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Display_ScreenIf you have $99.99 do you have a Benjamin? Nope
Well 99.99 is not the same as 99.999999999 repeating
Simplest proof 1/3=.3333 repeating
1/3 + 1/3 + 1/3 =3/3 =1
.333 repeating +.333 repeating +.333 repeating =.9999 repeating
Because 1/3 =.333 repeating, the left side of bothe equations are the same therefore the right side needs to be equal meaning .999 repeating =1
However 0.(9) === 1. (I wanted the equals sign with three lines, also number in brackets indicates recurrence)
Prove it? sure.
I've always liked this proof far more than the others, which can be too wordy. It's elegant.
x = 0.(9)
10x = 9.(9)
= 9 + 0.(9)
= 9 + x
9x = 9
x = 1
Your proof makes no sense at all.
Where's the proof stating that 10x= 9.(9)?
Oh yeah, there is none, because it doesn't exist. Saying .9 repeating is equal to one is the SAME EXACT THING as saying .9 rounds to 1. The only reason people say it equals 1 is because once it gets that many decimal places, the .000...0001 is so small that it doesn't matter.
Oh yeah, there is none, because it doesn't exist. Saying .9 repeating is equal to one is the SAME EXACT THING as saying .9 rounds to 1. The only reason people say it equals 1 is because once it gets that many decimal places, the .000...0001 is so small that it doesn't matter.
*DUMBCAN*Are you going to prove they aren't equal, or is that neurone of yours overheating at the possibility of things not being immediately obvious?
No, the "proof" that shows how they are supposedly equal, but it is just a trick, the number goes on infintely and will NEVER reach 1. Anyone who says otherwise just sees a proof that is a simple trick. Just manipulating the numbers to look like that way.
You forget as a human race we like them try to make everything seem nice and round. So it can fit perfectly into a box. That's not how life works you're going to have errors and measurements that don't 100% add to a number without a decimal but we round it anyway. decimals can be infinite you fill two cups of water up to the top then way them you can keep it going and eventually it'll change.
.undetectedNo, the "proof" that shows how they are supposedly equal, but it is just a trick, the number goes on infintely and will NEVER reach 1. Anyone who says otherwise just sees a proof that is a simple trick. Just manipulating the numbers to look like that way.
...but I haven't manipulated the numbers to look that way.
I haven't broken a single principle, fundamental or otherwise, in my proof. I used only basic maths, and I proved 0.(9) = 1.
But if you can't handle that, read up on convergence theorem and here's another.
We can write 0.(9) as 0.999... , or as (remember BIDMAS)
9*(1/10) + 9*(1/10)^2 + 9*(1/10)^3 + ...
which can be written generally as:
ar + ar^2 + ar^3 + ... where a = 9 and r = (1/10)
Convergence theorem (it's proven, 100% indisputable) states that
*DUMBCAN*...but I haven't manipulated the numbers to look that way.
I haven't broken a single principle, fundamental or otherwise, in my proof. I used only basic maths, and I proved 0.(9) = 1.
But if you can't handle that, read up on convergence theorem and here's another.
We can write 0.(9) as 0.999... , or as (remember BIDMAS)
9*(1/10) + 9*(1/10)^2 + 9*(1/10)^3 + ...
which can be written generally as:
ar + ar^2 + ar^3 + ... where a = 9 and r = (1/10)
Convergence theorem (it's proven, 100% indisputable) states that
ar + ar^2 + ar^3 + ... = (ar)/(1-r) providing |r|
Its hard to explain, Im not disagreeing with the math, I understand how exactly it is proved .9=1, I even understand that it is accepted in the math community that .9=1, but it is just simple manipulation of numbers
That is not a 'proof', it is making a claim without proving it and then claiming that doing such proves a parallel claim. Claiming that 1/3=.333... is the same as claiming that 1=.999... Of course if one is true then both are true, but you did nothing to prove that they are true. .333... is an approximation of 1/3 in the same way that .999... is approximately equal to 1. In all real situations, they can be used interchangably because the difference is infinitely small, but that doesn't make them literally equal, only equal in practice.
They are different infinities, see above. I will also post the following (someone else's words, not my own) because this is a good explanation as well.
'this "proof" uses the ambiguity of infinity (a man made construct, not an actual number) to try and ignore magnitudes.
The "..." after 0.999 is not equal to the "...." 9.9999. The "..." after 9.9999 is TEN TIMES LARGER, yet you act like it is exactly the same. It is not. Going out to infinity (again, not an actual number) wont stop this fact.'
Point being not all infinities are the same (i.e. the infinity 9s in .999... and the infinity 9s after the decimal in 9.999...). The same thing comes up in plenty of other situations as well. For example, with differentiation, integration, and limits, not all infinities are equal.
The referenced post has been removed.
Since when do people fill up their car with integer numbers of gallons of gas?
cool_nameWell 99.99 is not the same as 99.999999999 repeating
Simplest proof 1/3=.3333 repeating
1/3 + 1/3 + 1/3 =3/3 =1
.333 repeating +.333 repeating +.333 repeating =.9999 repeating
Because 1/3 =.333 repeating, the left side of bothe equations are the same therefore the right side needs to be equal meaning .999 repeating =1
You do realize that that 1/3 does not equal .3333 repeating? .3333 repeating is a numerical approximation. Therefore it can be said that .9999 repeating is an approximation of 1, however it does not equal 1.
In my last post, the first post I quoted was the 1/3=.333... 3/3=.999... 'proof'
Second post I quoted was x=.999..., 10x-x=9, x=1
Third was same as second from someone else
Fourth was has anyone gotten 1/10 cent back from gas station.
Like I say I have no basis to come down on either side of this issue but it's pretty interesting.
From what I have seen, all arguments claiming that .999...=1 fall into one of the following categories:
1) They disregard the fact that not all infinities are equal.
2) They claim that if nothing exists between two numbers, then those two numbers are equal, despite not having a basis for such a claim. (If two quantum particles, both being infinitely small, occupy two adjacent Planck spaces, therefore having nothing--not even space--between them, does that make them the same particle?)
3) They claim that .999... and 1 are equal because they are interchangable and can be treated as equal in all real situations.
4) They take a parallel concept (i.e. '1/3=.333...') without proving it and use that to 'prove' that .999...=1, when in truth the 'fact' they used to prove it is not true.
Maybe I've missed something. That's all I can remember.
I think the reason for all of the debate comes from how the situation is being looked at. If you just look at it mathematically, it is reasonable to conclude that .999...=1 because the two numbers are interchangable in practice in any mathematical situation without exception, and because often when .999... is produced as a solution to a problem, the solution is actually 1 (for example, if you take 1/3 into decimal form then multiply it by 3; however, .333... is not quite 1/3 so it's not exact).
However, infinity itself is not even really a number. It is more of a concept. To really understand what .999... means you can't just put it in an equation, you have to look at what the concept of infinity even means, and what the idea of .999... even means.
Something I typed a long time ago that explains my reasoning:
.999... is not a number. It is a limit as the variable approaches infinity [lim(n->infinity)(1-10^(-n))]. Infinity does not exist. Therefore, .999... does not exist. It exists in theory because we treat infinity as a number. That's fine, because it works. However, when we treat infinity as a number in theoretical applications such as limits, derivation, and integration, we have to remember that we are treating infinity as a number, and can therefore manipulate it as a number with operations such as addition and multiplication. That means we can take a hypothetical number n=1-[1/(10^infinity)], which is equal to .999.... If .999...=1, then 1-[1/(10^infinity)]=1, which means 1/(10^infinity)=0, which means 1=0(10^infinity). This is clearly incorrect, since 0x, with x being a number, is always equal to 0, and infinity is being treated as a number in this situation. Of course infinity is not a number and cannot be manipulated like this in real life, but infinity doesn't exist, so using infinity in limits also doesn't work in real life. It only works in theory when infinity is treated as a number. Claiming that .999...=1 is allowing infinity to be manipulated as a number in some ways that do not work in real life but not in others, all within the same theoretical situation. The difference between .999... and 1 is negligable to the extent that they can be treated as the same number in practical applications, but they are, by definition, not equal.
Sh4dowFrom what I have seen, all arguments claiming that .999...=1 fall into one of the following categories:
1) They disregard the fact that not all infinities are equal.
2) They claim that if nothing exists between two numbers, then those two numbers are equal, despite not having a basis for such a claim. (If two quantum particles, both being infinitely small, occupy two adjacent Planck spaces, therefore having nothing--not even space--between them, does that make them the same particle?)
3) They claim that .999... and 1 are equal because they are interchangable and can be treated as equal in all real situations.
4) They take a parallel concept (i.e. '1/3=.333...') without proving it and use that to 'prove' that .999...=1, when in truth the 'fact' they used to prove it is not true.
Maybe I've missed something. That's all I can remember.
I think the reason for all of the debate comes from how the situation is being looked at. If you just look at it mathematically, it is reasonable to conclude that .999...=1 because the two numbers are interchangable in practice in any mathematical situation without exception, and because often when .999... is produced as a solution to a problem, the solution is actually 1 (for example, if you take 1/3 into decimal form then multiply it by 3; however, .333... is not quite 1/3 so it's not exact).
However, infinity itself is not even really a number. It is more of a concept. To really understand what .999... means you can't just put it in an equation, you have to look at what the concept of infinity even means, and what the idea of .999... even means.
Something I typed a long time ago that explains my reasoning:
.999... is not a number. It is a limit as the variable approaches infinity [lim(n->infinity)(1-10^(-n))]. Infinity does not exist. Therefore, .999... does not exist. It exists in theory because we treat infinity as a number. That's fine, because it works. However, when we treat infinity as a number in theoretical applications such as limits, derivation, and integration, we have to remember that we are treating infinity as a number, and can therefore manipulate it as a number with operations such as addition and multiplication. That means we can take a hypothetical number n=1-[1/(10^infinity)], which is equal to .999.... If .999...=1, then 1-[1/(10^infinity)]=1, which means 1/(10^infinity)=0, which means 1=0(10^infinity). This is clearly incorrect, since 0x, with x being a number, is always equal to 0, and infinity is being treated as a number in this situation. Of course infinity is not a number and cannot be manipulated like this in real life, but infinity doesn't exist, so using infinity in limits also doesn't work in real life. It only works in theory when infinity is treated as a number. Claiming that .999...=1 is allowing infinity to be manipulated as a number in some ways that do not work in real life but not in others, all within the same theoretical situation. The difference between .999... and 1 is negligable to the extent that they can be treated as the same number in practical applications, but they are, by definition, not equal.
cool_nameWell 99.99 is not the same as 99.999999999 repeating
Simplest proof 1/3=.3333 repeating
1/3 + 1/3 + 1/3 =3/3 =1
.333 repeating +.333 repeating +.333 repeating =.9999 repeating
Because 1/3 =.333 repeating, the left side of bothe equations are the same therefore the right side needs to be equal meaning .999 repeating =1
1/3 ≠ .(3)
.(3) is an approximation of 1/3
I understand that if a value is approached infinitely, then it equals that value, but then there's no difference between y>x and y>=x
If we collapse to the nearest integer, then those two are equivalent, since then there's no number to include that would be infinitely close to x, but not x. Or if there is, we can't represent it. It's basically saying that .(9) is different than actually writing out infinite 9s. It's writing out how many significant figures you want, but why would mathematics on the scale of infinity need significant figures when you're using precise numbers?
Miomo1/3 ≠ .(3)
.(3) is an approximation of 1/3
I understand that if a value is approached infinitely, then it equals that value, but then there's no difference between y>x and y>=x
If we collapse to the nearest integer, then those two are equivalent, since then there's no number to include that would be infinitely close to x, but not x. Or if there is, we can't represent it. It's basically saying that .(9) is different than actually writing out infinite 9s. It's writing out how many significant figures you want, but why would mathematics on the scale of infinity need significant figures when you're using precise numbers?
no matter what you do .999.... will never be = 1 yeah you can talk about all your algebra shit and find clever ways to try and make .99... =1 but its not, its always going to be something in-between that and 1.
No, as you pile on more 9's you will get closer and closer and closer to 1, but there will always be an infinitesimally small margin between the two numbers. That's one of the main ideas behind how calculus works.
