blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 Yeah, it's hard to put it in simple terms... How about this: When you have the number 0.999... and the number 1, for there to be a difference between the two there has to be a number between them. For example, take 0.9 and 1. There are many many numbers between them (there are actually infinite numbers between them). Examples of these are the average between them, 0.95, which is directly between them, but there is also 0.91, 0.92, 0.915... etc. If you take any two numbers that are different, there are infinitely many numbers that you can have between them. Here's a silly example: I could take 0.4549552956210029872 and 0.45495529562100298721, and I could pick: 0.454955295621002987205 0.454955295621002987201 0.454955295621002987206 0.4549552956210029872001 0.454955295621002987209 0.45495529562100298720025568262549174562 ...and it just keeps going, I can pick as many numbers between them as I so choose. Now, consider 0.999... and 1 again. Can you think of a number between them? It's 9s all the way, so I can't just chuck in any more decimals... Before we tried the average of the two numbers to find one between them, but as I did in that image I made, the average of 0.999... and 1 is actually 0.999..., that is, the average is the same as one of them. By definition this means that they have to be the same number. This is essentially what I was demonstrating, in an admittedly complex way by the end (or the start actually). Hopefully that's understandable. Just about, like Dav says the difference is barely anything the smaller the decimal number travels. Like with seconds, 0.999 is barely anything different to 1.0. Add another number on, 0.9999 (is it tenths, hundredths, thousandths, ten thousandths etc..) the difference becomes debatable. The further you go, into hundred thousanths, millionths eventually into billionths is it a human thing that we disregard this and call it a round 1? How much do calculators owe their programming to humans? Like you said earlier in the thread, they round up 0.333 x 3 into 1. It's 0.999. Although my computers calculator came up with 0.999, is that because I'm doing it wrong and can't put the recuring figure in? When scientists began discovering the atom back in the 17th or 18th century, they declared they'd found the smallest thing in the universe (or something to that effect) and now we're looking (or have discovered) for the 'higgs boson'. I just think (probably naively) that to round it up to a singular 1, is a human error. Link to comment Share on other sites More sharing options...
Albert Posted October 14, 2012 Share Posted October 14, 2012 whats this number 356345645694564562756256934867356367456453276534295624756745724578320 Lots Just about, like Dav says the difference is barely anything the smaller the decimal number travels. Like with seconds, 0.999 is barely anything different to 1.0. Add another number on, 0.9999 (is it tenths, hundredths, thousandths, ten thousandths etc..) the difference becomes debatable. The further you go, into hundred thousanths, millionths eventually into billionths is it a human thing that we disregard this and call it a round 1? How much do calculators owe their programming to humans? Like you said earlier in the thread, they round up 0.333 x 3 into 1. It's 0.999. Although my computers calculator came up with 0.999, is that because I'm doing it wrong and can't put the recuring figure in? When scientists began discovering the atom back in the 17th or 18th century, they declared they'd found the smallest thing in the universe (or something to that effect) and now we're looking (or have discovered) for the 'higgs boson'. I just think (probably naively) that to round it up to a singular 1, is a human error. Again, it has nothing to do with rounding, 0.999... is equivalent to 1. They are the same number. This is not a debating point, they are the same number. The question is how this becomes intuitive to people. We're not talking about calculators, I don't even get why you got onto calculators... Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 Lots Again, it has nothing to do with rounding, 0.999... is equivalent to 1. They are the same number. This is not a debating point, they are the same number. The question is how this becomes intuitive to people. We're not talking about calculators, I don't even get why you got onto calculators... Well, I tried. May have not got anywhere with it. It just seems impossible to me. Link to comment Share on other sites More sharing options...
Albert Posted October 14, 2012 Share Posted October 14, 2012 What seems impossible about it? Think about it another way, for two numbers to be different there has to be a difference between them. That is, if 0.999... is not equal to 1 then: 1-0.999...= some number What do you think this number is? Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 They are just two different numbers, to me anyway. 0.999 and 1. 1 + 0.999 = 1.999. Link to comment Share on other sites More sharing options...
Albert Posted October 14, 2012 Share Posted October 14, 2012 They are just two different numbers, to me anyway. 0.999 and 1. 1 + 0.999 = 1.999. I will again point out that 0.999... means that the nines go on forever, like 1/3=0.333... Why did you add the two? What do you think the difference is then. 2 and 1 are different numbers with a difference of 1, 0.999 and 1 are different numbers with a difference of 0.001, what is the difference between 0.999... and 1 if there is one? Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 I will again point out that 0.999... means that the nines go on forever, like 1/3=0.333... Why did you add the two? What do you think the difference is then. 2 and 1 are different numbers with a difference of 1, 0.999 and 1 are different numbers with a difference of 0.001, what is the difference between 0.999... and 1 if there is one? 0.001, like you say. Link to comment Share on other sites More sharing options...
Albert Posted October 14, 2012 Share Posted October 14, 2012 0.001, like you say. Again, 0.999... is not 0.999, it's 0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999... and so forth, hence the ellipsis. It just keeps going. So no, 0.001 isn't the difference. There is no difference as the 9s never stop. Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 Again, 0.999... is not 0.999, it's 0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999... and so forth, hence the ellipsis. It just keeps going. So no, 0.001 isn't the difference. There is no difference as the 9s never stop. Dammit, that's what those dots mean. The difference between 1 and 0.999... is 0.001, 0.0001, 0.00001, 0.000001 and so on. In numerical terms, they are never the same. In a physical sense they are. At least, this is the only way my puny mind can 'picture' it. Does 0.000...1 work? Link to comment Share on other sites More sharing options...
Albert Posted October 14, 2012 Share Posted October 14, 2012 Dammit, that's what those dots mean. The difference between 1 and 0.999... is 0.001, 0.0001, 0.00001, 0.000001 and so on. In numerical terms, they are never the same. In a physical sense they are. At least, this is the only way my puny mind can 'picture' it. Does 0.000...1 work? Well... as I said, in numerical terms they are the same number. Again, there's no getting around this. This isn't about the physical sense of anything, this is entirely about the numbers. Also, no mind is puny, these sorts of things take a bit of thought. Think about the number you gave: 0.000...1 What do you think is interesting here? Think about it, you are leaving a 1 on the end after an infinite number of 0s, but does this make sense? If there was an end, this by definition means that it's not an infinite number of 0s, so that means it wasn't an infinite number of 9s, so its not recurring at all. That's the key here, they keep going, they never end. Think about it this way: 1-0.999...=0.000... There is no end to the 0s because there is no end to the 9s. Of course, 0.000... is just 0, but maybe this way of looking at it might help. Interestingly though, the idea of 0.000...1 actually goes to the heart of this. That's part of what was talked about, although in a more complicated manner. Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 Well... as I said, in numerical terms they are the same number. Again, there's no getting around this. This isn't about the physical sense of anything, this is entirely about the numbers. Also, no mind is puny, these sorts of things take a bit of thought. Think about the number you gave: 0.000...1 What do you think is interesting here? Think about it, you are leaving a 1 on the end after an infinite number of 0s, but does this make sense? If there was an end, this by definition means that it's not an infinite number of 0s, so that means it wasn't an infinite number of 9s, so its not recurring at all. That's the key here, they keep going, they never end. Think about it this way: 1-0.999...=0.000... There is no end to the 0s because there is no end to the 9s. Of course, 0.000... is just 0, but maybe this way of looking at it might help. Interestingly though, the idea of 0.000...1 actually goes to the heart of this. That's part of what was talked about, although in a more complicated manner. There is no end. 0.000...1, and the 1 is always constantly expanding away from the 0. beginning. Maybe that's my error. Thinking of it in expansion terms. Link to comment Share on other sites More sharing options...
Albert Posted October 14, 2012 Share Posted October 14, 2012 There is no end. 0.000...1, and the 1 is always constantly expanding away from the 0. beginning. Maybe that's my error. Thinking of it in expansion terms. ...wait... expanding? 0.999... isn't changing, it just keeps going. There's no change, its not like its moving or something. It's not that there is a large number of 9s following the decimal point, there is an infinite number of them, there was an infinite number of them and there will continue to be an infinite number of them. The value of 1 isn't changing. Just remember, 0.000...1 implies that there is an end. The decimal is recurring, so by definition there isn't. Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 ...wait... expanding? 0.999... isn't changing, it just keeps going. There's no change, its not like its moving or something. It's not that there is a large number of 9s following the decimal point, there is an infinite number of them, there was an infinite number of them and there will continue to be an infinite number of them. The value of 1 isn't changing. Just remember, 0.000...1 implies that there is an end. The decimal is recurring, so by definition there isn't. I was going to say it's permanent, rather than expanding, but I didn't want to frustrate you further with my physical representation of figures. Ha. Link to comment Share on other sites More sharing options...
mozza Posted October 14, 2012 Share Posted October 14, 2012 is it still sunday ? Link to comment Share on other sites More sharing options...
GboroRam Posted October 14, 2012 Share Posted October 14, 2012 I think it's the concept of infinite that people can't grasp. Link to comment Share on other sites More sharing options...
-JW- Posted October 14, 2012 Share Posted October 14, 2012 I bloody hate international breaks. Link to comment Share on other sites More sharing options...
Oxram Posted October 14, 2012 Share Posted October 14, 2012 I think the problem maybe the words being used you say 0.9999 is the same number as 1 rather than they both represent a whole unit. 0.33333 isn't truly a third (a failure of the decimal system) but is used to represent a third. Hence 0.99999 represents 3/3 and as such is equal to 1 just represent reoccurring decimals as fractions and everyone will get along i don't follow your no number between them argument, surely, an infinite number of decimal places down, every number is inseparable from the next and they would all merge into one indistinguishable number? what is the next smallest number after 0.99999 or does that merge to 0.99999 as there is nothing between them? surely this would repeat out of control Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 is it still sunday ? It's only twenty past six, aside from trying to fathom infinity and bloody numbers I've been writing a 1500 word essay. It's only bloody twenty past six, I feel like this Sunday has been eternal. Link to comment Share on other sites More sharing options...
davenportram Posted October 14, 2012 Author Share Posted October 14, 2012 It's only twenty past six, aside from trying to fathom infinity and bloody numbers I've been writing a 1500 word essay. It's only bloody twenty past six, I feel like this Sunday has been eternal. Surely that's a good thing if your writing an essay. On my way to st Ives I met a man with seven wives. Each wife had seven brothers and each brother had seven wives. How many men did I meet. Link to comment Share on other sites More sharing options...
blackNwhites Posted October 14, 2012 Share Posted October 14, 2012 Surely that's a good thing if your writing an essay. On my way to st Ives I met a man with seven wives. Each wife had seven brothers and each brother had seven wives. How many men did I meet. 50, 350, 2450...? Or [size=2]∞[/size] Link to comment Share on other sites More sharing options...
Recommended Posts
Archived
This topic is now archived and is closed to further replies.