is simply ∞2.
Natually, one can instantly think of a larger number: ∞∞. But then, that got me to thinking about ∞.
Now, by non-standard, but logical, mathematical rules of conventions regarding reciprocal and powers notation, it can be argued that:
∞ = 1/0 = 0-1
and
-∞ = -1/0 = -0-1
This naturally leads to the formula:
∞∞ = 0(0-1)
However, due to those rules governing mixtures of powers, I quickly realized that:
∞∞ ≠ 0(0-1)
Because
0(0-1) = 0∞ = 0 ! (& no, I’m not intending that to represent a factorial) :tongue:
Therefore, to properly notate the equation, it must go like this:
∞∞ = 0-(0-1) = 0-∞ ! (& no, I’m not intending that to represent a double-factorial) :tongue2:
Which results in the surprising conclusion:
∞ = -∞
Furthermore:
1/0 = -1/0
And
1 = -1
At the end, one kid replies “∞ times ∞”, which of course is simply ∞2.
Natually, one can instantly think of a larger number: ∞∞. But then, that got me to thinking about ∞.
Now, by non-standard, but logical, mathematical rules of conventions regarding reciprocal and powers notation, it can be argued that:
∞ = 1/0 = 0-1
and
-∞ = -1/0 = -0-1
This naturally leads to the formula:
∞∞ = 0(0-1)
However, due to those rules governing mixtures of powers, I quickly realized that:
∞∞ ≠ 0(0-1)
Because
0(0-1) = 0∞ = 0 ! (& no, I’m not intending that to represent a factorial) :tongue:
Therefore, to properly notate the equation, it must go like this:
∞∞ = 0-(0-1) = 0-∞ ! (& no, I’m not intending that to represent a double-factorial) :tongue2:
Which results in the surprising conclusion:
∞ = -∞
Furthermore:
1/0 = -1/0
And
1 = -1
(Of course, there is an error in the above—the negative in the exponent of 0 is the act of making the reciprocal of 0, NOT ∞.
It’s essentially the same error that folks often make when proving 1=0. Besides, even if there was no math error in my original calculation, there’s still an error in logic, as the entire exercise is merely a matter of mathematical notation, which is not the same as actual numbers.
But, ironically, it is quite interesting that the graphing of nearly any function that involves the value of ∞, there is a corresponding value to those functions that implies that ∞ does indeed equal -∞.)
It’s essentially the same error that folks often make when proving 1=0. Besides, even if there was no math error in my original calculation, there’s still an error in logic, as the entire exercise is merely a matter of mathematical notation, which is not the same as actual numbers.
But, ironically, it is quite interesting that the graphing of nearly any function that involves the value of ∞, there is a corresponding value to those functions that implies that ∞ does indeed equal -∞.)
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