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I have recently come up with a proof of 1/0 = ∞. Here is the proof:

We will first write 1 and 0 as powers of 10.

1 = 10^0

Since the limit of 10^x as x approaches -∞ is equal to 0, 0 = 10^-∞

Therefore, 1/0 = (10^0)/(10^-∞) = 10^(0-(-∞)) = 10^∞ = ∞

Is this proof correct?

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    In this context, infinity is not a number, but a shorthand notation for a limit. So you can't really do math with it, or prove anything about it if you don't use limits. Take your equation and multiply both sides by $0$ an you get $1=0\cdot\infty$. Shouldn't the right hand side equal zero since it's zero times something? I – B. Goddard Jun 22 '21 at 14:23
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    All those arithemtic operations in your proof (division, exponentiation, subtraction) are ordinarily defined only for numbers. Since infinity is not a number, your arithmetic operations are invalid, unless you can explain to us how you are defining them. – Lee Mosher Jun 22 '21 at 14:24
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    What will be 1/? Or what about $\times $ ? The point is that $\infty$ is just a symbol and on its own (independently) it has no meaning. – Koro Jun 22 '21 at 14:28
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    @LeeMosher If you use the extended real number line, ∞ and -∞ are treated like actual numbers. –  Jun 22 '21 at 14:35
  • You could use the same "proof" to show that $1/0 = - \infty$, since $0 = (-1) \cdot 0$. I won't continue the rest of your argument, just like the name of the Dark Lord should not be pronounced. So it's all nonsense. – Hans Engler Jun 22 '21 at 14:51
  • Read https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations, it has a paragraph that explains why $1/0$ is still undefined even in the extended real number line (which by the way is the only place where you can reasonably perform the kind of operations you've performed here). – SeraPhim Jun 22 '21 at 14:55
  • Obligatory wheel theory reference. – Shaun Jun 22 '21 at 15:57

3 Answers3

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Well... yes and no. It depends on the context in which you set yourself.

For example, consult the operations with the infinity element of the Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere#Arithmetic_operations

(Note that on the Riemann Sphere, also the real projective line, you don't make a difference between $-\infty$ and $+\infty$.)

In simple $\mathbb{R}$, where infinity is not an accepted element, this doesn't work, because you're exiting your base set. It's like trying to prove that $-2$ is a natural number, since $5 - 7 = -2$. However, this is not how $\mathbb{N}$ works: $-$ is not a closed operator in $\Bbb N$.

Tristan Duquesne
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Who told you that $\frac{1}{0}$ is $\infty$ !!

This expression is simply undefined as division by zero is undefined.

However following is correct

$\displaystyle\lim_{x \to 0^+}\frac{1}{x}=\infty$

As you already said in question that

$\displaystyle\lim_{x \to -\infty}10^{-x}=0$

then how can you say that $10^{-\infty}=0$. Both the expressions are different.

Lalit Tolani
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  • You would need to fix the limit to only include the right side: $\lim_{x \to 0} \frac{1}{x}$ does not exist. – Kman3 Jun 22 '21 at 15:07
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Since the limit of $10^x$ as $x$ approaches $-∞$ is equal to $0$, $0 = 10^{-∞}$

This is not correct. Infinity is not a number, so you cannot perform operations like exponentiation with it.

Kman3
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    "Infinity is not a number" is nonsense. – plop Jun 22 '21 at 14:36
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    @plop How is it nonsense? Would you like to define $10^{-\infty}$? – Kman3 Jun 22 '21 at 14:37
  • It is nonsense because it is not saying anything. What is a number? Is $\infty$ really not an element of none of the multiple answers that the first question can have? Moreover, having a concept of "number" is useless in mathematics. – plop Jun 22 '21 at 14:42
  • @plop When I use "number" in this context, I'm referring to a quantity which can be manipulated through arithmetical operations. – Kman3 Jun 22 '21 at 14:52
  • Yes, more nonsense. $\infty$ doesn't necessarily lack that property. – plop Jun 22 '21 at 14:53
  • @plop You still have yet to prove me wrong. What is $1+\infty$? What is $10^{-\infty}$? – Kman3 Jun 22 '21 at 14:54
  • You are the one making an affirmative claim. You are the one that has something to prove. I am just saving you the trouble, by telling you that "number" is not a concept in mathematics. – plop Jun 22 '21 at 14:55
  • @plop Did I not just tell you my reasoning? I define a "number" in this context as a quantity that can be manipulated through arithmetical operations. Infinity cannot, so it does not fit this definition and is thus not a number. – Kman3 Jun 22 '21 at 14:56
  • @Kman3 If you use the extended real number line, then you can perform operations on ∞ and -∞. –  Jun 22 '21 at 18:34
  • @MathGeek Sure, but $\infty$ and $-\infty$ in that case are being treated like numbers, and the results of each operation are constructed with that in mind. With that being said, it is also problematic for $\frac{1}{0}=\infty$ because what is $\frac{2}{0}$? Also infinity? $2\infty$? Then $2\infty=\infty$. – Kman3 Jun 22 '21 at 19:05
  • @Kman3 If 1/0 = ∞, then any number divided by 0 is equal to ∞. –  Jun 22 '21 at 20:34
  • @MathGeek Which means $\frac{1}{0}=\frac{2}{0}$, leading to the contradiction I stated. – Kman3 Jun 22 '21 at 20:58
  • @Kman3 Yes, but if you cross multiply the two fractions, you get 0=0, which is a true statement. –  Jun 22 '21 at 21:00
  • @MathGeek But then by your logic you've just done $0 \times \infty$, which is not defined in your extended real number line. – Kman3 Jun 22 '21 at 21:01
  • You've also done $1 \times \frac{0}{0} = 0$, which also defines $\frac{0}{0}=0$ when it is not defined. – Kman3 Jun 22 '21 at 21:03
  • If you cross multiply, you get 10 = 20, which is equivalent to 0=0. –  Jun 22 '21 at 21:03
  • Let us continue this discussion in chat. –  Jun 22 '21 at 21:04