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Post by drystyx on Dec 3, 2020 1:19:16 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic.
A: 1/3 = .3333___
B: 2/3 = .6666___
C: 1/3 + 2/3 = .3333___ +.6666___
D: 1/3 + 2/3 = 1.0
D: .3333___ + .6666___ = .9999___
E: Therefore, .3333___ +.6666___ = 1.0
I realize there is a flaw in logic, but what is the flaw? Or is it logical?
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Post by Arlon10 on Dec 3, 2020 5:46:10 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic. A: 1/3 = .3333___ B: 2/3 = .6666___ C: 1/3 + 2/3 = .3333___ +.6666___ D: 1/3 + 2/3 = 1.0 D: .3333___ + .6666___ = .9999___ E: Therefore, .3333___ +.6666___ = 1.0 I realize there is a flaw in logic, but what is the flaw? Or is it logical? There is a problem with algebraic manipulation of infinity. It doesn't work most of the time. Infinity plus seven is not larger than infinity. Two times infinity is not larger than infinity. Yet it can seem logical to some people that infinity minus infinity is zero. The problem with your algebra is that 1/3 is not .3333 or .33333333 or .333333333333. That "___" at the end signifies a never ending process just as in the above examples, not a fixed quantity. Algebra manipulates quantities, not ongoing processes. The source of confusion is that infinity is not a quantity and there is the hazard of treating it like one. In calculus the confusion is avoided by not using the word "equal." The expression .3333___ is not described as being "equal" to 1/3. The quantity 1/3 (which is indeed a quantity) is described as the " limit" of the sum of 3×10 -n as n " approaches" infinity from a starting value of 1. That is because infinity can only be approached, never reached. The decimal notation introduces a never ending process that is avoided otherwise. The good news is that such small numbers as would be involved far down the line of that sum have no real world examples and can be ignored. That is exactly what hoi polloi usually do. They use the word "equal" (though calculus does not) because for all practical purposes it is equal. Elementary school teachers often use the expression "one to one correspondence" to justify the algebraic manipulation you used. There is one '3' digit for every one '6' digit. That is true but it avoids the question of where you got those and how you get from a string of 9s to 1. You are jumping into and out of essentially different notations. The difference between .9999___ and "1" might described as 1/∞ and if 1/∞ is zero then .9999___ = 1. Again though the algebraic manipulation of infinity is not allowed. The expression "1/∞" is also not allowed.
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Post by moviemouth on Dec 3, 2020 18:23:01 GMT
It seems pointless to treat infinity as a quantity.
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Post by The Lost One on Dec 3, 2020 18:48:30 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic. A: 1/3 = .3333___ B: 2/3 = .6666___ C: 1/3 + 2/3 = .3333___ +.6666___ D: 1/3 + 2/3 = 1.0 D: .3333___ + .6666___ = .9999___ E: Therefore, .3333___ +.6666___ = 1.0 I realize there is a flaw in logic, but what is the flaw? Or is it logical? I feel D is wrong: 0.3333_ and 0.6666_ add up to 1.0, not 0.9999_ |
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Post by OpiateOfTheMasses on Dec 3, 2020 23:47:08 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic. A: 1/3 = .3333___ B: 2/3 = .6666___ C: 1/3 + 2/3 = .3333___ +.6666___ D: 1/3 + 2/3 = 1.0 D: .3333___ + .6666___ = .9999___ E: Therefore, .3333___ +.6666___ = 1.0 I realize there is a flaw in logic, but what is the flaw? Or is it logical? I feel D is wrong: 0.3333_ and 0.6666_ add up to 1.0, not 0.9999_ That's correct. Another way of representing 0.333_ and 0.666_ would be as fractions: 1/3 and 2/3. And if you add those together nobody would contest that they equal 1. Infinity is not a Real number. It's not even an Imaginary number. It's a concept. There are branches of mathematics that deal with the use of infinity but it gets fairly theoretical quite quickly as you can't treat like you treat Real numbers. But the example posted is not actually dealing with infinity - it's dealing with rounding errors/not displaying enough information.
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Post by Catman 猫的主人 on Dec 4, 2020 0:39:18 GMT
Catman recalls a lawsuit against Microsoft because Excel would round 0.9999999999 up to 1. Or something like that.
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Post by permutojoe on Dec 4, 2020 1:37:08 GMT
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Post by general313 on Dec 5, 2020 17:10:11 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic. A: 1/3 = .3333___ B: 2/3 = .6666___ C: 1/3 + 2/3 = .3333___ +.6666___ D: 1/3 + 2/3 = 1.0 D: .3333___ + .6666___ = .9999___ E: Therefore, .3333___ +.6666___ = 1.0 I realize there is a flaw in logic, but what is the flaw? Or is it logical? There is no flaw. 1.0 and 0.99999... are both representations of the same number, just as (1/3 + 2/3) and 1 are representations of the same number. Arguments that "infinity is not a number" confuse infinite quantities with finite sums of infinite series. The latter, being finite, are obviously suitable for algebraic manipulation, regardless of whether they are expressed as a sum of finite or infinite number of terms.
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Post by Arlon10 on Dec 5, 2020 18:34:43 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic. A: 1/3 = .3333___ B: 2/3 = .6666___ C: 1/3 + 2/3 = .3333___ +.6666___ D: 1/3 + 2/3 = 1.0 D: .3333___ + .6666___ = .9999___ E: Therefore, .3333___ +.6666___ = 1.0 I realize there is a flaw in logic, but what is the flaw? Or is it logical? There is no flaw. 1.0 and 0.99999... are both representations of the same number, just as (1/3 + 2/3) and 1 are representations of the same number. Arguments that "infinity is not a number" confuse infinite quantities with finite sums of infinite series. The latter, being finite, are obviously suitable for algebraic manipulation, regardless of whether they are expressed as a sum of finite or infinite number of terms. ^^ hoi polloi ^^
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Post by general313 on Dec 5, 2020 18:58:47 GMT
There is no flaw. 1.0 and 0.99999... are both representations of the same number, just as (1/3 + 2/3) and 1 are representations of the same number. Arguments that "infinity is not a number" confuse infinite quantities with finite sums of infinite series. The latter, being finite, are obviously suitable for algebraic manipulation, regardless of whether they are expressed as a sum of finite or infinite number of terms. ^^ hoi polloi ^^ ^^ Thinks that the math taught in schools is wrong. ^^ Why am I not surprised?
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Post by Arlon10 on Dec 5, 2020 19:29:17 GMT
^^ Thinks that the math taught in schools is wrong. ^^ Why am I not surprised? ^^ Thinks "hoi polloi" teach anything. Failed calculus, ^^
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Post by gameboy on Dec 5, 2020 20:26:23 GMT
Not sure if it's a fallacy, but in school, we learned the basic concept that .9999... =1.0 because of logic. A: 1/3 = .3333___ B: 2/3 = .6666___ C: 1/3 + 2/3 = .3333___ +.6666___ D: 1/3 + 2/3 = 1.0 D: .3333___ + .6666___ = .9999___ E: Therefore, .3333___ +.6666___ = 1.0 I realize there is a flaw in logic, but what is the flaw? Or is it logical? There is no flaw. 1.0 and 0.99999... are both representations of the same number, just as (1/3 + 2/3) and 1 are representations of the same number. Arguments that "infinity is not a number" confuse infinite quantities with finite sums of infinite series. The latter, being finite, are obviously suitable for algebraic manipulation, regardless of whether they are expressed as a sum of finite or infinite number of terms. In what universe are 1.0 and 0.99999 the same number? The difference of 0.00001 does not exist? I can give you examples as proof of its existence. This entire thread began with an inaccuracy.
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Post by general313 on Dec 5, 2020 23:04:26 GMT
There is no flaw. 1.0 and 0.99999... are both representations of the same number, just as (1/3 + 2/3) and 1 are representations of the same number. Arguments that "infinity is not a number" confuse infinite quantities with finite sums of infinite series. The latter, being finite, are obviously suitable for algebraic manipulation, regardless of whether they are expressed as a sum of finite or infinite number of terms. In what universe are 1.0 and 0.99999 the same number? The difference of 0.00001 does not exist? I can give you examples as proof of its existence. This entire thread began with an inaccuracy. That would be the same universe where 0.33333... = 1/3. You left out the ellipsis in your mention of "0.99999", perhaps that's the source of your misinterpretation here. Your argument is the same as: "In what universe are 1/3 and 0.33333 the same number? The difference of 0.00000333... does not exist?" 1/3 is not equal to 0.33333, however it is equal to 0.33333....
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Post by Arlon10 on Dec 6, 2020 1:15:32 GMT
In what universe are 1.0 and 0.99999 the same number? The difference of 0.00001 does not exist? I can give you examples as proof of its existence. This entire thread began with an inaccuracy. That would be the same universe where 0.33333... = 1/3. You left out the ellipsis in your mention of "0.99999", perhaps that's the source of your misinterpretation here. Your argument is the same as: "In what universe are 1/3 and 0.33333 the same number? The difference of 0.00000333... does not exist?" 1/3 is not equal to 0.33333, however it is equal to 0.33333.... Although I'm certain you attended inferior schools, I am also certain even they told you that infinity is not a fixed quantity, and that it may not be used in algebraic manipulations because it is not a fixed quantity. The fact that an infinite string of threes "represents" one third is irrelevant since there isn't enough paper in the world to write down that many threes and since that many threes won't fit in any computer no matter how super. Just as there is no real world example of anything so small as the difference between one third and an infinite string of threes after a decimal point, there is no real world calculation that can register an infinite number of threes. What this means
It means that any decimal representation of one third in the real world cannot "equal" one third. Furthermore when the calculations are prolonged the more error is introduced.
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Post by gameboy on Dec 6, 2020 9:17:18 GMT
In what universe are 1.0 and 0.99999 the same number? The difference of 0.00001 does not exist? I can give you examples as proof of its existence. This entire thread began with an inaccuracy. That would be the same universe where 0.33333... = 1/3. You left out the ellipsis in your mention of "0.99999", perhaps that's the source of your misinterpretation here. Your argument is the same as: "In what universe are 1/3 and 0.33333 the same number? The difference of 0.00000333... does not exist?" 1/3 is not equal to 0.33333, however it is equal to 0.33333.... No, 1/3 is merely a number which mathematicians can never calculate correctly in decimals because the .3333333333 goes on infinitely. That's a different problem involving the enigma of infinity.
But he said "in school, we learned the basic concept that .9999... =1.0 because of logic." I never learned that when I studied mathematics or logic at UCLA.
0.9999999999 ad infinitum will never equal the whole number one. It's a different quantity, almost one, but never quite. That was my ancillary point.
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Post by thefleetsin on Dec 6, 2020 17:20:06 GMT
there's a void some see when glaring into space.
very few discover what is there place.
most get lost in a never ending race.
as supplicants flail to an amazing grace.
never knowing god is one more basket case.
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Post by general313 on Dec 6, 2020 19:41:43 GMT
That would be the same universe where 0.33333... = 1/3. You left out the ellipsis in your mention of "0.99999", perhaps that's the source of your misinterpretation here. Your argument is the same as: "In what universe are 1/3 and 0.33333 the same number? The difference of 0.00000333... does not exist?" 1/3 is not equal to 0.33333, however it is equal to 0.33333.... No, 1/3 is merely a number which mathematicians can never calculate correctly in decimals because the .3333333333 goes on infinitely. That's a different problem involving the enigma of infinity.
But he said "in school, we learned the basic concept that .9999... =1.0 because of logic." I never learned that when I studied mathematics or logic at UCLA.
0.9999999999 ad infinitum will never equal the whole number one. It's a different quantity, almost one, but never quite. That was my ancillary point.
I take it that your math courses at UCLA were too basic to cover (infinite) series, or perhaps you forgot about them. Here's a basic refresher: en.wikipedia.org/wiki/Series_(mathematics)Just read the third and fourth paragraph for the purposes of this discussion, and especially note the equivalence between the expression ending with an ellipsis (...) and the summation sign, and the equation with the summation on the left and the limit on the right. It is possible to express the fraction 1/3 as an infinite series: 3/10 + 3/100 + 3/1000 + ... This can also be expressed in summation notation where each term is 3/(10^i) -- that is, 3 divided by 10 to the i-th power. The bottom of the summation begins with i = 1, and the top has an infinity symbol, and is by definition equal to the limit expression with a summation with n on the top, with the limit of n-> +infinity. These are mathematical definitions: the expression ending with an ellipsis is equivalent to the infinite series and the limit of the finite series. They hold for any sum that is convergent (i.e. the limit exists). In the case of the simple fractions we're talking about here, they clearly do. This is true for 1/3 = 3/10 + 3/100 + 3/1000 + ... as well as for 3/3 = 9/10 + 9/100 + 9/1000 + ... Note that these are equations, using an equals sign, not an "approximately equals" sign. I learned about this for the first time in my high school math classes, and again on multiple occasions in college.
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Post by gameboy on Dec 6, 2020 20:06:59 GMT
No, 1/3 is merely a number which mathematicians can never calculate correctly in decimals because the .3333333333 goes on infinitely. That's a different problem involving the enigma of infinity.
But he said "in school, we learned the basic concept that .9999... =1.0 because of logic." I never learned that when I studied mathematics or logic at UCLA.
0.9999999999 ad infinitum will never equal the whole number one. It's a different quantity, almost one, but never quite. That was my ancillary point.
I take it that your math courses at UCLA were too basic to cover (infinite) series, or perhaps you forgot about them. Here's a basic refresher: en.wikipedia.org/wiki/Series_(mathematics)Just read the third and fourth paragraph for the purposes of this discussion, and especially note the equivalence between the expression ending with an ellipsis (...) and the summation sign, and the equation with the summation on the left and the limit on the right. It is possible to express the fraction 1/3 as an infinite series: 3/10 + 3/100 + 3/1000 + ... This can also be expressed in summation notation where each term is 3/(10^i) -- that is, 3 divided by 10 to the i-th power. The bottom of the summation begins with i = 1, and the top has an infinity symbol, and is by definition equal to the limit expression with a summation with n on the top, with the limit of n-> +infinity. These are mathematical definitions: the expression ending with an ellipsis is equivalent to the infinite series and the limit of the finite series. They hold for any sum that is convergent (i.e. the limit exists). In the case of the simple fractions we're talking about here, they clearly do. This is true for 1/3 = 3/10 + 3/100 + 3/1000 + ... as well as for 3/3 = 9/10 + 9/100 + 9/1000 + ... Note that these are equations, using an equals sign, not an "approximately equals" sign. I learned about this for the first time in my high school math classes, and again on multiple occasions in college. Lol. I said my point is ancillary. I only took up to calculus at UCLA. I was a business major and only wanted the basics. I don't claim to be a math genius like others on this thread.
My main point is that 0.9999999... does not equal one. Can you justify the OP's statement? It seems he's building his premise on an error.
As for "1/3 = 3/10 + 3/100 + 3/1000 + ...", the ellipse still represents an infinite sequence of additions. 1/3 cannot be written finitely in decimals, as I already said.
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Post by general313 on Dec 6, 2020 20:31:14 GMT
I take it that your math courses at UCLA were too basic to cover (infinite) series, or perhaps you forgot about them. Here's a basic refresher: en.wikipedia.org/wiki/Series_(mathematics)Just read the third and fourth paragraph for the purposes of this discussion, and especially note the equivalence between the expression ending with an ellipsis (...) and the summation sign, and the equation with the summation on the left and the limit on the right. It is possible to express the fraction 1/3 as an infinite series: 3/10 + 3/100 + 3/1000 + ... This can also be expressed in summation notation where each term is 3/(10^i) -- that is, 3 divided by 10 to the i-th power. The bottom of the summation begins with i = 1, and the top has an infinity symbol, and is by definition equal to the limit expression with a summation with n on the top, with the limit of n-> +infinity. These are mathematical definitions: the expression ending with an ellipsis is equivalent to the infinite series and the limit of the finite series. They hold for any sum that is convergent (i.e. the limit exists). In the case of the simple fractions we're talking about here, they clearly do. This is true for 1/3 = 3/10 + 3/100 + 3/1000 + ... as well as for 3/3 = 9/10 + 9/100 + 9/1000 + ... Note that these are equations, using an equals sign, not an "approximately equals" sign. I learned about this for the first time in my high school math classes, and again on multiple occasions in college. Lol. I said my point is ancillary. I only took up to calculus at UCLA. I was a business major and only wanted the basics. I don't claim to be a math genius like others on this thread.
My main point is that 0.9999999... does not equal one. Can you justify the OP's statement? It seems he's building his premise on an error.
As for "1/3 = 3/10 + 3/100 + 3/1000 + ...", the ellipse still represents an infinite sequence of additions. 1/3 cannot be written finitely in decimals, as I already said.
And my point is that 0.9999999... does equal one, because it is equivalent to the infinite series expression 9/10 + 9/100 + 9/1000 + ..., as can 1/3 = 3/10 + 3/100 + 3/1000 + .... It's very simple: infinite series can have finite sums. The OP's premise is sound and error-free.
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Post by gameboy on Dec 6, 2020 22:15:26 GMT
Lol. I said my point is ancillary. I only took up to calculus at UCLA. I was a business major and only wanted the basics. I don't claim to be a math genius like others on this thread.
My main point is that 0.9999999... does not equal one. Can you justify the OP's statement? It seems he's building his premise on an error.
As for "1/3 = 3/10 + 3/100 + 3/1000 + ...", the ellipse still represents an infinite sequence of additions. 1/3 cannot be written finitely in decimals, as I already said.
And my point is that 0.9999999... does equal one, because it is equivalent to the infinite series expression 9/10 + 9/100 + 9/1000 + ..., as can 1/3 = 3/10 + 3/100 + 3/1000 + .... It's very simple: infinite series can have finite sums. The OP's premise is sound and error-free. Huh? You can never add enough 9th fractions to equal one. You'll always be some one fractional unit short. How is the sum ever finite? The number must always be expressed by decimal as .9999999999999999999999999999999999999999 and on forever.
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