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So, I was going through some calculus videos, and what I found was that it said that $\,0\cdot\infty\,$ is undefined. I'm confused why. Like, from an intuition stand point, it is obsviously $0$. Multiplication is simply repeated addition, and let's look at both sides. Let's say we are adding $\infty$ $0$ times to itself, the result is obviously $0$, as there is nothing to add, and it's simply blank, which means it's $0$. Let's take it the other way around, adding $0$ to itself an infinite number of times. This will again be $0$, as no matter how many you add $0$s together, it'll always remain $0$. So it clearly seems to be $0$, but why is it left undefined then?

Angelo
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Tsar Asterov XVII
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    Why is it obviously $0$? Consider $f(x)=\frac 1x$ and $g(x)=x^2$. Then $f(x)\to 0$ for large $x$ and $g(x)\to \infty$. But $f(x)\times g(x)\to \infty$, not $0$. – lulu Dec 31 '22 at 14:19
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    It's probably talking about limits. – Hypernova Dec 31 '22 at 14:21
  • @lulu but functions aside, from taking the basic understandings of multiplication, it feels clearly 0. – Tsar Asterov XVII Dec 31 '22 at 14:21
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    $\infty$ is no number, you cannot multiply it with $0$. – Peter Dec 31 '22 at 14:22
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    The question only makes sense in the context of limits. – lulu Dec 31 '22 at 14:23
  • @Peter Even if infinity is no number, multiplication is simply repeated addition, so it basically means 0 added infinitely, and that would clearly be 0, yes? – Tsar Asterov XVII Dec 31 '22 at 14:24
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    A line has an infinite set of points, all of lemgth zero. What is its lemgth ? – Empy2 Dec 31 '22 at 14:24
  • @Empy2 Wait, you're right. I got what you mean. Now I see why it's a contradiction. Sorry for the annoyance, and thank you for the explanation. – Tsar Asterov XVII Dec 31 '22 at 14:26
  • The infinite sum of zeros is $0$ because of the definition of the limit. Since every (finite) partial sum is $0$, the limit is $0$. However, we cannot actually add infinite many zeros. – Peter Dec 31 '22 at 14:31
  • There is no real flaw with your logic, with the only exception that $\infty$ is not a number and therefore the usual addition and multiplication laws may not behave the same as they do for natural numbers. In practice, $\infty\cdot 0$ is undefined because the convention is not to define it. There are reasons behind this that are just as good as your reason for $\infty\cdot 0 = 0$, but at the end of the day its a question about the conventions. – Yanko Dec 31 '22 at 14:37
  • I've taken classes in real analysis where it's taken to be equal to $0$, and many more where it's undefined for reasons like lulu said. – Mark S. Dec 31 '22 at 14:59
  • Thanks to everyone for clearing up my doubt. – Tsar Asterov XVII Dec 31 '22 at 14:59

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