<!--QuoteBegin-xect+Jun 20 2004, 10:38 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (xect @ Jun 20 2004, 10:38 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Step 2 and 3 in "I gorged..."'s socalled 'proof' is a fallacy.
Can't juggle with infinite numbers unless you're dealing with limit values (at least I think that's what it's called, not sure of the english term)
0,<u>9999</u> != 1
What you're basically proving is that
lim(x) = 1 x-->1-
And well, that's pretty obvious.
Unless you want me to prove that the hyperbola f(x)=1/x collides with the x-axis in infinity? Which pretty much defies definitions. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> it does... That's the definition of limits. IF you could reach infinity (which you can't) then 1/X would be 0. However, we cannot reach infinity, thus the hyperbola does NOT collide with the x axis in any forseeable future. However, it does hit it in infinity, except for the fact that you can't reach infinity.
because space probes crash on mars with such mistakes <!--emo&:0--><img src='http://www.unknownworlds.com/forums/html//emoticons/wow.gif' border='0' style='vertical-align:middle' alt='wow.gif' /><!--endemo-->
Exactly. Fractions made for repeating numbers can be exact, also. Decimals cannot. For example, pi = 22/7, IIRC. Something like that. However, it does not equal the 3.14 and so on thing. That is just an approximation. The only way to be exact is to use a fraction or the symbol.
There is a property that says a number is always equal to itself and NOTHING else.
And this proof is all you SHOULD need to believe that point nine repeating equals one. Because point nine repeating IS, by DEFNITION,
lim x->inf σ(i=0,x) 1/(9*10^i)
Do you want the delta-epsilon proof? I can offer that. Do you want to see the math worked out? I can offer that. 9/9 = 1. 9/9 != an approximation of 1, it is, by defnition, 1. .99999999999999999999999999999 is an approximation. .9 repeating IS 1.
Definition of the word IS in this context: there is no difference between.
There is no difference between .9 repeating and 1. Literally.
Subtract .9 from 1 (difference). You get 0.01. Now subtract .99 from 1. That equals .001. You can see an instant pattern. Count the number of "9"s you're subtracting, and put that many 0s in the answer, and then tag on a 1 at the end.
Subtract point nine repeating from 1. This means that the answer would be an unending stream of zeros, with a 1 at the end. Wait a minute... unending... end? If we just said an UNENDING amount of zeros, then there IS no end. Thus, there IS NO DIFFERENCE between .9repeating and 1.
Verbal that there is no "smallest number greater than zero": <!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->hey, what's the smallest number greater than zero? <i>think of a number greater than zero.</i> is this like a magic trick? <i>in a way, yes.</i> cool! i like magic tricks! alright, got one! <i>halve it</i> uhuh <i>halve it again</i> yup <i>halve it again</i> ok...
<b>*** several minutes later ***</b>
<i>halve it again</i> um, how long is this going to take?
<b>*** several hours later ***</b>
<i>halve it again</i> are you following me?!?!??!
<b>*** several weeks later ***</b>
<i>halve it again</i> please! i can't take it anymore! can't we just call it quits and round it down to zero?!?!?! <i>certainly not! halve it again...</i> arghhhhhhh!!!!!!<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Just as there is no biggest number, because you can always add 1, there is no smallest number greater than zero, because you can always halve it. Or quarter it for that matter, or decimate (divide by 10) it, etc etc etc. Since said smallest number doesn't exist, point nine repeating DOES equal 1.
Here's some other interesting things that are related.
To be perfectly clear right here and now: Infinity is a word that indicates endlessness.
Where c is any constant, real number
∞ ± c = ∞ Yes, this means that ∞ + 1 is no bigger than ∞, sorry to disillusion your childhood there. For that matter, ∞ - 1 is no smaller than ∞ either.
∞ ± ∞ = ∞ This may come to a surprise to some of you, who are still thinking that ∞ is a discrete number that is in any way entitled to obey the rules that every real number in existence obeys. Get over this idea fast. Consider the following example: take the set of all real integers, which equals ∞. Now remove all the odd numbers. Since the set of all odd numbers also equals ∞, you have just subtracted ∞ from ∞, and yet you still have the set of all even numbers left, which is also equal to ∞. Going the other way, if you take the set of real integers and add all the halves in, you get a set which is still just ∞, no bigger or smaller than ∞
In a way, when we're subtracting all the odd numbers from the set of all real integers, we're really just crossing out ever other number, which is the same thing as diving by two. And the reverse, which is the same thing as doubling. Which leads directly into:
∞ / c = ∞ ∞ * c = ∞ But it gets better. Going back to the example of adding in all the halves in between, so the set would go 1, 1.5, 2, 2.5, ..., ∞, you could continue on by adding all the thirds, and fourths, and fifths, and sixths, and so on forever, and yet still equal infinity. As such: ∞ * ∞ = ∞ ∞ / ∞ = ∞ Please note that the above does not include any limits. Limits are covered later in the node. For now, we're talking conceptually about pure infinity in the manner that I described above. Anyway, it follows from this that...
∞^c = ∞
Why does it matter? Well, do you, the simpleton, it doesn't. But if you want to MAKE something... say, a rocketship that goes to the moon, or a partical beam weapon, or try and figure out if a time machine is possible, and if it is, build one, or if you want to make your DVD player smaller and faster, or if you want to make an aimbot for the real world to equip your tanks and soldiers with, you'd BETTER KNOW THIS STUFF or you'll fail miserably at your attempts. Want proof of THIS statement? Go to college. Take a physics course. Yes. It is needed. That's why I care.
By the way...
∞^∞ > ∞
Just to blow your mind away. This is saying that the realm of 2d is smaller than the realm of 3d, and 4d is larger than 3d, etc etc etc onwards forever.
Oh, I just found an amusing little anecdote... <!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Using base-12 arithmetic: 1/3 = 4/12 = 0.4 2/3 = 8/12 = 0.8 3/3 = 12/12 = 1.0 God only knows what this proves, except that 1 = 0.999 recurring is just a stupid artefact of our tiny minds and our poorly selected numerical system. If we just had an extra finger on each hand we might have avoided that particular one. Let us all work in duodecimal from now on and never speak of this again.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> While this is true... (and since we're using duodecimal, A=10, B=11)
.B repeating equals 1.
In trinary, .2 repeating = 1.
In octal, .7 repeating = 1.
In hex, .F repeating = 1.
Any base system is capable of having repeating digit numbers of this nature. They all have a statement where something repeating = 1. That's the way it goes.
<!--emo&:0--><img src='http://www.unknownworlds.com/forums/html//emoticons/wow.gif' border='0' style='vertical-align:middle' alt='wow.gif' /><!--endemo--> that was time well spent Fieari
<!--QuoteBegin-I Gorged Your Mom+Jun 20 2004, 01:20 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (I Gorged Your Mom @ Jun 20 2004, 01:20 AM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> 3. subtract x from the left, 0.9 repeating from the right, giving
9x = 9
4. divide both by nine, giving
x = 1
5. yet x = 0.9 repeating 6. therefore
0.9 repeating = 1 <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> You cant do that. You cannot subtract x from one side and 0.99... from the other side, you'll have to subtract either 0.99... from both sides or x from both sides, since it IS a mathematical rule that you can only do THE EXACT SAME THING on both sides, and -x != -0.99..., even if they DO represent the exact same value. This is since x is a unknown value. That means that you'll have to use either the unknown variable x OR the number that x represent, you cannot use one in each lead, you'll have to use the same in both.
So this means that step 3 would be a) 9x = 9.99.. -x b) 10x - 0.99... = 9.99... - 0.99... = 10x - 0.99... = 9
If you want to solve this equation then it will be 10x - 0.99... + 0.99... = 9.99... -> 10x/10 = 9.99.../10 -> x = 0.99 and we're back where we started.
This is just a way to fool people's logical-thinking by using incorrect mathematical rules.
GG I win now please lock the thread. <!--emo&::nerdy::--><img src='http://www.unknownworlds.com/forums/html//emoticons/nerd.gif' border='0' style='vertical-align:middle' alt='nerd.gif' /><!--endemo-->
<!--QuoteBegin-Zig+Jun 21 2004, 01:01 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (Zig @ Jun 21 2004, 01:01 AM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> i don't know why people are saying .33_ and .66_ and .99_ don't exist.
use a compass and measure a circle into three equal parts.
wow.
how hard was that. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> Each segment of the circle wont be 0.33_, it will be 1/3. 0.33_ != 1/3, but however it is the value that is beeing used for expressing 1/3 in decimals.
You see, 0.33_ isnt 1/3 because; 1/3 + 1/3 + 1/3 = 3/3 = 1/1 = 1 while 0.33_ + 0.33_ + 0.33_ = 0.99_ and 0.99_ != 1 as I explained in my previous post.
Of course the values exist, but they arent equal to 1/3, 2/3 or 3/3.
<!--QuoteBegin-coris+Jun 21 2004, 12:43 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (coris @ Jun 21 2004, 12:43 AM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> <!--QuoteBegin-Zig+Jun 21 2004, 01:01 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (Zig @ Jun 21 2004, 01:01 AM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> i don't know why people are saying .33_ and .66_ and .99_ don't exist.
use a compass and measure a circle into three equal parts.
wow.
how hard was that. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> Each segment of the circle wont be 0.33_, it will be 1/3. 0.33_ != 1/3, but however it is the value that is beeing used for expressing 1/3 in decimals.
You see, 0.33_ isnt 1/3 because; 1/3 + 1/3 + 1/3 = 3/3 = 1/1 = 1 while 0.33_ + 0.33_ + 0.33_ = 0.99_ and 0.99_ != 1 as I explained in my previous post.
Of course the values exist, but they arent equal to 1/3, 2/3 or 3/3. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> Coris, .3 repeating DOES equal 1/3. There is no approximation, no fuzzyness. .333 is an aproximation, .333333333333333 is an approximation, .33333333333333333333333333333333333333333333333333333333333333333333333333333333 is an approximation, but .3 repeat FOREVER is equal to EXACTLY one third.
Just as 4/12 is EXACTLY 1/3, and is represented by .4 in a duodecimal system.
Coris, do you know what a limit is? It's a concept that calculus was invented for, and it makes physics possible. We went to the moon because of calculus. It is EMPIRICALLY verified. The numbers are made to match the world, not the other way around. The rules are rules.
The limit of a function as x approaches n means that as x becomes closer and closer to n, the value of the function evaluated at x is closer and closer to the value of the limit. Get out your graphing calculator... graph "y=x+1"
As x gets closer and closer to 0, y gets closer and closer to 1. You can check this. Go to trace, type in .1, and you'll see a value pretty close to 1, but not exact. Type in .01, and you'll get a number closer to 1. .001 will get a number closer to 1 still. Come up with ANY number close to 0, and I can come up with a number even closer to zero and get a value closer to 1. We need never actually make x=0 for this to work.
What you now need to understand is, that any "Repeating" number IS A LIMIT. Specifically, if "r" is the repeating digit, than it's the limit represented by:
lim x x->inf Σ (1/(r*10^i) ) i=0
No approximation here, this is LITTERALLY what .r repeating is, by the definition of Σ, which is:<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->If am, am+1, am+2, ... an are real numbers and m and n are integers such that m ≤ n, then n Σ a[sub]i[/sub] = a[sub]m[/sub] + a[sub]m[/sub]+1 + a[sub]m[/sub]+2 + ... a[sub]n[/sub]-1 + a[sub]n[/sub] i=m<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Understand that when dealing with limits, your dealing with rules WE UNDERSTAND and CAN MANIPULATE. This is no chicanery, this is observable fact. The fact that I can quickly answer min/max problems, the fact that I can calculate tragectories, the fact that acceleration is based on velocity is based on position... all follow these same rules. The rules can't just apply to some things and not others.
We can explain the fact that .9 repeating is in fact, equal to EXACTLY 1 in all kinds of ways, but the fundamental thing that makes it SURE, is the fact that calculus works. Denying that .9repeating equals 1 is denying the existance of calculus. You can't do that! Calculus works! It -always- works.
<!--QuoteBegin-Omegaman!+Jun 20 2004, 10:27 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (Omegaman! @ Jun 20 2004, 10:27 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Dang, people. I really didnt expect the thread to live this long. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> This is the NS.org forums, they are unlike any other.
<i>Assume nothing.</i> ------------------ LOL Sorry I just always wanted to say that.
The way i see it, .9repeating IS one, as the "repeating" concept is a limit.
Limits are cool things. Limits and integrals can find the area under a curve of a line that is undefined on the x axis... that's pretty cool.
It's the idea of "it never goes here, but if it ever did, here i as far as it would go." and .99999999 never goes to 1, but the "repeating" concept means that if it could jump up to it's limit, it would be 1.
And this proof is all you SHOULD need to believe that point nine repeating equals one. Because point nine repeating IS, by DEFNITION,
lim x->inf σ(i=0,x) 1/(9*10^i)
Do you want the delta-epsilon proof? I can offer that. Do you want to see the math worked out? I can offer that. 9/9 = 1. 9/9 != an approximation of 1, it is, by defnition, 1. .99999999999999999999999999999 is an approximation. .9 repeating IS 1.
Definition of the word IS in this context: there is no difference between.
There is no difference between .9 repeating and 1. Literally.
Subtract .9 from 1 (difference). You get 0.01. Now subtract .99 from 1. That equals .001. You can see an instant pattern. Count the number of "9"s you're subtracting, and put that many 0s in the answer, and then tag on a 1 at the end.
Subtract point nine repeating from 1. This means that the answer would be an unending stream of zeros, with a 1 at the end. Wait a minute... unending... end? If we just said an UNENDING amount of zeros, then there IS no end. Thus, there IS NO DIFFERENCE between .9repeating and 1.
Verbal that there is no "smallest number greater than zero": <!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->hey, what's the smallest number greater than zero? <i>think of a number greater than zero.</i> is this like a magic trick? <i>in a way, yes.</i> cool! i like magic tricks! alright, got one! <i>halve it</i> uhuh <i>halve it again</i> yup <i>halve it again</i> ok...
<b>*** several minutes later ***</b>
<i>halve it again</i> um, how long is this going to take?
<b>*** several hours later ***</b>
<i>halve it again</i> are you following me?!?!??!
<b>*** several weeks later ***</b>
<i>halve it again</i> please! i can't take it anymore! can't we just call it quits and round it down to zero?!?!?! <i>certainly not! halve it again...</i> arghhhhhhh!!!!!!<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Just as there is no biggest number, because you can always add 1, there is no smallest number greater than zero, because you can always halve it. Or quarter it for that matter, or decimate (divide by 10) it, etc etc etc. Since said smallest number doesn't exist, point nine repeating DOES equal 1.
Here's some other interesting things that are related.
To be perfectly clear right here and now: Infinity is a word that indicates endlessness.
Where c is any constant, real number
∞ ± c = ∞ Yes, this means that ∞ + 1 is no bigger than ∞, sorry to disillusion your childhood there. For that matter, ∞ - 1 is no smaller than ∞ either.
∞ ± ∞ = ∞ This may come to a surprise to some of you, who are still thinking that ∞ is a discrete number that is in any way entitled to obey the rules that every real number in existence obeys. Get over this idea fast. Consider the following example: take the set of all real integers, which equals ∞. Now remove all the odd numbers. Since the set of all odd numbers also equals ∞, you have just subtracted ∞ from ∞, and yet you still have the set of all even numbers left, which is also equal to ∞. Going the other way, if you take the set of real integers and add all the halves in, you get a set which is still just ∞, no bigger or smaller than ∞
In a way, when we're subtracting all the odd numbers from the set of all real integers, we're really just crossing out ever other number, which is the same thing as diving by two. And the reverse, which is the same thing as doubling. Which leads directly into:
∞ / c = ∞ ∞ * c = ∞ But it gets better. Going back to the example of adding in all the halves in between, so the set would go 1, 1.5, 2, 2.5, ..., ∞, you could continue on by adding all the thirds, and fourths, and fifths, and sixths, and so on forever, and yet still equal infinity. As such: ∞ * ∞ = ∞ ∞ / ∞ = ∞ Please note that the above does not include any limits. Limits are covered later in the node. For now, we're talking conceptually about pure infinity in the manner that I described above. Anyway, it follows from this that...
∞^c = ∞
Why does it matter? Well, do you, the simpleton, it doesn't. But if you want to MAKE something... say, a rocketship that goes to the moon, or a partical beam weapon, or try and figure out if a time machine is possible, and if it is, build one, or if you want to make your DVD player smaller and faster, or if you want to make an aimbot for the real world to equip your tanks and soldiers with, you'd BETTER KNOW THIS STUFF or you'll fail miserably at your attempts. Want proof of THIS statement? Go to college. Take a physics course. Yes. It is needed. That's why I care.
By the way...
∞^∞ > ∞
Just to blow your mind away. This is saying that the realm of 2d is smaller than the realm of 3d, and 4d is larger than 3d, etc etc etc onwards forever.
Oh, I just found an amusing little anecdote... <!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Using base-12 arithmetic: 1/3 = 4/12 = 0.4 2/3 = 8/12 = 0.8 3/3 = 12/12 = 1.0 God only knows what this proves, except that 1 = 0.999 recurring is just a stupid artefact of our tiny minds and our poorly selected numerical system. If we just had an extra finger on each hand we might have avoided that particular one. Let us all work in duodecimal from now on and never speak of this again.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> While this is true... (and since we're using duodecimal, A=10, B=11)
.B repeating equals 1.
In trinary, .2 repeating = 1.
In octal, .7 repeating = 1.
In hex, .F repeating = 1.
Any base system is capable of having repeating digit numbers of this nature. They all have a statement where something repeating = 1. That's the way it goes. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> <!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->x->inf σ(i=0,x) 1/(9*10^i) = 1<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> ..... that's 0.<u>1111</u> + 0.0<u>1111</u> + 0.00<u>1111</u> + 0.000<u>1111</u>...... It doesn't even get close to 1
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->There is no difference between .9 repeating and 1. Literally.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> A lie, there is a difference, but is infinitessimaly small (like a singularity.... you know... those black hole things)
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Subtract .9 from 1 (difference). You get 0.01. Now subtract .99 from 1. That equals .001. You can see an instant pattern. Count the number of "9"s you're subtracting, and put that many 0s in the answer, and then tag on a 1 at the end.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> 1 - .9 != 0.01 ....... 1 - .9 = 0.1 1 - .99 != 0.001 .... 1 - .99 = 0.01
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->∞ ± c = ∞<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> You have just proven my point that there are such things as <i>conceptual</i> numbers, and that you cannot perform arithmetic on them.
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->∞^∞ > ∞<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> What orifice did you pull that one out of? ∞^∞ boils down to the following operation done infinitely many times ∞ * ∞ = x->∞ σ(i=0,x) ∞ (which leads to "∞ + ∞" done infinitely many times since "∞ + ∞" = ∞)
and since the base step of ∞ * ∞ has no effect, then the entire system has no effect.
This entire argument hinges on one thing critical point. The people that believe that 0.<u>9999</u> = 1 are those that blindly accept everything they read without thinking about the underlying systems. The rest of us are people that realize that even mathematicians will sometimes have to go "Oh crap, we can't work with that.... Kludge it." (say hello to imaginary numbers and complex hyperplanes) So they came up with systems of approximation to call it "good enough."
EDIT: More inconsistencies found in your original post.
<!--QuoteBegin-DOOManiac+Jun 19 2004, 06:34 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (DOOManiac @ Jun 19 2004, 06:34 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Anything between .9999 and 1 is = 1. Otherwise you have too much time on your hands to worry about that sort of crap. Go do something. :P <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> I stand behind my answer. :)
Fieari, 0.33_ will be very VERY very close to 1/3, but it will not be exactly 1/3, it will be slightly smaller, since 0.33_+0.33_0.33_ = 0.99_ and 0.99_ != 1.
But [WHO]Them is absolutly right, you actually cant count with infinite numbers since any result you get will be infinite.
But I dont know how many times I have to prove my point, 0.99 in infinite isn't one, though, it is as close you can get to 1.
DOOManiac is right.
"Anything between .9999 and 1 is = 1. Otherwise you have too much time on your hands to worry about that sort of crap. Go do something. <!--emo&:p--><img src='http://www.unknownworlds.com/forums/html//emoticons/tounge.gif' border='0' style='vertical-align:middle' alt='tounge.gif' /><!--endemo-->"
you're only thinking as if it's stopping at some time. It doesn't. That's what infinity is, never stopping. And thus 1/3 = 0.333 repeating forever and never stopping. And that is a precise value as long as you never stop.
You can count with infinitely long numbers. Such as Pi
1x Pi = Pi and not 1x Pi = infinity
0.99 in infinity is one. It's hard to picture, but it is. There is no "as close as you can get to 1". There is exactly one, or not one.
And [WHO]Them , I think I'd still trust my teachers and people from universities with some sort of diploma or something. So far I'm convinced they are equal. Saying "they are different" isn't quite as convincing than a full demonstration.
Comments
Can't juggle with infinite numbers unless you're dealing with limit values (at least I think that's what it's called, not sure of the english term)
0,<u>9999</u> != 1
What you're basically proving is that
lim(x) = 1
x-->1-
And well, that's pretty obvious.
Unless you want me to prove that the hyperbola f(x)=1/x collides with the x-axis in infinity? Which pretty much defies definitions. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
it does... That's the definition of limits. IF you could reach infinity (which you can't) then 1/X would be 0. However, we cannot reach infinity, thus the hyperbola does NOT collide with the x axis in any forseeable future. However, it does hit it in infinity, except for the fact that you can't reach infinity.
1 = 1
Enough said.
There is a property that says a number is always equal to itself and NOTHING else.
lim
x->inf σ(i=0,x) 1/(9*10^i) = 1
And this proof is all you SHOULD need to believe that point nine repeating equals one. Because point nine repeating IS, by DEFNITION,
lim
x->inf σ(i=0,x) 1/(9*10^i)
Do you want the delta-epsilon proof? I can offer that. Do you want to see the math worked out? I can offer that. 9/9 = 1. 9/9 != an approximation of 1, it is, by defnition, 1. .99999999999999999999999999999 is an approximation. .9 repeating IS 1.
Definition of the word IS in this context: there is no difference between.
There is no difference between .9 repeating and 1. Literally.
Subtract .9 from 1 (difference). You get 0.01. Now subtract .99 from 1. That equals .001. You can see an instant pattern. Count the number of "9"s you're subtracting, and put that many 0s in the answer, and then tag on a 1 at the end.
Subtract point nine repeating from 1. This means that the answer would be an unending stream of zeros, with a 1 at the end. Wait a minute... unending... end? If we just said an UNENDING amount of zeros, then there IS no end. Thus, there IS NO DIFFERENCE between .9repeating and 1.
Verbal that there is no "smallest number greater than zero":
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->hey, what's the smallest number greater than zero?
<i>think of a number greater than zero.</i>
is this like a magic trick?
<i>in a way, yes.</i>
cool! i like magic tricks! alright, got one!
<i>halve it</i>
uhuh
<i>halve it again</i>
yup
<i>halve it again</i>
ok...
<b>*** several minutes later ***</b>
<i>halve it again</i>
um, how long is this going to take?
<b>*** several hours later ***</b>
<i>halve it again</i>
are you following me?!?!??!
<b>*** several weeks later ***</b>
<i>halve it again</i>
please! i can't take it anymore! can't we just call it quits and round it down to zero?!?!?!
<i>certainly not! halve it again...</i>
arghhhhhhh!!!!!!<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Just as there is no biggest number, because you can always add 1, there is no smallest number greater than zero, because you can always halve it. Or quarter it for that matter, or decimate (divide by 10) it, etc etc etc. Since said smallest number doesn't exist, point nine repeating DOES equal 1.
Here's some other interesting things that are related.
To be perfectly clear right here and now: Infinity is a word that indicates endlessness.
Where c is any constant, real number
∞ ± c = ∞
Yes, this means that ∞ + 1 is no bigger than ∞, sorry to disillusion your childhood there. For that matter, ∞ - 1 is no smaller than ∞ either.
∞ ± ∞ = ∞
This may come to a surprise to some of you, who are still thinking that ∞ is a discrete number that is in any way entitled to obey the rules that every real number in existence obeys. Get over this idea fast. Consider the following example: take the set of all real integers, which equals ∞. Now remove all the odd numbers. Since the set of all odd numbers also equals ∞, you have just subtracted ∞ from ∞, and yet you still have the set of all even numbers left, which is also equal to ∞. Going the other way, if you take the set of real integers and add all the halves in, you get a set which is still just ∞, no bigger or smaller than ∞
In a way, when we're subtracting all the odd numbers from the set of all real integers, we're really just crossing out ever other number, which is the same thing as diving by two. And the reverse, which is the same thing as doubling. Which leads directly into:
∞ / c = ∞
∞ * c = ∞
But it gets better. Going back to the example of adding in all the halves in between, so the set would go 1, 1.5, 2, 2.5, ..., ∞, you could continue on by adding all the thirds, and fourths, and fifths, and sixths, and so on forever, and yet still equal infinity. As such:
∞ * ∞ = ∞
∞ / ∞ = ∞
Please note that the above does not include any limits. Limits are covered later in the node. For now, we're talking conceptually about pure infinity in the manner that I described above.
Anyway, it follows from this that...
∞^c = ∞
Why does it matter? Well, do you, the simpleton, it doesn't. But if you want to MAKE something... say, a rocketship that goes to the moon, or a partical beam weapon, or try and figure out if a time machine is possible, and if it is, build one, or if you want to make your DVD player smaller and faster, or if you want to make an aimbot for the real world to equip your tanks and soldiers with, you'd BETTER KNOW THIS STUFF or you'll fail miserably at your attempts. Want proof of THIS statement? Go to college. Take a physics course. Yes. It is needed. That's why I care.
By the way...
∞^∞ > ∞
Just to blow your mind away. This is saying that the realm of 2d is smaller than the realm of 3d, and 4d is larger than 3d, etc etc etc onwards forever.
Oh, I just found an amusing little anecdote...
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Using base-12 arithmetic:
1/3 = 4/12 = 0.4
2/3 = 8/12 = 0.8
3/3 = 12/12 = 1.0
God only knows what this proves, except that 1 = 0.999 recurring is just a stupid artefact of our tiny minds and our poorly selected numerical system. If we just had an extra finger on each hand we might have avoided that particular one.
Let us all work in duodecimal from now on and never speak of this again.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
While this is true... (and since we're using duodecimal, A=10, B=11)
.B repeating equals 1.
In trinary, .2 repeating = 1.
In octal, .7 repeating = 1.
In hex, .F repeating = 1.
Any base system is capable of having repeating digit numbers of this nature. They all have a statement where something repeating = 1. That's the way it goes.
from the right, giving
9x = 9
4. divide both by nine, giving
x = 1
5. yet x = 0.9 repeating
6. therefore
0.9 repeating = 1 <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
You cant do that.
You cannot subtract x from one side and 0.99... from the other side, you'll have to subtract either 0.99... from both sides or x from both sides, since it IS a mathematical rule that you can only do THE EXACT SAME THING on both sides, and -x != -0.99..., even if they DO represent the exact same value. This is since x is a unknown value. That means that you'll have to use either the unknown variable x OR the number that x represent, you cannot use one in each lead, you'll have to use the same in both.
So this means that step 3 would be
a) 9x = 9.99.. -x
b) 10x - 0.99... = 9.99... - 0.99... = 10x - 0.99... = 9
If you want to solve this equation then it will be
10x - 0.99... + 0.99... = 9.99... -> 10x/10 = 9.99.../10 -> x = 0.99 and we're back where we started.
This is just a way to fool people's logical-thinking by using incorrect mathematical rules.
GG I win now please lock the thread. <!--emo&::nerdy::--><img src='http://www.unknownworlds.com/forums/html//emoticons/nerd.gif' border='0' style='vertical-align:middle' alt='nerd.gif' /><!--endemo-->
use a compass and measure a circle into three equal parts.
wow.
how hard was that.
use a compass and measure a circle into three equal parts.
wow.
how hard was that. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
Each segment of the circle wont be 0.33_, it will be 1/3. 0.33_ != 1/3, but however it is the value that is beeing used for expressing 1/3 in decimals.
You see, 0.33_ isnt 1/3 because;
1/3 + 1/3 + 1/3 = 3/3 = 1/1 = 1
while
0.33_ + 0.33_ + 0.33_ = 0.99_ and 0.99_ != 1 as I explained in my previous post.
Of course the values exist, but they arent equal to 1/3, 2/3 or 3/3.
use a compass and measure a circle into three equal parts.
wow.
how hard was that. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Each segment of the circle wont be 0.33_, it will be 1/3. 0.33_ != 1/3, but however it is the value that is beeing used for expressing 1/3 in decimals.
You see, 0.33_ isnt 1/3 because;
1/3 + 1/3 + 1/3 = 3/3 = 1/1 = 1
while
0.33_ + 0.33_ + 0.33_ = 0.99_ and 0.99_ != 1 as I explained in my previous post.
Of course the values exist, but they arent equal to 1/3, 2/3 or 3/3. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
Coris, .3 repeating DOES equal 1/3. There is no approximation, no fuzzyness. .333 is an aproximation, .333333333333333 is an approximation, .33333333333333333333333333333333333333333333333333333333333333333333333333333333 is an approximation, but .3 repeat FOREVER is equal to EXACTLY one third.
Just as 4/12 is EXACTLY 1/3, and is represented by .4 in a duodecimal system.
Coris, do you know what a limit is? It's a concept that calculus was invented for, and it makes physics possible. We went to the moon because of calculus. It is EMPIRICALLY verified. The numbers are made to match the world, not the other way around. The rules are rules.
The limit of a function as x approaches n means that as x becomes closer and closer to n, the value of the function evaluated at x is closer and closer to the value of the limit. Get out your graphing calculator... graph "y=x+1"
As x gets closer and closer to 0, y gets closer and closer to 1. You can check this. Go to trace, type in .1, and you'll see a value pretty close to 1, but not exact. Type in .01, and you'll get a number closer to 1. .001 will get a number closer to 1 still. Come up with ANY number close to 0, and I can come up with a number even closer to zero and get a value closer to 1. We need never actually make x=0 for this to work.
What you now need to understand is, that any "Repeating" number IS A LIMIT. Specifically, if "r" is the repeating digit, than it's the limit represented by:
lim x
x->inf Σ (1/(r*10^i) )
i=0
No approximation here, this is LITTERALLY what .r repeating is, by the definition of Σ, which is:<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->If am, am+1, am+2, ... an are real numbers and m and n are integers such that m ≤ n, then
n
Σ a[sub]i[/sub] = a[sub]m[/sub] + a[sub]m[/sub]+1 + a[sub]m[/sub]+2 + ... a[sub]n[/sub]-1 + a[sub]n[/sub]
i=m<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Understand that when dealing with limits, your dealing with rules WE UNDERSTAND and CAN MANIPULATE. This is no chicanery, this is observable fact. The fact that I can quickly answer min/max problems, the fact that I can calculate tragectories, the fact that acceleration is based on velocity is based on position... all follow these same rules. The rules can't just apply to some things and not others.
We can explain the fact that .9 repeating is in fact, equal to EXACTLY 1 in all kinds of ways, but the fundamental thing that makes it SURE, is the fact that calculus works. Denying that .9repeating equals 1 is denying the existance of calculus. You can't do that! Calculus works! It -always- works.
This is the NS.org forums, they are unlike any other.
<i>Assume nothing.</i>
------------------
LOL Sorry I just always wanted to say that.
Limits are cool things. Limits and integrals can find the area under a curve of a line that is undefined on the x axis... that's pretty cool.
It's the idea of "it never goes here, but if it ever did, here i as far as it would go." and .99999999 never goes to 1, but the "repeating" concept means that if it could jump up to it's limit, it would be 1.
x->inf σ(i=0,x) 1/(9*10^i) = 1
And this proof is all you SHOULD need to believe that point nine repeating equals one. Because point nine repeating IS, by DEFNITION,
lim
x->inf σ(i=0,x) 1/(9*10^i)
Do you want the delta-epsilon proof? I can offer that. Do you want to see the math worked out? I can offer that. 9/9 = 1. 9/9 != an approximation of 1, it is, by defnition, 1. .99999999999999999999999999999 is an approximation. .9 repeating IS 1.
Definition of the word IS in this context: there is no difference between.
There is no difference between .9 repeating and 1. Literally.
Subtract .9 from 1 (difference). You get 0.01. Now subtract .99 from 1. That equals .001. You can see an instant pattern. Count the number of "9"s you're subtracting, and put that many 0s in the answer, and then tag on a 1 at the end.
Subtract point nine repeating from 1. This means that the answer would be an unending stream of zeros, with a 1 at the end. Wait a minute... unending... end? If we just said an UNENDING amount of zeros, then there IS no end. Thus, there IS NO DIFFERENCE between .9repeating and 1.
Verbal that there is no "smallest number greater than zero":
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->hey, what's the smallest number greater than zero?
<i>think of a number greater than zero.</i>
is this like a magic trick?
<i>in a way, yes.</i>
cool! i like magic tricks! alright, got one!
<i>halve it</i>
uhuh
<i>halve it again</i>
yup
<i>halve it again</i>
ok...
<b>*** several minutes later ***</b>
<i>halve it again</i>
um, how long is this going to take?
<b>*** several hours later ***</b>
<i>halve it again</i>
are you following me?!?!??!
<b>*** several weeks later ***</b>
<i>halve it again</i>
please! i can't take it anymore! can't we just call it quits and round it down to zero?!?!?!
<i>certainly not! halve it again...</i>
arghhhhhhh!!!!!!<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
Just as there is no biggest number, because you can always add 1, there is no smallest number greater than zero, because you can always halve it. Or quarter it for that matter, or decimate (divide by 10) it, etc etc etc. Since said smallest number doesn't exist, point nine repeating DOES equal 1.
Here's some other interesting things that are related.
To be perfectly clear right here and now: Infinity is a word that indicates endlessness.
Where c is any constant, real number
∞ ± c = ∞
Yes, this means that ∞ + 1 is no bigger than ∞, sorry to disillusion your childhood there. For that matter, ∞ - 1 is no smaller than ∞ either.
∞ ± ∞ = ∞
This may come to a surprise to some of you, who are still thinking that ∞ is a discrete number that is in any way entitled to obey the rules that every real number in existence obeys. Get over this idea fast. Consider the following example: take the set of all real integers, which equals ∞. Now remove all the odd numbers. Since the set of all odd numbers also equals ∞, you have just subtracted ∞ from ∞, and yet you still have the set of all even numbers left, which is also equal to ∞. Going the other way, if you take the set of real integers and add all the halves in, you get a set which is still just ∞, no bigger or smaller than ∞
In a way, when we're subtracting all the odd numbers from the set of all real integers, we're really just crossing out ever other number, which is the same thing as diving by two. And the reverse, which is the same thing as doubling. Which leads directly into:
∞ / c = ∞
∞ * c = ∞
But it gets better. Going back to the example of adding in all the halves in between, so the set would go 1, 1.5, 2, 2.5, ..., ∞, you could continue on by adding all the thirds, and fourths, and fifths, and sixths, and so on forever, and yet still equal infinity. As such:
∞ * ∞ = ∞
∞ / ∞ = ∞
Please note that the above does not include any limits. Limits are covered later in the node. For now, we're talking conceptually about pure infinity in the manner that I described above.
Anyway, it follows from this that...
∞^c = ∞
Why does it matter? Well, do you, the simpleton, it doesn't. But if you want to MAKE something... say, a rocketship that goes to the moon, or a partical beam weapon, or try and figure out if a time machine is possible, and if it is, build one, or if you want to make your DVD player smaller and faster, or if you want to make an aimbot for the real world to equip your tanks and soldiers with, you'd BETTER KNOW THIS STUFF or you'll fail miserably at your attempts. Want proof of THIS statement? Go to college. Take a physics course. Yes. It is needed. That's why I care.
By the way...
∞^∞ > ∞
Just to blow your mind away. This is saying that the realm of 2d is smaller than the realm of 3d, and 4d is larger than 3d, etc etc etc onwards forever.
Oh, I just found an amusing little anecdote...
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Using base-12 arithmetic:
1/3 = 4/12 = 0.4
2/3 = 8/12 = 0.8
3/3 = 12/12 = 1.0
God only knows what this proves, except that 1 = 0.999 recurring is just a stupid artefact of our tiny minds and our poorly selected numerical system. If we just had an extra finger on each hand we might have avoided that particular one.
Let us all work in duodecimal from now on and never speak of this again.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
While this is true... (and since we're using duodecimal, A=10, B=11)
.B repeating equals 1.
In trinary, .2 repeating = 1.
In octal, .7 repeating = 1.
In hex, .F repeating = 1.
Any base system is capable of having repeating digit numbers of this nature. They all have a statement where something repeating = 1. That's the way it goes. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->x->inf σ(i=0,x) 1/(9*10^i) = 1<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
..... that's 0.<u>1111</u> + 0.0<u>1111</u> + 0.00<u>1111</u> + 0.000<u>1111</u>...... It doesn't even get close to 1
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->There is no difference between .9 repeating and 1. Literally.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
A lie, there is a difference, but is infinitessimaly small (like a singularity.... you know... those black hole things)
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->Subtract .9 from 1 (difference). You get 0.01. Now subtract .99 from 1. That equals .001. You can see an instant pattern. Count the number of "9"s you're subtracting, and put that many 0s in the answer, and then tag on a 1 at the end.<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
1 - .9 != 0.01 ....... 1 - .9 = 0.1
1 - .99 != 0.001 .... 1 - .99 = 0.01
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->∞ ± c = ∞<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
You have just proven my point that there are such things as <i>conceptual</i> numbers, and that you cannot perform arithmetic on them.
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->∞^∞ > ∞<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
What orifice did you pull that one out of?
∞^∞ boils down to the following operation done infinitely many times
∞ * ∞ = x->∞ σ(i=0,x) ∞ (which leads to "∞ + ∞" done infinitely many times since "∞ + ∞" = ∞)
and since the base step of ∞ * ∞ has no effect, then the entire system has no effect.
This entire argument hinges on one thing critical point. The people that believe that 0.<u>9999</u> = 1 are those that blindly accept everything they read without thinking about the underlying systems. The rest of us are people that realize that even mathematicians will sometimes have to go "Oh crap, we can't work with that.... Kludge it." (say hello to imaginary numbers and complex hyperplanes) So they came up with systems of approximation to call it "good enough."
EDIT: More inconsistencies found in your original post.
.......
.....
Im too scared to try and figure this out one way or another. it was a good read tho
I stand behind my answer. :)
<!--emo&:0--><img src='http://www.unknownworlds.com/forums/html//emoticons/wow.gif' border='0' style='vertical-align:middle' alt='wow.gif' /><!--endemo-->
<!--emo&:0--><img src='http://www.unknownworlds.com/forums/html//emoticons/wow.gif' border='0' style='vertical-align:middle' alt='wow.gif' /><!--endemo--> <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
<a href='http://www.google.com/search?hl=en&lr=&ie=UTF-8&q=real+ufo+abductions' target='_blank'>http://www.google.com/search?hl=en&lr=&ie=...+ufo+abductions</a>
Not being an expert has never stopped anyone from writing about something.
But [WHO]Them is absolutly right, you actually cant count with infinite numbers since any result you get will be infinite.
But I dont know how many times I have to prove my point, 0.99 in infinite isn't one, though, it is as close you can get to 1.
DOOManiac is right.
"Anything between .9999 and 1 is = 1. Otherwise you have too much time on your hands to worry about that sort of crap. Go do something. <!--emo&:p--><img src='http://www.unknownworlds.com/forums/html//emoticons/tounge.gif' border='0' style='vertical-align:middle' alt='tounge.gif' /><!--endemo-->"
You can count with infinitely long numbers. Such as Pi
1x Pi = Pi and not 1x Pi = infinity
0.99 in infinity is one. It's hard to picture, but it is. There is no "as close as you can get to 1". There is exactly one, or not one.
And [WHO]Them , I think I'd still trust my teachers and people from universities with some sort of diploma or something. So far I'm convinced they are equal. Saying "they are different" isn't quite as convincing than a full demonstration.
1/3 = 0.33_ so it will be as close as you get one third. BUT 0.33_ + 0.33_ + 0.33_ = 0.99_
I still want to see someone giving me a mathematical PROOF that 0.99_ is EXACTLY 1, because no-one has proveded a calculation that is correct.
0.99_ in infinity is as close to 1 you can get, but it will never be exactly one if you do not add 0.00_1 to it.
You just have to realise that this is the way it is, I dont care about your opinions, can you give me some actual proof that strengthens your theory?