.99999... does in fact equal 1 and can be easily mathematically proved.
Let x=.9999999......
Therefore 10x=9.99999.......
10x-x =9x. == 9.99999.....-.999999....=9
So 9x=9
9x÷9 = 9÷9
Therefore x=1 which was equal to .999999... originally.
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.99999... does in fact equal 1 and can be easily mathematically proved.
Let x=.9999999......
Therefore 10x=9.99999.......
10x-x =9x. == 9.99999.....-.999999....=9
So 9x=9
9x÷9 = 9÷9
Therefore x=1 which was equal to .999999... originally.
Seems feasable...can you say the same of .22222222222......=1?
.99999... does in fact equal 1 and can be easily mathematically proved.
Let x=.9999999......
Therefore 10x=9.99999.......
10x-x =9x. == 9.99999.....-.999999....=9
So 9x=9
9x÷9 = 9÷9
Therefore x=1 which was equal to .999999... originally.
Following the same set if steps, you would end up with 9x =2. Thus x =2\9, which is indeed what. 2222222..... is equal to.
Tell me if I'm wrong here:
x=.22222222......
10x= 2.22222222......
10x-x=9x 2.22222222.....-.2222222222= 2
2=9x
divide each side by 9
2/9=x
x=.2222222222222
back to where we started
ah, I think I get it...
.99999999.....would be the only number equal to 1?
Tell me if I'm wrong here:
x=.22222222......
10x= 2.22222222......
10x-x=9x 2.22222222.....-.2222222222= 2
2=9x
divide each side by 9
2/9=x
x=.2222222222222
back to where we started
ah, I think I get it...
.99999999.....would be the only number equal to 1?
Correct, it is an odd situation that comes up in mathematics. There others out there too, but I will spare everyone the messy details because the proofs involve higher mathematics and aren't easy to type. This method shown above can be used to figure out the fraction for any repeating decimal. Pretty nifty I'd say.
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It depends on the setting really. If your manufacturing gears with a wire edm then .9999 is still .0001 underside from 1.00000. now if you have a geometric tolerance of lets say +- .00001 than that dimention being at .9999 would bring you out of tolerance by .00009 rendering the part non conforming and scrap.
It depends on the setting really. If your manufacturing gears with a wire edm then .9999 is still .0001 underside from 1.00000. now if you have a geometric tolerance of lets say +- .00001 than that dimention being at .9999 would bring you out of tolerance by .00009 rendering the part non conforming and scrap.
Partially true. .9999 itself is not equal to 1. But the idea behind the question and the proof is for .9999...., meaning repeating infinitely forever and is absolutely 100% true.
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thats only true if you assume 0.3333... = 1/3
which (depending on context, may or may not be true
1/3 is exactly = .333... repeated for ever and you can easily discover that for yourself by dividing 1 by 3. If you do it manually using the standard you will constantly get repeating 3's forever until you decide you've had enough punishment.
Quote:
Originally Posted by Josepho1997
Your proofs are all incorrect. .999999 infinitely gets closer to one, but it never gets there.
I wanted to avoid the use of higher mathematics, but the reason behind the equality is stated right in the wiki article:
Quote:
The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis.
Furthermore:
Quote:
The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it, commonly enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education
Whether anyone chooses to accept it or not is their choice, but nonetheless, it is 100% true. I often run into non-believers when I teach this topic in the Infinite Series part of Calculus II, granted the proof for that class is different giving the context of the class.
Last edited by lunatic59; November 26th, 2012 at 12:03 PM.
Reason: fixed tag
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thats only true if you assume 0.3333... = 1/3
which (depending on context, may or may not be true
1/3 is exactly = .333... repeated for ever and you can easily discover that for yourself by dividing 1 by 3. If you do it manually using the standard you will constantly get repeating 3's forever until you decide you've had enough punishment.
I wanted to avoid the use of higher mathematics, but the reason behind the equality is stated right in the wiki article:
Furthermore:
Whether anyone chooses to accept it or not is their choice, but nonetheless, it is 100% true. I often run into non-believers when I teach this topic in the Infinite Series part of Calculus II, granted the proof for that class is different giving the context of the class.
I still don't believe it. That's wikipedia. Anybody can edit it. Give me a real reason.
I could give you the real reason, but would require the use of real analysis which is usually a 4th year mathematics course. As I stated earlier, it has to deal with the fact that there is no non-zero infinitesimal between .999... and 1.0. In other words, there is no number between .999... and 1.
For the purpose of a proof, lets assume they are distinct numbers. For any number not ending in a infinite sequence of 9's, you can always find another number in between two different numbers by simply finding the midpoint (x1+x2)/2. This doesn't work with numbers ending in the infinite sequence of 9's because the midpoint between .9 and 1 is .95. so if you could take the midpoint of the two, following that idea, the last number would have to be a 5, which contradicts the whole notion because .999 is an infinite sequence of 9's and would actually be greater than our midpoints. This creates an absurdity, which many mathematicians call a proof by contradiction. So since our original idea about them being two different numbers let to a contradiction, that means that assumption must be false, and thus means they have to be the same number.
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Without a doubt .999 =1 it can be mathematicaly proved as shown above. .222 does not =1 nor does .3333 .444 etc. Again all this was said above but i thought i would verify it.
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Quote:
Originally Posted by jhawkkw
1/3 is exactly = .333... repeated for ever and you can easily discover that for yourself by dividing 1 by 3. If you do it manually using the standard you will constantly get repeating 3's forever until you decide you've had enough punishment.
.
Thats a pretty interesting read... and having read it I now think I get whats going on. It appears that 0.999... in the way that mathematicians currently understand it, doesn't actually mean what I thought it did.
So i'm fine with the whole 0.999... =1 thing. Which is completely different from 0.999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 9999999999999 =1 which it clearly isn't.
Who says you can't learn anything in pointless conversations
With the math soltions and skills being demonstrated in this thread by some people, you will also need a good tax lawyer because you be a going to prision.
.999999 does not equal 1! And .9999999999999999999999999999999999999999999999999 9999999 does not equal 1, and .9999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 9999999999999999999 does not equal 1 either. Neither does .99999 (on to infinity) equal 1. And remember, infinity is not a number, it is a concept.
With the math soltions and skills being demonstrated in this thread by some people, you will also need a good tax lawyer because you be a going to prision.
.999999 does not equal 1! And .9999999999999999999999999999999999999999999999999 9999999 does not equal 1, and .9999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 9999999999999999999 does not equal 1 either. Neither does .99999 (on to infinity) equal 1. And remember, infinity is not a number, it is a concept.
If my math skills are wrong, please identify where they are wrong because I'm curious as to why you think so.
You can't do the maths with infinite or reccurring numbers so you must start with a set amount of digits to prove a point. An absolute number.
Multiply by 10 simply moves the decimal point. It can't conveniently make more digits or decimal places
X = .999999
10x = 9.99999
10x - x = 9x
Or
9.99999 - .999999 = 8.999991
Its not so much that your maths is wrong as it is that maths is wrong in general.
This as written is correct, but fails to take into account that .999... a number of infinite length. Thus require the use of an infinite limit. I'll post the calculus proof by use of infinite series, since there is still non believers once I finish teaching my last class tonight.
yeah.. i got musical theme to your post... funny :P
but it confers that i have something wrong with my understanding.
and it should refer to "recurring" not "non-recurring" numbers, since the issue is over "recurring" numbers (.9999...)
This as written is correct, but fails to take into account that .999... a number of infinite length.
Actually words 7 and 9 of my post take this into account when combined with the rest of the first sentence
If you look how your maths is written, you have .9999 ad infinitum but when you multiply it by 10, you magically have add infinitum + 1 extra digit, which is why your maths looks to work. But we all know .99999 blah is not actually 1.
Since infinity isn't really a number, it can't really be used in maths to form any cohesive proof that 1 isn't 1.
If you massage numbers correctly, you can "prove" anything and you did a good job of massaging them. Well done. But joking aside, .9 reoccurring is nothing except what it is, even if you can make it appear so on paper. Maths is the last bastion of comedy.
Actually words 7 and 9 of my post take this into account when combined with the rest of the first sentence
If you look how your maths is written, you have .9999 ad infinitum but when you multiply it by 10, you magically have add infinitum + 1 extra digit, which is why your maths looks to work. But we all know .99999 blah is not actually 1.
Since infinity isn't really a number, it can't really be used in maths to form any cohesive proof that 1 isn't 1.
If you massage numbers correctly, you can "prove" anything and you did a good job of massaging them. Well done. But joking aside, .9 reoccurring is nothing except what it is, even if you can make it appear so on paper. Maths is the last bastion of comedy.
You certainly use infinity in math, it's one of the entire foundations of Calculus. The ability to take limits as a variable goes to positive or negative infinity is a subject that comes up in multiple topics such as horizontal asymptotes, anti-derivatives/integrals, improper integrals, and sums of infinite series. For the record, I am a college mathematics professor who teaches these topics in Calculus I & II. I never claimed to be good with taxes, just with mathematical theory. I noticed that people picked apart my algebraic proof with skepticism, but flat out ignored my proof through analysis most likely because very few ever take 4th year mathematics courses when single variable calculus is considered to be first year. As promised, I am going to provide the proof that anyone who has taken calculus II could do by use of the Geometric Series. Just so people don't think I pull formulas out of thin air, I'm actually going to use a formula in this proof that you can look up in any calculus textbook, and I will even cite one, James Stewart's Calculus Early Transcendentals 5th Edition(ISBN:0-534-39321-7). Because the proof requires the use of symbols not easily typed, I have hand written it and attached it to this. Hopefully this will be much more convincing.
yeah.. i got musical theme to your post... funny :P
but it confers that i have something wrong with my understanding.
and it should refer to "recurring" not "non-recurring" numbers, since the issue is over "recurring" numbers (.9999...)
So the recurring number in question IS one ( .999... = 1)
But a non-recurring number ain't "one" (.999 =/= 1)
Which is what you correctly stated.
I've heard a puppy gets strangled every second a bad joke gets dragged out so I better cease and deactivate.
You certainly use infinity in math, it's one of the entire foundations of Calculus. The ability to take limits as a variable goes to positive or negative infinity is a subject that comes up in multiple topics such as horizontal asymptotes, anti-derivatives/integrals, improper integrals, and sums of infinite series. For the record, I am a college mathematics professor who teaches these topics in Calculus I & II. I never claimed to be good with taxes, just with mathematical theory. I noticed that people picked apart my algebraic proof with skepticism, but flat out ignored my proof through analysis most likely because very few ever take 4th year mathematics courses when single variable calculus is considered to be first year. As promised, I am going to provide the proof that anyone who has taken calculus II could do by use of the Geometric Series. Just so people don't think I pull formulas out of thin air, I'm actually going to use a formula in this proof that you can look up in any calculus textbook, and I will even cite one, James Stewart's Calculus Early Transcendentals 5th Edition(ISBN:0-534-39321-7). Because the proof requires the use of symbols not easily typed, I have hand written it and attached it to this. Hopefully this will be much more convincing.
Thanks for posting that. I know that infinity is used in maths. I think you're missing my point. I did not say your maths was wrong. I said maths in general is wrong. I am completely comfortable arguing this with a college mathematics professor. In fact, thats all the more poignant.
1 is 1. .9999' is not 1. I am using ' as the symbol for recurring as there isn't one.
We cannot use infinity to prove something that is not 1 is actually 1. Infinity is not a number. It is s concept. Recurring numbers do not actually exist. If I have 9 cakes, I can share them between 3 people equally. If I have 2 people, cutting the cake is possible. If I have 10 cakes and 3 people, cutting the cake wont work. It is physically not possible.
Recurring numbers exist so from a mathematical stand point, it is possible when in reality we know the contrary to be true.
So now we have a series of equations that attempt to prove .9999' is equal to 1. The mathematics behind it stands up. My point is, that is irrelevant. Using a non-existant number in an equation to prove that something that really exists is in fact something else, wont wash.
Mathematics is hugely flawed and even the greatest mathematical minds of our time still do not understand it fully. We know 1 is 1. We know .9999' is not 1. We cannot use mathematical flaws (the use of things that cannot exist) to prove otherwise. If all it is doing is "proving" that the opposite of reality is true (incorrectly) then it's use of proving anything lessens significantly. Its credibility is lost.
So you see, its not me against you. Its reality against maths. .9999' is the closest you can get in maths to 1 without being 1 and no amount of mathematical loop holes will change that.
Maths exists in reality and has its uses in everyday life. I do not deny. How else will I share cakes fairly or count in binary?! However as infinity is a concept and not a reality, regardless as to whether we can use it to show something that doesn't exist is the same as something that does exist, doesn't make it conclusive or even correct. You can only use existing numbers in reality to prove conclusively, which I've already shown proves the opposite and that my good man, is my point.
There are 10 types of people in this world. Those that understand binary, those that don't and those that didn't expect this joke to be in base-3.
Nice spin Xyro, I don't think I've ever seen that classic that way. I'll have to remember that alternate.
Quote:
Originally Posted by SUroot
1)If I have 10 cakes and 3 people, cutting the cake wont work. It is physically not possible.
2)Recurring numbers exist so from a mathematical stand point, it is possible when in reality we know the contrary to be true.
3)Mathematics is hugely flawed and even the greatest mathematical minds of our time still do not understand it fully. We know 1 is 1. We know .9999' is not 1.
1st point: you might not have intended or realized this, but this indirectly proves my point. I'll get to it after number 2.
2nd point: repeating numbers exist because of division. You slightly hint at this with the cake reference, but is an unwanted consequence of dividing by certain prime numbers. Also, not every decimal that is infinite in length is a repeating number. The two most famous ones being PI (the ratio between a circle's circumference and diameter), or e (the base of the natural logarithm).
Back to number 1, which is an expansion on a slightly less rigorous method shown earlier.. The cake problem is exactly how defining 1/3 comes about. If you attempt to define 1/3 in decimal form, it becomes a never ending attempt of dividing 3 into 10. You will always get 3 with a remainder of 1, and thus have to bring down another 0 and create a brand new 10, and the resulting decimal is .333... or .333` as you last wrote. Extrapolate that problem to 2/3. This ones again turns into a never ending problem, but this time trying to divide 20 by 3, which results in 6 with a remainder of 2, forcing you to bring down the next zero. These numbers do not terminate because the division never terminates, not because we don't want them to or because it is convenient for them to continue on forever, it's because that's how the algorithm is done. In conclusion, simple addition of 1/3 + 2/3 would get us 3/3 or 1. since .333... and .666... only contain 3s or 6s and. There is no 4 in 1/3 or 7 in 2/3 at the end of the tunnel,and 3+6 is nine, so the result of adding .333... (1/3)and .666... (2/3)would give us .999... (3/3=1). And while not everything can be easily cut into thirds, one thing that can be cut into thirds are circles into into 3 120 degree parts.
Point 3) All of the great mathematical minds accept the fact that .999... = 1, which is why this particular problem in relation to accepting it to be true is studied, because it puzzles us mathematicians why people cannot accept it to be true when the math is 100 percent sound. In fact, math is one of the only things that is ABSOLUTE. The 5 basic operators(+-*/^) don't just work in some cases and not in others. If you follow the rules of math, it will always work out and will always be correct, granted, not everyone knows all the rules as evident in many papers I grade . If you drive at a constant rate of 60mph, for exactly 2 hours, you will drive 120 miles. not 119, not 121, but exactly 120. . Mathematicians might not have discovered or proved everything in math, such as the P=NP question, the Riemann Hypothesis, or any of the other millennium questions, but what has been proven is FACT.
Accepting things to be true that don't intuitively seem to be true is tough. I was the same way when I was in school with certain topics, such as the total number of integers and rational numbers (fractions) are equal until I saw the proof using the Cantor pairing function (it's actually quite elegant) .
Nice spin Xyro, I don't think I've ever seen that classic that way. I'll have to remember that alternate.
1st point: you might not have intended or realized this, but this indirectly proves my point. I'll get to it after number 2.
2nd point: repeating numbers exist because of division. You slightly hint at this with the cake reference, but is an unwanted consequence of dividing by certain prime numbers. Also, not every decimal that is infinite in length is a repeating number. The two most famous ones being PI (the ratio between a circle's circumference and diameter), or e (the base of the natural logarithm).
Back to number 1, which is an expansion on a slightly less rigorous method shown earlier.. The cake problem is exactly how defining 1/3 comes about. If you attempt to define 1/3 in decimal form, it becomes a never ending attempt of dividing 3 into 10. You will always get 3 with a remainder of 1, and thus have to bring down another 0 and create a brand new 10, and the resulting decimal is .333... or .333` as you last wrote. Extrapolate that problem to 2/3. This ones again turns into a never ending problem, but this time trying to divide 20 by 3, which results in 6 with a remainder of 2, forcing you to bring down the next zero. These numbers do not terminate because the division never terminates, not because we don't want them to or because it is convenient for them to continue on forever, it's because that's how the algorithm is done. In conclusion, simple addition of 1/3 + 2/3 would get us 3/3 or 1. since .333... and .666... only contain 3s or 6s and. There is no 4 in 1/3 or 7 in 2/3 at the end of the tunnel,and 3+6 is nine, so the result of adding .333... (1/3)and .666... (2/3)would give us .999... (3/3=1). And while not everything can be easily cut into thirds, one thing that can be cut into thirds are circles into into 3 120 degree parts.
Point 3) All of the great mathematical minds accept the fact that .999... = 1, which is why this particular problem in relation to accepting it to be true is studied, because it puzzles us mathematicians why people cannot accept it to be true when the math is 100 percent sound. In fact, math is one of the only things that is ABSOLUTE. The 5 basic operators(+-*/^) don't just work in some cases and not in others. If you follow the rules of math, it will always work out and will always be correct, granted, not everyone knows all the rules as evident in many papers I grade . If you drive at a constant rate of 60mph, for exactly 2 hours, you will drive 120 miles. not 119, not 121, but exactly 120. . Mathematicians might not have discovered or proved everything in math, such as the P=NP question, the Riemann Hypothesis, or any of the other millennium questions, but what has been proven is FACT.
Accepting things to be true that don't intuitively seem to be true is tough. I was the same way when I was in school with certain topics, such as the total number of integers and rational numbers (fractions) are equal until I saw the proof using the Cantor pairing function (it's actually quite elegant) .
In some cases it can be done easily, like a cake of length 12 inches can be cut into 3rds with length 4, or a circular cake (like a wedding cake) of 120 degrees. But for lengths that are not a multiple of 3, we has humans do not tools precise enough to make the cut exactly equal 3rds.
In the end, the practicality of the argument does seem worthless. But just because something is worthless doesn't make it untrue. What an ugly double negative.
In some cases it can be done easily, like a cake of length 12 inches can be cut into 3rds with length 4, or a circular cake (like a wedding cake) of 120 degrees. But for lengths that are not a multiple of 3, we has humans do not tools precise enough to make the cut exactly equal 3rds.
In the end, the practicality of the argument does seem worthless. But just because something is worthless doesn't make it untrue. What an ugly double negative.
Thanks for posting that. I know that infinity is used in maths. I think you're missing my point. I did not say your maths was wrong. I said maths in general is wrong. I am completely comfortable arguing this with a college mathematics professor. In fact, thats all the more poignant.
1 is 1. .9999' is not 1. I am using ' as the symbol for recurring as there isn't one.
We cannot use infinity to prove something that is not 1 is actually 1. Infinity is not a number. It is s concept. Recurring numbers do not actually exist. If I have 9 cakes, I can share them between 3 people equally. If I have 2 people, cutting the cake is possible. If I have 10 cakes and 3 people, cutting the cake wont work. It is physically not possible.
Recurring numbers exist so from a mathematical stand point, it is possible when in reality we know the contrary to be true.
So now we have a series of equations that attempt to prove .9999' is equal to 1. The mathematics behind it stands up. My point is, that is irrelevant. Using a non-existant number in an equation to prove that something that really exists is in fact something else, wont wash.
Mathematics is hugely flawed and even the greatest mathematical minds of our time still do not understand it fully. We know 1 is 1. We know .9999' is not 1. We cannot use mathematical flaws (the use of things that cannot exist) to prove otherwise. If all it is doing is "proving" that the opposite of reality is true (incorrectly) then it's use of proving anything lessens significantly. Its credibility is lost.
So you see, its not me against you. Its reality against maths. .9999' is the closest you can get in maths to 1 without being 1 and no amount of mathematical loop holes will change that.
Maths exists in reality and has its uses in everyday life. I do not deny. How else will I share cakes fairly or count in binary?! However as infinity is a concept and not a reality, regardless as to whether we can use it to show something that doesn't exist is the same as something that does exist, doesn't make it conclusive or even correct. You can only use existing numbers in reality to prove conclusively, which I've already shown proves the opposite and that my good man, is my point.
Part of your problem is that you think there are non-existent numbers.
1/3 is not a non-existent number. It is impossible to represent finitely as a decimal number. That's a simple way of saying that it is impossible to represent as a finite sum of powers of 10. If, however, you were to think in terms of base-3, 1/3 is neatly represented as .1
To say that something is not a number simply because it can't be finitely represented in the base system of your choice would be to end computer math as we know it, because there are a *lot* of numbers that can't be finitely represented as powers of 2. 1/5 (.2) is one such number. The nice and neat .2 in decimal is represented by the infinitely repeating (and in your world, "nonexistant") .0011001100110011...
(note, I am well aware that computers don't represent floating point numbers this way).
Man, it's really going to blow some of your minds when you learn that e^i*pi = -1, where i is the square root of -1.
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.9999...=1
. 
to show it.
