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Are there any axiomatic or mathematical paradoxes with assuming that any number divided by zero equals that number multiplied by infinity, if we also assumed concepts such as:

$\infty+\infty$$=2\infty$

$2\infty-\infty=\infty$

$\infty-\infty=0$

$\frac{\infty}{\infty}=1$

ect...

Also, $\frac{1}{0}=\infty$

$-\frac{1}{0}=-\infty$

and

since $\frac{1}{0}=\infty$, $\frac{1}{\infty}=0$ therefore $\infty \cdot 0= \frac{\infty}{\infty}=1$

Because if we did then $\frac{x}{0}=x\cdot \infty$ therefore $\frac{0}{0}= 0\cdot \infty$--> $\frac{0}{0}=1$. Not only do we get rid of the strange {} empty set nonnumber as well as the bizaar $\frac{0}{0}$ omni-number thing, but we just made math a lot cleaner by deciding that $\frac{x}{x}$ ALWAYS equals $1$. Is there any paradoxes with these definitions?

Sigma6RPU
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jacobj
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    So $2=2\times 1 = 2 \times \dfrac{\infty}{\infty}=\dfrac{2 \times \infty}{\infty}=\dfrac{\infty+ \infty}{\infty}=\dfrac{\infty}{\infty}=1$ ? – Henry Aug 11 '16 at 23:52
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    and similarly $3=3\times 1 = 3 \times \dfrac{0}{0}=\dfrac{3\times 0}{0}=\dfrac{0+0+0}{0}=\dfrac{0}{0}=1$ ? – Henry Aug 11 '16 at 23:56
  • @Henry or just for giggles $/frac{0+0+0}{0}=\frac{0}{0}+\frac{0}{0}+\frac{0}{0}=1+1+1=3$ – Sigma6RPU Aug 12 '16 at 00:09
  • @Henry I messed up in my previous comment the beginning part should read $\frac{0+0+0}{0}$ – Sigma6RPU Aug 12 '16 at 00:43
  • No... I am saying inf plus inf equals 2 inf and is twice the size of regular inf. Also, the second example shouldnt work because it doesnt obey the order of operations i wouldnt think, although i see room for me to be convinced otherwise. – jacobj Aug 12 '16 at 11:03

1 Answers1

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Let's say we play along. We lose a lot of nice properties of multiplication and addition, like associativity and distributivity.

For instance, $2\cdot(0\cdot \infty)=2\cdot1=2$, but $(2\cdot 0)\cdot\infty=0\cdot\infty=1$.

Another example: $(1-1)\cdot\infty=0\cdot\infty$, but multiplying out the brackets we get $\infty-\infty$, which could be anything.

In the end, it's best to let $\infty$ stay where it belongs, which is nowhere near arithmetic.

If you really want to see how to make this work, look into ordinal arithmetic or non-standard analysis. If you're really brave, try to take on the surreal numbers.

Arthur
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  • I dont understand what a surreal number is. – jacobj Aug 12 '16 at 11:04
  • That's not so strange. It took me several hours of staring at definitions and examples before I began to understand. I still don't comprehend multiplication for surreal numbers. But they contain the real numbers, they contain infinite numbers, and they have division and square roots. – Arthur Aug 12 '16 at 11:57
  • What is an example of a surreal number? Is this the same as a hyperreal number? – jacobj Aug 12 '16 at 23:20
  • The surreal numbers are created from scratch. They consist of a pair of sets of previously created surreal numbers, called the left set and the right set (with some restrictions), written $\langle L\mid R\rangle$. The first number created is $0$, with the representation $0=\langle \mid \rangle$. Then comes $1$ and $-1$, written as $\langle 0\mid\rangle$ and $\langle\mid0\rangle$, then the next generation consisting of $-2=\langle\mid -1\rangle$, \frac12=\langle -1\mid0\rangle$, and so on, beyond infinitely many generations. There are so many surreals that there is no set containing them all. – Arthur Aug 13 '16 at 09:17
  • Then you have the first infinity, called $\omega$, represented as $\langle 1,2,3,\ldots\mid{}\rangle$, and $1/\omega=\langle{}\mid 1,\frac12,\frac14,\frac18,\ldots\rangle$. You still have $0\cdot\omega=0$ and $0/0$ undefined, though. Few good things can come from a system that changes those rules. – Arthur Aug 13 '16 at 09:47