For example, $$\frac{1}{1}=1\quad \frac{1}{2}=0.5\quad \frac{1}{3}=0.\overline3\quad \frac{1}{10}=0.1$$ so the larger the denominator is, the smaller the number is.

Would this mean that $\frac{1}{\infty}=0$, or what else could it be?

Also, $$\frac{1}{0.5}=2\quad \frac{1}{0.\overline3}=3\quad \frac{1}{0.1}=10\quad \frac{1}{0.001}=1000$$ and so on. As the numbers in the denominator get smaller, the value of the answer gets larger.

This leads me to the conclusion that $\frac{1}{0}=\infty$. Would this be correct?

  • $\begingroup$ + or - infinity ? $\endgroup$ – M. Di Feb 12 '19 at 16:24
  • 4
    $\begingroup$ Short answer: no. Any value you try will destroy some important algebraic identity - usually the distributive law. There are many questions on this site asking the same thing. math.stackexchange.com/questions/260876/… $\endgroup$ – Ethan Bolker Feb 12 '19 at 16:24
  • $\begingroup$ Your examples sort of answer the question. It's that infinity itself is a concept of limiting behavior. There is no value to give to infinity, because by definition it is the characterization of "what happens when you get closer and closer to something" $\endgroup$ – NazimJ Feb 12 '19 at 16:29

I made a comment to your post, but let me add this counterexample:

Let's say infinity was a number, $\infty$. Well we know by definition that for any natural numbers $n$ and $m$ (say $n=3$ and $m=5$), $$3\infty = 5\infty$$ And if infinity is a number, we divide on both sides to get $$3=5$$ In fact it is very easy to come up with examples like this. Reason being is that infinity is a concept of what happens in the limit when you approach something

  • $\begingroup$ There are things called transfinite numbers(infinite cardinals and infinite ordinals). So 3*infinity is not equal to 5*infinity when it comes to ordinal numbers. However, this is true when you are dealing with cardinal numbers. The cofinality of the positive real numbers is the set of natural numbers. $\endgroup$ – Mr X Jul 22 '19 at 19:22

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