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.
Last edited by rileywkong (2016-06-20 11:45:39)
"Life is like a bicycle; to keep your balance, you have to move." - Albert Einstein
But Wait! It's actually 9.999...8!
Didn't follow that. . Why an 8 ?
I thought we never reached that last digit.
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Besides, the notion of 0.999... doesn't even make sense, i.e., it is not a valid decimal representation.
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.