Treat $\infty$ carefully!

It is generally believed that infinity cannot be treated as a number. Otherwise, we would arrive at contradictions like this famous one:

If $\infty = \infty + 1 = \infty + 2$ ,

And we subtract $\infty$ from each equation,

Then $0 = 1 = 2$

However, why did we subtract only $\infty$ from each equation? If $\infty = \infty + 1 = \infty + 2$ as we first said, then couldn't we subtract $\infty$ from the first equation, $\infty + 1$ from the second, and $\infty + 2$ from the third? We wouldn't be breaking an algebraic rules, because all of these amounts are equal to each other. Then, we would get $0 = 0 = 0$ which is perfectly true!

The moral of the story: treat infinity carefully! It isn't exactly a number, but if we take certain precautions, then we can use as if it were a number.

Please note however that this again represents curious exploration, not rigid proof.

I would welcome any thoughts, comments, or objections!

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

$∞ -∞$ is indeterminate, you can’t divide by zero, you can’t subtract ∞, you can’t divide by ∞, you can’t add ∞, can you multiply by ∞ without creating loopholes???

Jason Gomez - 3 weeks, 5 days ago

Not sure I understand. Are you disproving, or asking a question?

David Stiff - 3 weeks, 5 days ago

The first part was facts (in the sense that they create loopholes in math if defined by the normal operations), the last part was a question

Jason Gomez - 3 weeks, 5 days ago

@Jason Gomez – I see. As I mentioned, this note was mainly speculation at the time I wrote it, but since then, I've learned that there are actual formalizations of mathematics which incorporate division by zero and infinity, such as the projectively extended real line, or the more thorough wheel theory. The people who came up with them know a lot more about it then me. :)

David Stiff - 3 weeks, 4 days ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Treat ∞\infty∞ carefully!

Comments

Treat $\infty$ carefully!