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| Posted By: Tracee Posted On: Nov 11, 2004 Views: 2166 | Dumb Thought/Question I have ok...so 1/3+2/3 of course equals 1. 1/3=.3333(infinatly) 2/3=.6666(infinatly) but shouldn't .3333(infinatly)+ .6666(infinatly)= .9999(infinatly). am I stupid and missing something very obvious, or isn't there something wrong with this? Does anyone understand what I am getting at? Any comments would be very helpful!!! |
| Posted By: Bill Posted On: Nov 13, 2004 Views: 2164 | RE: Dumb Thought/Question I have First, there are no dumb questions. =) I see it as this... you are either approaching 1 or 0 in your description. This is a kind of domain or range. I forget... What you're describing is something like focusing in on one tiny particular region of the whole and focusing down infinitely. So as I see it, you're thinking about 1 dimension of infinite degrees. |
| Posted By: Tracee Posted On: Nov 14, 2004 Views: 2161 | RE: Dumb Thought/Question I have Thanx for trying to answer my question, but I beleive I did a very poor job explaining exactly what my question is...let me see if I can re-explain. ok, if 1/3+2/3=1, why is it that if 1/3=.3repeating and 2/3=.6repeating the following equation would not agree: .3repeating+.6repeating DOES NOT equal exactly 1, but instead .9repeating. Does this make any sense as to what I am asking, I am sure the answer is obvious, math is not my forte. |
| Posted By: Keith Mayes Posted On: Nov 15, 2004 Views: 2159 | RE: Dumb Thought/Question I have Hi Tracee, The only reason your sums do not appear to make sense is because we are making approximations when we use decimals. Look at it in a simple way, it makes it easier to understand, and easier for me to explain. 1/3 of something is precisely one third. So divide a cake into three equal portions and each portion is exactly one third of the whole cake. If however, you decide to change your method of calculation and go decimal, it will not quite work. You can't exactly express 1/3 as a decimal, because as you say it is 0.3333333'. Therefore two slices is 0.6666666'. Because this conversion to decimal is not precise, it looks as if the sums are wrong, but they are not, its just that decimals are not fractions. The three slices still make a complete whole cake, nothing has gone missing of course, its just the degree of accuracy used is different. Fractions are exact, 1/3 means 1 divided by 3. Decimals are a more useful way of expressing and handling numbers, but technically are not as accurate. Keith |
| Posted By: Tracee Posted On: Nov 15, 2004 Views: 2155 | RE: Dumb Thought/Question I have Thank you for the explanation! |
| Posted By: Pyrovus Posted On: Dec 21, 2004 Views: 2131 | RE: Dumb Thought/Question I have Actually it can be proven that .9999' exactly equals 1: let x = .99999' = 9/10 + 9/100 + 9/1000 + 9/10000 . . . Multiply both sides by 10: :. 10x= 9 + 9/10 + 9/100 + 9/1000 + 9/10000 . . . substitute 9/10 + 9/100 + 9/1000 . . . = x: :. 10x= 9 + x :. 9x= 9 :. x= 1 |
| Posted By: Keith Mayes Posted On: Dec 21, 2004 Views: 2129 | RE: Dumb Thought/Question I have Hi Pyrovus, Nice try, but the fatal flaw in your maths is screaming out from the page. You begin by saying.... let x = .99999 Okay, why not. Then four lines down you change the value of X........ substitute 9/10 + 9/100 + 9/1000 . . . = x Err, sorry no, we cant do that. What you have done is to begin by giving x a decimal value, then later on change your mind and it give fractional value. How then we can be surprised that by doing this you show that the decimal is equal to the fraction? Sorry, but 0.999999 does not equal 1, and no amount of tinkering will ever make it so. |
| Posted By: Pyrovus Posted On: Jan 3, 2005 Views: 2118 | RE: Dumb Thought/Question I have Except that both values of x are the same. When we write a number, say 12,345, as a result of the way place value notation works, what we are really writing is 1x10^4 + 2x10^3 +3x10^2 +4x10^1 +5x10^0 (read 1 lot of ten thousand plus 2 lots of one thousand etc.) Likewise, decimal numbers like .1724 can be rewritten as 1x10^-1 + 7x10^-2 + 2x10^-3 + 4x10^-4, or 1x0.1 + 7x0.01 + 2x.001 + 4x.0001, or 1/10 + 7/100 + 2/1000 + 4/10000 (.1 = 1/10, .01 = 1/100 etc). All these forms are equivalent. Going back to .9999999' We can write this as an infinite sum: .9999999'=.9 +.09 +.009 +.0009 . . . forever and as .9=9/10, .09=9/100 etc then .999999'= 9/10 + 9/100 + 9/1000 etc They both have the same value, even though they are written differently, much in the same way that sin(pi/2) and cos(0), both represent the same number and therefore may be interchanged. And another thing. The interval between any two different numbers can be infinitely divided eg. there are infinitely many numbers between one and two. However, there are no numbers between 0.999' and 1, so they cannot be different numbers. |
| Posted By: Keith Mayes Posted On: Jan 3, 2005 Views: 2115 | RE: Dumb Thought/Question I have I don't know why you are having such a problem with this, it's very simple. I think that your quote below sums up where you are going wrong, and it is basically the same as you said before. "Going back to .9999999' We can write this as an infinite sum: .9999999'=.9 +.09 +.009 +.0009 . . . forever and as .9=9/10, .09=9/100 etc then .999999'= 9/10 + 9/100 + 9/1000 etc" We CAN say 0.9 = 9/10. because 9/10 simply means 9 divided by ten, and the answer is a real and precise number of 0.9. It is called a whole number, even though in this example it is expressed as a decimal. The whole number 0.9 is equal to the fraction 9 divided by 10. Fine. What you CANNOT do is compare a whole number with an irrational number. An irrational number is one that has no end, such as pi for example. This is because a fraction means what it says, such as 22/7. It represents whatever value 22 divided by 7 is. The answer, when you actually do the division, results in a decimal that can be expressed as 3.1428571... and has no end, it is an irrational number. The longer you keep dividing and the longer the answer becomes, the closer it is getting to be an accurate answer to 22 divided by 7, but it can never be exactly right, can never be exactly the same, because it never ends. It is obvious for example that 9/10 is not the same as 1. It is obvious that if we add to it 9/100 the answer still isnt 1. It should also be obvious that no matter how long we continue this process we can see that the answer can never be 1. What you are saying is that at some point it does become 1, and as it equals 0.99999999etc this means that also equals 1. Cant you see that is seriously flawed? Your entire argument is wrong because you think that a whole number is equal to an irrational number. It isnt, ask a mathematician. Why don't you look it up? You really do not to look up rational and irrational numbers because you do not understand them. You could also read my section on mathematics. |
| Posted By: Pyrovus Posted On: Jan 3, 2005 Views: 2112 | RE: Dumb Thought/Question I have Perhaps you should check up a bit about infinite sums. Quote: "The longer you keep dividing and the longer the answer becomes, the closer it is getting to be an accurate answer to 22 divided by 7, but it can never be exactly right, can never be exactly the same, because it never ends." I agree that the more digits you calculate 22/7 to, the less the error between the calculated value and the actual. You can never write down the exact value with a finite number of terms. However, mathematics is capable of dealing with an infinite number of terms, for instance 22/7 can be represented by: infinity 3+ Ó 142857/10^6n n=1 You were arguing that I cannot represent .99999' as a sum of fractions because it's value can never be defined by a finite number of terms. However, I was using an infinite number of terms. Oh, BTW 22/7 is not an irrational number, as it can be expressed as a fraction in which numerator and denominator are both integers. Irrational numbers CAN also be represented by an infinite number of rational terms using Maclaurin series. For instance, the Maclaurin series for e^x is 1 + x + (1/2!)x^2 +(1/3!)x^3 +(1/4!)x^4 . . . The more terms we take the closer it gets to e^x, and if we take an infinite number of terms it equals e^x. Quote: "It is obvious for example that 9/10 is not the same as 1. It is obvious that if we add to it 9/100 the answer still isnt 1. It should also be obvious that no matter how long we continue this process we can see that the answer can never be 1. What you are saying is that at some point it does become 1, and as it equals 0.99999999etc this means that also equals 1. Cant you see that is seriously flawed?" I am not saying that as soon as you calculate .99999' to say, the 10^96th digit it would suddenly equal one; rather that if calculated to an infinite number of digits .99999'=1. The reason why you can't get one by continually adding 9/successively high powers of ten is that it is simply not possible to do such a process forever. However, if you could, you would get one. How about we examine what 1 - 0.9999999' is: We can do this by first noting that 1-.9 = 10^-1, 1-.99 = 10^-2, 1-.999 = 10^-3 etc. So in general, 1 - .9(to n places) = 10^-n As we make n larger and larger, we get closer and closer to 1 - .999999'. So in other words, to find 1 - .99999', we must find Lim n->infinity 1/10^n = 0 Or in other words, .99999'=1. |
| Posted By: Pyrovus Posted On: Jan 3, 2005 Views: 2110 | RE: Dumb Thought/Question I have CORRECTION Just with this bit the Ó is supposed to be a sigma. The software didn't seem to like it. infinity 3+ Ó 142857/10^6n n=1 |
| Posted By: Keith Mayes Posted On: Jan 4, 2005 Views: 2106 | RE: Dumb Thought/Question I have Good grief, you do go on and on! Okay, I am sure you must be right. OF COURSE 0.99999999' is equal to 1 Every idiot knows that. |
| Posted By: .9 repeating does NOT equal 1 Posted On: Feb 4, 2005 Views: 2063 | RE: Dumb Thought/Question I have You made a fatal mistake in your theory. 1 != .999999999' As we learn from derivatives and anti-derivatives, every time you derive you either lose a constant or add one (+ C). You failed to account for this. Your answer should be: 3x^2 = 6x, but the anti-derivative of 6x = 3x^2 + C |
| Posted By: Keith Mayes Posted On: Feb 24, 2005 Views: 2033 | RE: Dumb Thought/Question I have Thanks for your confirmation that 0.99999999' does not equal 1 I am still at a loss to understand how anyone could think otherwise! No amount of maths can make 1 equal to something that by defintion is less than 1. |
| Posted By: Alan Posted On: Mar 1, 2005 Views: 2021 | RE: Dumb Thought/Question I have WOAH!!! ARE YOU TRYING TO MAKE MY HEAD EXPLODE!? |
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