Math Is Fun Forum

Hi all,

I'd like to suggest an anonymous/guest feature for *ahem* sensitive or otherwise anonymous topics.

Hi all,

Reading the many "0.999... = 1" topics, I suddenly realize that really, infinity is hard to grasp.

Laymen think of infinity as the "largest number in the number system". Meaning, if you count from 1, 2, 3, 4.... the last number you count to is infinity.

Of course, you can't count that high. No one can.

So, our friend Cantor devised a clever notation: aleph-null, or ℵ₀ (the first letter of the Semitic alphabet, btw), denoting the cardinality of the natural numbers (what I said above). And, this extends to 0.999..., for example. 0.999... has aleph-null trailing 9's behind it.

What happens when we multiply 0.999... by 10? We get 9.999... But Wait! It's actually 9.999...8! After aleph-null 9's, there's an 8! (edit: I meant 0.999...0 . )

You may think that spells the end of the commonly-used proof of 0.999... = 1. But no. Aleph-null + 1 = aleph-null.

Hence, the proof is sound.

Now what if we extend aleph-null to include pi, e, 1/2, and so on? We get aleph-one, or ℵ₁.

I'll leave that to you to muddle over.

Cheers!