High School Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1?

[Infrared]

Given that this thread is marked 'B', discussions about the extended reals and the Riemann sphere are way off topic.

This could well be the case, but then I don't think it's fair to say that wolfram alpha is wrong or inconsistent. It doesn't know about the limited usage which may or may not be relevant here.

I don't plan to message more on this issue, for fear of further derailing the thread.

 

[Office_Shredder]

I think a natural way to see the problem is to notice that $2/\infty =0$, so $\infty \times 0=2$. You can pick any number in the numerator here. So the multiplication clearly doesn't make sense even if the original equation has meaning to it. It doesn't matter what definition you try to attach to these operations

 

[martinbn]

Just to add to the fun, there are contexts in which $\infty \times 0 = 0$.

 

[Infrared]

Just to add to the fun, there are contexts in which $\infty \times 0 = 0$.

I think I mentioned this already in post 25 :)

 

[frost_zero]

1/∞ is not zero, it tends towards zero because the difference between numerator and denominator is huge, it certainly is undefined in common mathematics, however in certain cases such as in physics it is taken as zero because the difference is huge and we are not aiming for much precision in physics; eg:- NASA uses only till 15 digits of pi after decimal point for some of its highest accuracy calculations.

 

[PeroK]

1/∞ is not zero, it tends towards zero because the difference between numerator and denominator is huge

There's a lot of imprecise terminology there.

, it certainly is undefined in common mathematics,

That is true.

however in certain cases such as in physics it is taken as zero because the difference is huge and we are not aiming for much precision in physics

It's not a question of precision. It's a question of physicists using a shorthand $1/\infty$ instead of writing out precise statements with limits. For example, in physics "at infinity" means "far enough away that the gravitational potential differs from zero by a negligible amount".

 

[mathman]

$\frac{1}{\infty}=0$, $\frac{2}{\infty}=0$,, $\frac{3}{\infty}=0$,, etc. Indeterminate when multiplied back.

 

[WWGD]

I believe in measure theory we accept $$\infty \cdot 0 =0; \infty \times \infty = \infty $$, in that we do accept values in $$\mathbb R \cup \{\infty \} $$.

And I guess the Riemann Sphere is the/a 1-point compactification. I believe there may be inequivalent types of 1-point compactifications.

 

[InkTide]

I think it's helpful to consider infinity itself to be undefined. Think of it less like a number and more like a special symbol. I say symbol and not variable because there is more than one kind of infinity, such as countable versus uncountable infinities, that the simple "1/∞ != ∞*0" formulation can't differentiate between even before you get to using limits.

It's a bit like the bar over a repeating decimal - an artifact of the limitations of our ability to express a mathematical concept without collapsing the blackboard into an infinite mass black hole from information density.

 

[WWGD]

I think it's helpful to consider infinity itself to be undefined. Think of it less like a number and more like a special symbol. I say symbol and not variable because there is more than one kind of infinity, such as countable versus uncountable infinities, that the simple "1/∞ != ∞*0" formulation can't differentiate between even before you get to using limits.

It's a bit like the bar over a repeating decimal - an artifact of the limitations of our ability to express a mathematical concept without collapsing the blackboard into an infinite mass black hole from information density.

I'd love to have a drink at that bar ;).

 

[frost_zero]

There's a lot of imprecise terminology there.

That is true.

It's not a question of precision. It's a question of physicists using a shorthand 1/ instead of writing out precise statements with limits. For example, in physics "at infinity" means "far enough away that the gravitational potential differs from zero by a negligible amount".

I agree with the last statement and to phrase my first statement better : as the difference in numerator and denominator increases the number becomes smaller and smaller, with 1/∞ the difference is possibly infinite which is why it is said to tend towards 0

 

[PeroK]

as the difference in numerator and denominator increases the number becomes smaller and smaller,

That's not true. A ratio does not directly depend on this difference. Consider $\frac n {2n}$ as $n$ increases.

with 1/∞ the difference is possibly infinite which is why it is said to tend towards 0

$\frac 1 \infty$ is simply undefined. Or, if you want to extend numbers to include $\infty$ then it may be defined to be precisely $0$.

 

[anuttarasammyak]

As we know those relations are true: if a/b = c, then a = b*c and b = a/c

Following your scienario
\lim_{a\rightarrow +0} a=+0
\lim_{a\rightarrow +0} \frac{1}{a}=+\infty
\lim_{a\rightarrow +0} a\cdot \frac{1}{a}=1
+0\cdot+\infty=1
Examples of other scenarios are
\lim_{a\rightarrow +0} 2a=+0
\lim_{a\rightarrow +0} \frac{1}{a}=+\infty
\lim_{a\rightarrow +0} 2a\cdot \frac{1}{a}=2
+0\cdot+\infty=2 and
\lim_{a\rightarrow +0} a^2=+0
\lim_{a\rightarrow +0} \frac{1}{a}=+\infty
\lim_{a\rightarrow +0} a^2\cdot \frac{1}{a}=+0
+0\cdot+\infty=+0
, and
\lim_{a\rightarrow +0} a=0
\lim_{a\rightarrow +0} \frac{1}{a^2}=+\infty
\lim_{a\rightarrow +0} a\cdot \frac{1}{a^2}=+\infty
+0\cdot+\infty=+\infty

I am afraid that we cannot choose unique scenario of yours.

 

[diegogarcia]

Pretty simple to demonstrate and Wolfram agrees.

First, what does Wolfram says about $0 \times \infty$: undefined

But Wolfram also says that $\frac{1}{\infty} = 0$. It is not undefined.

There is mathematica and there is IEEE754.

IEEE754 is a standard for digital floating point arithmetic. It's purpose is to ensure that every programmed math operation has a well-defined result, even if the operation, such as division by zero, is mathematically untenable.

In IEEE754, infinity is any number that exceeds the maximum floating point value, even by a tiny fractional value, is infinity, but it also raises an overflow exception.

IEEE754 allows division by zero and other operations defined by limiting values, such as log(0), and it will default to infinity as an answer (because generall it makes the most sense), buit it will also raise an exception (it will flag the result as a possible error).

IEEE754 also has +0 and -0, i.e. signed zeros, which do not occur in ordionary arithmetic.

https://en.wikipedia.org/wiki/Signed_zero

Do not confuse mathematics with IEEE754.

 

[jack action]

There is mathematica and there is IEEE754.

IEEE754 is a standard for digital floating point arithmetic. It's purpose is to ensure that every programmed math operation has a well-defined result, even if the operation, such as division by zero, is mathematically untenable.

In IEEE754, infinity is any number that exceeds the maximum floating point value, even by a tiny fractional value, is infinity, but it also raises an overflow exception.

IEEE754 allows division by zero and other operations defined by limiting values, such as log(0), and it will default to infinity as an answer (because generall it makes the most sense), buit it will also raise an exception (it will flag the result as a possible error).

IEEE754 also has +0 and -0, i.e. signed zeros, which do not occur in ordionary arithmetic.

https://en.wikipedia.org/wiki/Signed_zero

Do not confuse mathematics with IEEE754.

I was using the extended real number line $\overline{\mathbb{R}}$, defined as:
$$\mathbb{R} \cup \{-\infty, +\infty\} \text{ where } -\infty < x < +\infty \text{ for all } x \in \mathbb{R}$$