Supergrunch Posted September 27, 2005 Posted September 27, 2005 Well... it should be possible to write down all numbers right? (to a degree- I'm kind of exluding irrational and complex numbers) If so, then the difference between 0.999rec and 1 must be a number you can write down, but you can't, therefore the difference doesn't exist and 1 = 0.999rec. This is a wordy version of a proof someone posted further up.
KingJoe Posted September 27, 2005 Posted September 27, 2005 People generally aren't comfortable with infinity and the main reason is that their experiences of life don'e give them much to draw on in the form of analogy to draw on. You can write complex numbers down (3+2i, anyone) and irrational numbers too (sqrt(2)). The point is that there is NO number between 0.9999... amd 1. They are the same number. 0.99... isn't someone beginning with 0.9 and writing forever. Most of you agree that the limit as we keep adding nines gets closer to 1. In more precise terms, if we are given a number (no matter how small) we can find some number of 9's in our expansion so that the difference between our expansion and 1 is smaller than that number (ie, if we wanted to make the difference smaller than 0.000000000002, we simply have 11 9's or more) we simply have more 9's than the number of 0's in the decimal expansion. Bear with me! Ok, so by adding more an more 9'm makes the difference smaller and smaller. so it makes the numbers closer and closer. 0.99... is itself a LIMIT. it is the limiting value of a sequence of more and more 9's. It is the value that adding more 9's gets closer to, because we are adding more 9's than have ever been concieved. INFINTY ISN'T JUST THE BIGGEST THING YOU CAN THINK OF. IT'S MUCH BIGGER. more 9's: closer to 1. An infinite number of 9's: infinitely close. Infinitely close : in the same place. This is a bit of a crude argument but if you haven't got it yet than I can't think of a way to explain it to you. If 2/3rds of the potential maths teachers on my course couldn't get it then there's not a huge shame in it. If anyone is interested in finding a bit more out about infinity then hava a look here http://math.youngzones.org/Hotel_Infinity.html or google a man called Gregor Cantor. edit: to whover wished I was their maths teacher, cheers!
Zakatu Posted September 27, 2005 Posted September 27, 2005 Ok, i understood the word explanation(s). Just made me click. Thanks. leaves me feeling a bit unsatisfied, meh. it doesn't seem right, perhaps because nothing really is infinite.
Mr_Odwin Posted September 27, 2005 Posted September 27, 2005 The point is that there is NO number between 0.9999... amd 1. I think that this is the most important bit Zakatu. There are an infinite number of numbers so whenever you pick two of them you can always always always pick out a number that is between them. However, in the case that we are looking at you can't, therefore they are the same number.
Zakatu Posted September 27, 2005 Posted September 27, 2005 yer, i understand. I just think it was all a bit misleading to begin with, stating x=0.999... when that is the same as one, may aswell of just said x = 1. None of the manipulation was neccessary at all (multiplying it by 10?) it was misleading. because nothing was changed. SHould of said what it was REALLY about from the start, i.e a recurring number is the same as the whole number it is near, as long as the recurring number goes on for infinite.
Supergrunch Posted September 27, 2005 Posted September 27, 2005 When I said you can't write complex numbers or irrationals, I meant that you couldn't write them out in full- i is simply a representation and irrationals vary forever. I suppose you could say that 0.999rec is a representation, but people can realistically see from it exactly what the number implies, to however many decimal places they want.
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