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calculus - What is infinity divided by infinity? - Mathematics Stack . . . Essentially, you gave the answer yourself: "infinity over infinity" is not defined just because it should be the result of limiting processes of different nature I e , since such a definition would be given for the sake of completeness and coherence with the fact "the limiting ratio is the ratio of the limits", your
One divided by Infinity? - Mathematics Stack Exchange Similarly, the reals and the complex numbers each exclude infinity, so arithmetic isn't defined for it You can extend those sets to include infinity - but then you have to extend the definition of the arithmetic operators, to cope with that extended set And then, you need to start thinking about arithmetic differently
limits - Can I subtract infinity from infinity? - Mathematics Stack . . . $\begingroup$ Can this interpretation ("subtract one infinity from another infinite quantity, that is twice large as the previous infinity") help us with things like $\lim_{n\to\infty}(1+x n)^n,$ or is it just a parlor trick for a much easier kind of limit? $\endgroup$ –
What exactly is infinity? - Mathematics Stack Exchange Infinity is not a natural number, or a real number: there should be no confusion about that We can use infinity as the upper limit of an integral as shorthand to say that all the reals greater than the lower limit are included - that is a conventional use - along with others involving arbitrarily large numbers
limits - Infinity divided by infinity - Mathematics Stack Exchange When we use straightforward approach, we get $$ \frac{\infty+1}{\infty} = \frac{\infty}{\infty} $$ In the process of investigating a limit, we know that both the numerator and denominator are going to infinity but we dont know the behaviour of each dynamics But if we investigate further we get : $$ 1 + \frac{1}{x} $$ Some other examples :
Why is $\\infty\\times 0$ indeterminate? - Mathematics Stack Exchange Your title says something else than "infinity times zero" It says "infinity to the zeroth power" It is also an indefinite form because $$\infty^0 = \exp(0\log \infty) $$ but $\log\infty=\infty$, so the argument of the exponential is the indeterminate form "zero times infinity" discussed at the beginning
Types of infinity - Mathematics Stack Exchange $\begingroup$ "Or that the infinity of the even numbers is the same as that of the natural numbers " - not necessary This depends on your definitions I would argue the infinity of natural numbers is by 1 2 less than the infinity of even numbers (positive, negative and zero) I men, not 1 2 times, but the difference $\endgroup$ –
What is the result of - Mathematics Stack Exchange Infinity does not lead to contradiction, but we can not conceptualize $\infty$ as a number The issue is similar to, what is $ + - \times$, where $-$ is the operator The answer is undefined, because $+$ and $\times$ are not the kind of mathematical objects that $-$ acts upon
What is imaginary infinity, - Mathematics Stack Exchange The infinity can somehow branch in a peculiar way, but I will not go any deeper here This is just to show that you can consider far more exotic infinities if you want to Let us then turn to the complex plane The most common compactification is the one-point one (known as the Riemann sphere), where a single infinity $\tilde\infty$ is added