From zero to infinity

Some mathematical oddities I never quite understood

Consider the following equations :

1 / 9 = 0.1111…1111

0.1111…1111 * 9 = 0.9999…9999

From this, we can conclude :

1 = 0.9999…9999

From this, we can conclude :

0 = 0.0000…0001

Now, consider this :

1 / ∞ = 0.0000…0001

– 1 / ∞ = – 0.0000…0001

This would lead to the following conclusion :

1 / ∞ = 0 = –1 / ∞

1 / 0 = ∞ = – ∞

When I draw a graph for 1 / x, that is precisely the result I would expect.

I never really understood why people keep saying that 1 / ∞ ≠ 0. I’be been taught this in high school, but it never made any sense to me at all. This, especially considering the the fact the “limit” of 1 / x is ∞ or -∞ for “x → 0”.

Why can’t we just use “limits” to determine the value of infinity in equations? Why can’t we just say that 1 / 0 = ∞ because ∞ is the resulting value of the limit of 1 / x with “x → 0” or that 1 / ∞ = 0 because 0 is the resulting value of the limit of 1 / x with “x → ∞”?

I guess the typical argument against the argumentation here-above, would be that the “limit” of 2 / x, 3 / x or 4 / x with “x → 0” are all also ∞. This, they argue, makes the argumentation here-above invalid. But does it really? I guess it does, if and only if we treat ∞ as some kind of garbage bin that represents any value that’s infinitely large. However, I’m not convinced that we should.

Why can’t we just define ∞ as eg. the number 100000000…000 (with an infinite number of zeroes)? Basically, you’d have something like ∞ = 10ⁿ, with n = ∞. If we would define ∞ like that, we could easily argue that 2 / x is equal to 2*10ⁿ (n = ∞) and thus 2∞. Or, if we would multiply two infinite values, we’d end up with 10²ⁿ (n = ∞), or ∞².

Basically, why can’t we just use ∞ as a normal number? Why can’t we have different orders of magnitude and express those by using a notation like 2∞ or ∞³? Doing so, I believe, would significantly improve our understanding of infinity because it would give us a broader understanding of the impact of multiple values of infinite magnitude have on each other.

Also, why must we treat the value of x / x as undefined for x = 0, when we intuitively feel it should be 1 (eg. when looking at its graph, or dividing by x at both the top and bottom of the fraction) and the limit of x / x for “x → 0” also gives us that value? We must we treat the value of 1 / x as undefined for x = 0, looking at its graph tells us it should be ∞ or -∞ and this is exactly what you get when calculating the limit? Why can’t we just use the limits as the actual value instead of some ackward workaround to deal with supposed undefineds? I find it hard to believe that I’m the only one who believes it makes far more sense to say that the value of 2 / x is 2∞ for x = 0 than to say that the “limit” of 2 / x is ∞ for x = 0? Right?!

Notations similar to those used for 10²ⁿ and 2*10ⁿ, with n = ∞, could also be used for n = -∞ (= infinitesimals). Whether n = -∞ or ∞, it always makes sense to preserve the differences between 10-ⁿ, 2*10ⁿ, 3*10ⁿ. Sure, these differences may seem irrelevant, but they become relevant whenever you multiple these numbers by 10⁻ⁿ. So why choose to lose this information? Why treat 10ⁿ * 10⁻ⁿ as “undefned” for -∞ and ∞, when we know that 10ⁿ * 10⁻ⁿ = 1 in every other situation as well as the “limits” for -∞ and ∞?

I had hoped the videos below would help me better understand why infinity is given such a strange treatment by mathematicians… but unfortunately they don’t! In fact, mathematics never made less sense to me than it did after watching both videos.