Does $\frac20 = \frac50$?

Question

If $\frac20$ and $\frac50$ are both equal to infinity, are these infinities the same "level of infinity?" Or can they be different cardinalities of infinity?

Answer 1

Just about the only system of numbers that allows division by zero is the Projectively extended real line. (Another one is the Riemann sphere, similar in many respects.) That system adds one number, called $\infty$, to the real numbers. In that system,

$$\frac 20=\infty \qquad \text{and} \qquad \frac 50=\infty$$

so in that system it is possible to say that

$$\frac 20 = \frac 50$$

However, even in that system you cannot subtract infinity from itself, so you cannot say that $\dfrac 20-\dfrac 50=0$. You can, however, say that

$$\frac 20\cdot\frac 52 = \frac 50$$

Do be careful what you try to do in that system.

In that system, there is only one infinity, so there are no "levels" of infinity. Cardinalities also do not exist in that system.

Note that this is not the extended real number system used in some versions of calculus--that would be the affinely extended real number system. Division by zero is not defined in that system, so my answer really does not apply to calculus.

Answer 2

Note that $\forall a\in \mathbb{R}$ the ratio

$$\frac{a}0$$

in not defined.

Otherwise for $\forall a\in \mathbb{R}$ with $a\neq 0$ and for $x\to 0$

$$\left|\frac{a}x\right|\to +\infty$$

In this case $+\infty$ is just a symbol to indicate th fact that the ratio becomes arbitrarly big as $x$ get closer to $0$.

Answer 3

In standard calculus, infinity is not a number, and neither is $\frac{2}{0}$, so it doesn't equal anything. You can extend arithmetic to work with infinities, if you are willing to sacrifice field properties, but that still doesn't mean all arithmetic combinations of $\infty$ are defined. In particular, $\infty - \infty$ is still not defined, so there is no such thing as $\infty = \infty$ in any case.

example

another example