Does $\frac {1}{∞} = 0$?

Question

Actually , I am not believing that $\frac {1}{∞} = \frac{0}{1}$ because simply $0 ≠ 1$ (we get this if we multiple numerator by denominator) , In the other hand it is very very very small , So I am confused

Your reasoning has to flaws: (1) $\infty$ is not a number, it is not governed by standard arithmetic (and actually can have multiple nonequivalent definitions) and (2) just because numerators are different doesn't mean numbers are, i.e. $\frac{1}{2}=\frac{2}{4}$. — freakish, Apr 19 '19 at 18:06
usually this notation is used when discussing limits, standard arithmetic may not apply — Vasili, Apr 19 '19 at 18:10
For sequences in $\Bbb R$, $\lim_{n\to\infty}x_n=0$ doesn't imply $\lim_{n\to\infty}\frac{1}{x_n}=\infty$. The latter limit could also be $-\infty$ or nonexistent. — J.G., Apr 19 '19 at 18:17

score 2 · Answer 1 · answered Apr 19 '19 at 18:03

2

Since $\infty$ is not a number, asking what is the quotient of the division of $1$ by $\infty$ doesn't make sense.

answered Apr 19 '19 at 18:03

José Carlos Santos

427,504

score 1 · Answer 2 · answered Apr 19 '19 at 18:09

1

In the extended complex plane (a.k.a the Riemann Sphere), the answer is yes.

answered Apr 19 '19 at 18:09

Shaun

44,997

score 1 · Answer 3 · answered Apr 19 '19 at 18:11

1

In the Extended Real Numbers, it is common to adopt

$$\dfrac{c}{\infty} = \dfrac{c}{-\infty} = 0$$

See: https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations

answered Apr 19 '19 at 18:11

David P

12,320

Does $\frac {1}{∞} = 0$?

3 Answers3