I have always thought about how confusing infinity is...
Is infinity equal to 0? My proof: Let infinity $= a, a = 2a,$so $0=a$.
- Is there something wrong with my proof?
-Explain why you chose what you chose for #1.

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I agree with @Zakir Husain. Infinity is not exactly a number. I like to think of infinity as "everything" and zero as "nothing".

However, if we're going to use either infinity or zero as numbers, like when doing calculations with other numbers for instance, we have to have special rules in place which will protect us from making a mistake.

For instance, though many would disagree, I would propose that rule #1 is:

$\frac{x}{0} = \infty$

where $x$ is any number except $0$.

And because of that condition, rule #2 would be:

$\frac{0}{0}$ and $\frac{\infty}{\infty}$ are undefined.

Here are the last few rules for working with $0$ and $\infty$. Sometimes $0$ behaves like $\infty$, but other times it does the exact opposite:

$x + 0 = x$ BUT $x + \infty = \infty$ (adding nothing to $x$ gives $x$, BUT adding everything to $x$ gives everything)

$x \cdot 0 = 0$ AND $x \cdot \infty = \infty$ (multiplying $x$ and nothing gives nothing AND multiplying $x$ and everything gives everything)

So in your original proof, it is impossible to subtract $a$ ($\infty$) from both sides and have $0$ on one side, and $\infty$ on the other. By rule 4, $2 \cdot \infty = \infty$, so we would be left with $0$ on both sides.

Your welcome! Just keep in mind that those rules are not ones mathematicians have formally agreed upon or anything. They're just basic math combined with my own propositions. :)

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## Comments

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TopNewestI agree with @Zakir Husain. Infinity is not exactly a number. I like to think of infinity as "everything" and zero as "nothing".

However, if we're going to use either infinity or zero as numbers, like when doing calculations with other numbers for instance, we have to have special rules in place which will protect us from making a mistake.

For instance, though many would disagree, I would propose that rule #1 is:

where $x$ is any number except $0$.

And because of that condition, rule #2 would be:

Here are the last few rules for working with $0$ and $\infty$. Sometimes $0$ behaves like $\infty$, but other times it does the exact opposite:

$x + 0 = x$ BUT $x + \infty = \infty$ (adding nothing to $x$ gives $x$, BUT adding everything to $x$ gives everything)

$x \cdot 0 = 0$ AND $x \cdot \infty = \infty$ (multiplying $x$ and nothing gives nothing AND multiplying $x$ and everything gives everything)

So in your original proof, it is impossible to subtract $a$ ($\infty$) from both sides and have $0$ on one side, and $\infty$ on the other. By rule 4, $2 \cdot \infty = \infty$, so we would be left with $0$ on both sides.

Hope this helps! Sorry if it was too long :)

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Thanks!

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Your welcome! Just keep in mind that those rules are not ones mathematicians have formally agreed upon or anything. They're just basic math combined with my own propositions. :)

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I like your explanation.

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Thanks Mei!

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Infinity is not a number, it is a concept in mathematics. You can's treat it as a number.

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Yes, I agree. Plus this reminds me of the Numberphile video you shared. "types of infinity"

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I always pictured infinity as the longest closed curve possible...in the real world of course

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I don't think you can apply normal mathematics to infinity, As @Zakir Husain Sir said, $\infty$ is not a number but a concept.

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