h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

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10x = 9.99999...
Subtract x from both sides:
9x = 9
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This is a bit odd.

It has been many years since my last math class, but I am quite sure that you subracted x from one side and 0.99999... from the other. Okay, so they are supposed to be equal. Otherwise the result would be 9x = 9.99999...-x or 10x-0.99999... = 9, which is not too informative. But can be easily simplified in an additional step making things a bit clearer.

Unless you state that you are subtracting the first line from the second, that would make more sense. At least to me.

Sometimes the logic of logic escapes me.

since X is given as 0.999... it can be subtracted from the side with the varible as X and from the side with the constant as 0.999...

addendum to my last post-
Does it matter? since we have no idea of what infinity means we have no use for things that use it. Infinite anything is useless because we don't have a concrete definition of what infinity is so we can't apply its "infinityness" to anything. Knowing that a black hole has infintite mass doesn't make it any less dangerous, does it? Having a number that we can't express does us no good. Why do you think that childern are taught to round numbers? It's because we don't care if we are exact or not. Fractions are the same type of thing. If we want exact numbers we use decimal. If we don't care if a number is 0.5 or .49999... we use 1/2. Anyhow, you can achieve a sum of 0.9999... with this simple equation- 0.3333...+0.6666....=0.9999... If you use fractions you get 1/3 + 2/3 = 1.

Just because we 'have no idea what infinity means' - or more accurately, because we cannot understand it, does not mean that we have no use for it. i, the square root of minus one, is impossible to express in real terms, but still proves mathematically useful for solving practical problems using imaginary numbers. Maths is not constrained by the limits of human perception!

Hate to be pedantic at you, but... there is indeed a concrete definition of infinite, and mathematicians (and physicists) can do quite a bit with it... black holes don't have infinite mass, they generally have about the mass of a star - they do have infinite density, however...

One of the more interesting bits of maths I've read is the fact that there is more than one type of infinity... the 'countable' infinity, which is how many integers there are, and the 'infinity of the continuum', which is how many real numbers there are. This second one is a whole lot 'larger' than the first.

Hmm, how about this for a definition of infinite:

10 GOTO 10

A little program which will run for an infinite amount of time...

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The flaw in the proof is actually the multiplication; how can you guarantee that the multiplication does not alter the structure of the repetend in some way? What happens at the n-th position may be different from what happens at the infinite-th position. When multiplying by 10 the digit at the position n takes on the value of the digit at position n+1. But what happens to the digit at the infinite-th position?

Playing with infinity is a little bit dangerous.

Also I'm afraid the program [[ 10 GOTO 10 ]] would not run for an infinite amount of time on any of my machines. Because ...

a) I'd kill it (ctrl-c or task manager or pull the plug out of the wall or whatever)

... and ...

b) I don't allow GOTO (see Djikstra's article "The GOTO Considered Harmful"

t-jx01

Well, the program wouldn't work in practice, but it does illustrate a fairly easy definition of infinity...

I believe Dijkstra said "It is practically impossible to teach good programming style to students that have had prior exposure to BASIC; as potential programmers they are mentally mutilated beyond hope of regeneration." I guess I've given myself away there ... but my programming isn't *that* bad...

And... hmm... *is* there an infinite-th digit? Surely there are just an infinite number of normal digits, on which multiplication works normally?

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