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12654 clicks; posted to Geek » on 03 Feb 2013 at 12:03 PM (4 years ago)   |   Favorite    |   share:    more»

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alexanderplatz: So, if you want to express the quantity one-third, that is exactly one third, but you have to denote it in a decimal form, how do you do that? You write 0.3333... indicating an infinite number of decimal points that are all occupied by 3. What you have expressed in decimal form is exactly equal to the fraction 1/3. It is not almost one-third; it is exactly one-third.

Similarly, if you write 0.9999... indicating an infinite number of decimal places all occupied by 9, the number depicted is exactly 1, not some number just a hair less than one. If it were a finite number of decimal points, it would be some number that was almost 1, but since it is an infinite number of decimal points then the value is indistinguishable from 1.

Or as I heard it explained by a 10 year old on NPR last week: Suppose you have a pizza and divide it into 3 equal parts. Each part is 0.3333... of the pizza, so if you add them all together you get 0.9999... But we also know that those three parts make up exactly one pizza. So 1 = 0.9999... QED

Therefore, (1) GoldDude is not smarter than a fifth grader and (2) everything is better with pizza.

Thank you timharrod for filling in the missing link between 0.333... and 0.999....

Next week we shall convince the great unwashed masses of the veracity of the Monty Hall problem.

GoldDude: 1 = 0.99999999...  X
Way wrong.
If this is probability, 1 is absolute certainty and anything less than 1 is not.
The sun will rise tomorrow?  Yeah, I'd give that 0.9999999999.... probability given that it is very unlikely the world will end, the earth will stop turning, the sun will burn out, etc. within the next 24 hours.  But it is not absolutely certain.

Since 0.9999.... = 1, then it is also an absolute certainty.

0.999999999999 is like a "tease" of 1.
You're infinitely close to it, but can't actually reach it.

Another similar way of looking at this is as follows:
Say you are standing 10 feet away from a line painted on the sidewalk.
Every minute, you walk half the distance remaining to the line... so 5.0 feet the first minute, 2.5 feet the second minutes, 1.25 feet the third minute, 0.625 feet the fourth minute, etc.
How long before you cross the line?
Never.  Because you are just infinitely moving closer to it, without ever reaching it.

While I agree with the infinite series and the foolishness of expressing anything in decimal form, I must semi agree from a probability standpoint that nothing equals one or zero.

The following exceptions prove the rule:

Death
Taxes
Chicks choose assholes instead of nice guys

/serious, decimal anything is bad math
// 22/7 is also bad math
/// 5318008

GoldDude: 0.999999999999 is like a "tease" of 1.
You're infinitely close to it, but can't actually reach it.

Another similar way of looking at this is as follows:
Say you are standing 10 feet away from a line painted on the sidewalk.
Every minute, you walk half the distance remaining to the line... so 5.0 feet the first minute, 2.5 feet the second minutes, 1.25 feet the third minute, 0.625 feet the fourth minute, etc.
How long before you cross the line?
Never.  Because you are just infinitely moving closer to it, without ever reaching it.

You don't understand the difference between a statement of equality and a limit "as it approaches" statement.

What you're missing is a precalc class that would fix this in about two class lessons. Or rereading through the first chapter of any calculus textbook.

Bottom line - you're showing you don't know what you're talking about by using an "approaching" analogy to disprove an equality statement.

GoldDude: 0.999999999999 is like a "tease" of 1.
You're infinitely close to it, but can't actually reach it.

Another similar way of looking at this is as follows:
Say you are standing 10 feet away from a line painted on the sidewalk.
Every minute, you walk half the distance remaining to the line... so 5.0 feet the first minute, 2.5 feet the second minutes, 1.25 feet the third minute, 0.625 feet the fourth minute, etc.
How long before you cross the line?
Never.  Because you are just infinitely moving closer to it, without ever reaching it.

Oh lordy, a limits troll.

Meh, I suppose they're rare enough, carry on.

sxacho: 1 = 0.999999999....

Here we go again.

From some perspectives a number cannot have an infinite series of digits. In this case one can't say that 1 = .999... Instead  one has to say for the series x1 = .9, x2 = .99, x3 = .999, etc. that as n grows larger xn approaches 1 and that only the limit of the series, not a particular member of it is 1. Sure this is a fussy technical distinction but it is important in handling all sorts of different limits in other areas.

Nothing like "beautiful equations" without ... you know ... the equations...

timharrod: alexanderplatz: So, if you want to express the quantity one-third, that is exactly one third, but you have to denote it in a decimal form, how do you do that? You write 0.3333... indicating an infinite number of decimal points that are all occupied by 3. What you have expressed in decimal form is exactly equal to the fraction 1/3. It is not almost one-third; it is exactly one-third.

Similarly, if you write 0.9999... indicating an infinite number of decimal places all occupied by 9, the number depicted is exactly 1, not some number just a hair less than one. If it were a finite number of decimal points, it would be some number that was almost 1, but since it is an infinite number of decimal points then the value is indistinguishable from 1.

And speaking of one-third, divide one by three.

Now divide 0.999999999... by three.

Same result. Proven equal.

Divide 0.99999... by one. Then divide one by one. Different results.

Not saying that "1=0.999..." is untrue. I just think the logic there is too inconsistent to serve as a proof of it.

Clockwork Kumquat: timharrod: alexanderplatz: So, if you want to express the quantity one-third, that is exactly one third, but you have to denote it in a decimal form, how do you do that? You write 0.3333... indicating an infinite number of decimal points that are all occupied by 3. What you have expressed in decimal form is exactly equal to the fraction 1/3. It is not almost one-third; it is exactly one-third.

Similarly, if you write 0.9999... indicating an infinite number of decimal places all occupied by 9, the number depicted is exactly 1, not some number just a hair less than one. If it were a finite number of decimal points, it would be some number that was almost 1, but since it is an infinite number of decimal points then the value is indistinguishable from 1.

And speaking of one-third, divide one by three.

Now divide 0.999999999... by three.

Same result. Proven equal.

Divide 0.99999... by one. Then divide one by one. Different results.

Not saying that "1=0.999..." is untrue. I just think the logic there is too inconsistent to serve as a proof of it.

And now I've found a Wikipedia article on 0.999..., which is actually kind of interesting, though everything beyond the Algebraic Proofs section might as well be the Voynich Manuscript to me. This sort of thing sometimes makes me wish I'd had more of an interest in math when I was a kid. Sometimes. I generally get over it.

Clockwork Kumquat: And now I've found a Wikipedia article on 0.999..., which is actually kind of interesting,

Thanks - I hadn't seen this one before.

Q: How many mathematicians does it take to screw in a lightbulb?
A: 0.999999....

Clockwork Kumquat: Divide 0.99999... by one. Then divide one by one. Different results.

I assume you are going like this?

0.99999.../1 = 0.99999...
1/1 = 1

In this case they  are the same result. Just with different notation. Try dividing by 3 (or 9):

0.99999.../3 = 0.33333...
1/3 = 0.33333...
Therefore:
0.99999.../3 = 1/3

And if a/x=b/x, then a=b. Therefore:

0.99999... = 1

Based on that, if we refer back to the original statements,

0.99999.../1 = 1/1

And we are all happy.

rubi_con_man: Nothing like "beautiful equations" without ... you know ... the equations...

It is only in the mysterious equation of love that any logical reasons can be found.

It's true by definition. 0.999999... is a limit, specifically:

limit _{n->\infinity} ( \sum_1^n 9 * 10^-n)

and that limit is 1. That's really the only way to see it.

0.999999 is like that Voyager episode where they became lizardsalamanders.

timharrod: Now divide 0.999999999... by three.

Same result. Proven equal.

They're just going to say that 0.33333333... isn't 1/3; it's just really close.

GoldDude: 1 = 0.99999999...  X
Way wrong.
If this is probability, 1 is absolute certainty and anything less than 1 is not.
The sun will rise tomorrow?  Yeah, I'd give that 0.9999999999.... probability given that it is very unlikely the world will end, the earth will stop turning, the sun will burn out, etc. within the next 24 hours.  But it is not absolutely certain.

This was something I learned in a math class - probably calc - when in college.

Since 1/3+1/3+1/3 = 1
and
since 1/3 = .33333333
then
.333...+.333...+.333...= 1.000
and NOT .999...

Great stuff.

aerojockey: timharrod: Now divide 0.999999999... by three.

Same result. Proven equal.

They're just going to say that 0.33333333... isn't 1/3; it's just really close.

(Shrug) Then they're responsible for identifying the difference.

Clockwork Kumquat:

Divide 0.99999... by one. Then divide one by one. Different results.

No... differently written results.

Ivo Shandor: Was this not covered yet, or did I just miss it?
[i.imgur.com image 100x30]

Came for this, but am shocked at the lack of Newton:

F = dp/dt

and Maxwell:

It only explains ALL of electrodynamics.

If 0.999... = 1, then 1/0 = infinity. Yes, you "might as well say they are both equal", but they are actually limits, and unfortunately, math hasn't actually created infinity as a real number yet.

Also, Euler's formula is beautiful in that it illustrates that the way we look at mathematics is flawed. A real number and an "imaginary" number together equals another real number? Hint: it's not that it's "imaginary"; it's that the number system we use is relative and cannot support the whole of mathematics.

I've always had a soft spot for propositional logic and, specifically Modus ponens, since it's the underpinning of pretty much all of reality. I took a number theory class as an elective in college and it help bridge the computational theory classes (heavy on formal logic and graph theory) I took with the other math courses I had to take ( all of calculas, differential equations, linear algebra ). At the root of it all was good'ole Modus ponens.

http://en.wikipedia.org/wiki/Modus_ponens

GoldDude: 1 = 0.99999999...  X
Way wrong.
If this is probability, 1 is absolute certainty and anything less than 1 is not.
The sun will rise tomorrow?  Yeah, I'd give that 0.9999999999.... probability given that it is very unlikely the world will end, the earth will stop turning, the sun will burn out, etc. within the next 24 hours.  But it is not absolutely certain.

By definition A = B if and only if A-B=0. Let's take a look at the difference between 1 and 0.999999 (which I'll just call 0.9*). Since we doing the long hand would be an exercise in futility, let us consider X(n) be 0.9 with n 9's. As n approaches infinity, X[n] approaches our 0.9*. Luckily 1 is a constant. 1-0.9* = limit 1-X[n] as n approaches infinity.

Choose E greater than zero. Select n > -log base 10 of E. In other words, write E as Ax10^B where A>0 and let n be greater than that exponent (-B when E is small). Then |1-X[n]| = 1 - 0.000(... n 0's total)1 < E. Since we picked E arbitrarily, this holds for all E>0. This is the definition of convergence. Therefore the sequence 1-X[n] converges to zero which in turn means that 1-0.9* = 0. By definition then, 1 = 0.9*. QED.

/RIGOR!
//Rigor would be a great middle name to inflict upon a child

blue_2501: If 0.999... = 1, then 1/0 = infinity. Yes, you "might as well say they are both equal", but they are actually limits, and unfortunately, math hasn't actually created infinity as a real number yet.

Also, Euler's formula is beautiful in that it illustrates that the way we look at mathematics is flawed. A real number and an "imaginary" number together equals another real number? Hint: it's not that it's "imaginary"; it's that the number system we use is relative and cannot support the whole of mathematics.

Rubbish. Euler's formula does no such thing.

In fact, everything you said above is completely meaningless word salad. You throw in words like "real" and "imaginary" and "infinity" and "limits" and even "equals" without seeming to realize that these words have specific and precise meanings to mathematicians.

Hint: People who don't know what they are talking about should refrain from trying to paint word pictures about mathematical concepts they demonstrably don't understand.

czetie: blue_2501: If 0.999... = 1, then 1/0 = infinity. Yes, you "might as well say they are both equal", but they are actually limits, and unfortunately, math hasn't actually created infinity as a real number yet.

Also, Euler's formula is beautiful in that it illustrates that the way we look at mathematics is flawed. A real number and an "imaginary" number together equals another real number? Hint: it's not that it's "imaginary"; it's that the number system we use is relative and cannot support the whole of mathematics.

Rubbish. Euler's formula does no such thing.

In fact, everything you said above is completely meaningless word salad. You throw in words like "real" and "imaginary" and "infinity" and "limits" and even "equals" without seeming to realize that these words have specific and precise meanings to mathematicians.

Hint: People who don't know what they are talking about should refrain from trying to paint word pictures about mathematical concepts they demonstrably don't understand.

I had a similar "what the I don't even" reaction to that vocabulary vomit as well.  Reminds me of the debates I'd have with many philosophy majors I knew in college.  They'd tell me that modern calculus was completely incorrect, based on their interpretation of Zeno's paradox, or some other semantic triviality.  They also don't handle the specifics of relativity or mass/energy duality well.  Three semester of physics training and about two semester of calculus, and those problems completely disappear.

Don't get me wrong, I love philosophy, even minored in it.  But it doesn't address technical aspects of math and physics particularly well.  Now, if we want to chat about ethical systems, or even the ethical application of the hard sciences, THAT'S a good conversation and well worth having.

sakanagai: GoldDude: 1 = 0.99999999...  X
Way wrong.
If this is probability, 1 is absolute certainty and anything less than 1 is not.
The sun will rise tomorrow?  Yeah, I'd give that 0.9999999999.... probability given that it is very unlikely the world will end, the earth will stop turning, the sun will burn out, etc. within the next 24 hours.  But it is not absolutely certain.

By definition A = B if and only if A-B=0. Let's take a look at the difference between 1 and 0.999999 (which I'll just call 0.9*). Since we doing the long hand would be an exercise in futility, let us consider X(n) be 0.9 with n 9's. As n approaches infinity, X[n] approaches our 0.9*. Luckily 1 is a constant. 1-0.9* = limit 1-X[n] as n approaches infinity.

Choose E greater than zero. Select n > -log base 10 of E. In other words, write E as Ax10^B where A>0 and let n be greater than that exponent (-B when E is small). Then |1-X[n]| = 1 - 0.000(... n 0's total)1 < E. Since we picked E arbitrarily, this holds for all E>0. This is the definition of convergence. Therefore the sequence 1-X[n] converges to zero which in turn means that 1-0.9* = 0. By definition then, 1 = 0.9*. QED.

/RIGOR!
//Rigor would be a great middle name to inflict upon a child

/QED = Quit, Enough, Done.

Khellendros: czetie: blue_2501: If 0.999... = 1, then 1/0 = infinity. Yes, you "might as well say they are both equal", but they are actually limits, and unfortunately, math hasn't actually created infinity as a real number yet.

Also, Euler's formula is beautiful in that it illustrates that the way we look at mathematics is flawed. A real number and an "imaginary" number together equals another real number? Hint: it's not that it's "imaginary"; it's that the number system we use is relative and cannot support the whole of mathematics.

Rubbish. Euler's formula does no such thing.

In fact, everything you said above is completely meaningless word salad. You throw in words like "real" and "imaginary" and "infinity" and "limits" and even "equals" without seeming to realize that these words have specific and precise meanings to mathematicians.

Hint: People who don't know what they are talking about should refrain from trying to paint word pictures about mathematical concepts they demonstrably don't understand.

I had a similar "what the I don't even" reaction to that vocabulary vomit as well.  Reminds me of the debates I'd have with many philosophy majors I knew in college.  They'd tell me that modern calculus was completely incorrect, based on their interpretation of Zeno's paradox, or some other semantic triviality.  They also don't handle the specifics of relativity or mass/energy duality well.  Three semester of physics training and about two semester of calculus, and those problems completely disappear.

Don't get me wrong, I love philosophy, even minored in it.  But it doesn't address technical aspects of math and physics particularly well.  Now, if we want to chat about ethical systems, or even the ethical application of the hard sciences, THAT'S a good conversation and well worth having.

Ahh, good ol' imaginary numbers.  Without fail, when I introduce the topic in algebra i get the question "Why do we study imaginary numbers if they are not real?"  Which is a valid question and for anyone who wants a way to try to explain the idea, feel free to borrow this:

If something is real, then you can draw a picture of it, correct?

Then draw me a picture of heat or wind.  Most will be unable to, Some creative types will draw a picture of a frying pan with the "heat squiggles" or a bending tree with "wind squiggles".  From here just keep pressing the idea that you can't see wind or wind doesn't have squiggle lines in reality.  Eventually you prod them to say something along the lines of 'you have to imagine the heat/wind'.  You can see the effects of heat/wind on other objects, but not heat/wind itself.  Same with imaginary numbers.  We may not be able to "see" imaginary numbers, but we can see their impact on real numbers.

Hyjamon: Khellendros: czetie: blue_2501: If 0.999... = 1, then 1/0 = infinity. Yes, you "might as well say they are both equal", but they are actually limits, and unfortunately, math hasn't actually created infinity as a real number yet.

Also, Euler's formula is beautiful in that it illustrates that the way we look at mathematics is flawed. A real number and an "imaginary" number together equals another real number? Hint: it's not that it's "imaginary"; it's that the number system we use is relative and cannot support the whole of mathematics.

Rubbish. Euler's formula does no such thing.

In fact, everything you said above is completely meaningless word salad. You throw in words like "real" and "imaginary" and "infinity" and "limits" and even "equals" without seeming to realize that these words have specific and precise meanings to mathematicians.

Hint: People who don't know what they are talking about should refrain from trying to paint word pictures about mathematical concepts they demonstrably don't understand.

I had a similar "what the I don't even" reaction to that vocabulary vomit as well.  Reminds me of the debates I'd have with many philosophy majors I knew in college.  They'd tell me that modern calculus was completely incorrect, based on their interpretation of Zeno's paradox, or some other semantic triviality.  They also don't handle the specifics of relativity or mass/energy duality well.  Three semester of physics training and about two semester of calculus, and those problems completely disappear.

Don't get me wrong, I love philosophy, even minored in it.  But it doesn't address technical aspects of math and physics particularly well.  Now, if we want to chat about ethical systems, or even the ethical application of the hard sciences, THAT'S a good conversation and well worth having.

Ahh, good ol' imaginary numbers.  Without fail, when I introduce the topic in algebra i get the question "Why do we study imaginary numbers if they are not real?"  Which is a valid question and for anyone who wants a way to try to explain the idea, feel free to borrow this:

If something is real, then you can draw a picture of it, correct?

Then draw me a picture of heat or wind.  Most will be unable to, Some creative types will draw a picture of a frying pan with the "heat squiggles" or a bending tree with "wind squiggles".  From here just keep pressing the idea that you can't see wind or wind doesn't have squiggle lines in reality.  Eventually you prod them to say something along the lines of 'you have to imagine the heat/wind'.  You can see the effects of heat/wind on other objects, but not heat/wind itself.  Same with imaginary numbers.  We may not be able to "see" imaginary numbers, but we can see their impact on real numbers.

Great explanation!

Khellendros: czetie: blue_2501: If 0.999... = 1, then 1/0 = infinity. Yes, you "might as well say they are both equal", but they are actually limits, and unfortunately, math hasn't actually created infinity as a real number yet.

Also, Euler's formula is beautiful in that it illustrates that the way we look at mathematics is flawed. A real number and an "imaginary" number together equals another real number? Hint: it's not that it's "imaginary"; it's that the number system we use is relative and cannot support the whole of mathematics.

Rubbish. Euler's formula does no such thing.

In fact, everything you said above is completely meaningless word salad. You throw in words like "real" and "imaginary" and "infinity" and "limits" and even "equals" without seeming to realize that these words have specific and precise meanings to mathematicians.

Hint: People who don't know what they are talking about should refrain from trying to paint word pictures about mathematical concepts they demonstrably don't understand.

I had a similar "what the I don't even" reaction to that vocabulary vomit as well.  Reminds me of the debates I'd have with many philosophy majors I knew in college.  They'd tell me that modern calculus was completely incorrect, based on their interpretation of Zeno's paradox, or some other semantic triviality.  They also don't handle the specifics of relativity or mass/energy duality well.  Three semester of physics training and about two semester of calculus, and those problems completely disappear.

Don't get me wrong, I love philosophy, even minored in it.  But it doesn't address technical aspects of math and physics particularly well.  Now, if we want to chat about ethical systems, or even the ethical application of the hard sciences, THAT'S a good conversation and well worth having.

Lol @ zeno's paradox. Makes me think of the comic "why engineers aren't allowed at philosophy conferences" after answering the clone question. "Easy, the person that came first is the real one, whoever comes later is a clone. Next! BTW will they all be this easy?"

wjllope: SevenizGud: PV=nRT

/because fark pure math

"fark pure math" after the ideal gas law?  nice

Hollie Maea: SevenizGud: PV=nRT

/because fark pure math

Considering the dumbass things I've seen you say in other threads, it seems appropriate that you would say "Fark pure math" and cite an equation that is never accurate.

All you haters will now please get on the same page.

SevenizGud: All you haters will now please get on the same page.

Those two responses were perfectly consistent... You mentioned rejecting "pure math"
and quoted a specific "physics" equation which implied your preference for something more
realistic/useful.

But PV=nRT, the  ideal gas law, is really only applicable for thought experiments and for
teaching high school students the very basics. Even the Van der Waals equation, which
extends PV=nRT in explicit and frankly trivial ways w/ additional parameters to include the
treatment of excluded volumes and pairwise attractive forces is still just an approximation.

Clearly, we read "because fark pure math" as something that was incongruous w/ your use
of an equally pure and generally inapplicable equation from physics...

cheers

SineSwiper: LrdPhoenix: GoldDude: 1 = 0.99999999...  X
Way wrong.
If this is probability, 1 is absolute certainty and anything less than 1 is not.
The sun will rise tomorrow?  Yeah, I'd give that 0.9999999999.... probability given that it is very unlikely the world will end, the earth will stop turning, the sun will burn out, etc. within the next 24 hours.  But it is not absolutely certain.

So, precisely how much less likely is 0.9999999999.... compared to 1?

0.01, with an overscore on the second zero.

Duh.

I like the cut of your genoa. I also agree with your answer which causes me to (see first sentence).

Hyjamon: Ahh, good ol' imaginary numbers.  Without fail, when I introduce the topic in algebra i get the question "Why do we study imaginary numbers if they are not real?"  Which is a valid question and for anyone who wants a way to try to explain the idea, feel free to borrow this:

If something is real, then you can draw a picture of it, correct?

Then draw me a picture of heat or wind.  Most will be unable to, Some creative types will draw a picture of a frying pan with the "heat squiggles" or a bending tree with "wind squiggles".  From here just keep pressing the idea that you can't see wind or wind doesn't have squiggle lines in reality.  Eventually you prod them to say something along the lines of 'you have to imagine the heat/wind'.  You can see the effects of heat/wind on other objects, but not heat/wind itself.  Same with imaginary numbers.  We may not be able to "see" imaginary numbers, but we can see their impact on real numbers.

I didn't say imaginary numbers aren't real, but that the fact that you can't solve for them is a problem with the way we view mathematics and our numeral system, not that it's "unsolvable".  The square root of -1 really does exist, and we use it to great effect.  The "i" is just a way of us trying to shoehorn the number into our system, and sidestep the problem of sqrt(-1) being a mathematically paradox.

And yes, I will continue to talk about this sort of thing  philosophically.  Philosophy is applied logic, and logic is applied mathematics, but it should make sense from both directions.

/yeah, I probably suck at trying to explain this; I do like Hyjamon's explanation of imaginary numbers

blue_2501: Hyjamon: Ahh, good ol' imaginary numbers.  Without fail, when I introduce the topic in algebra i get the question "Why do we study imaginary numbers if they are not real?"  Which is a valid question and for anyone who wants a way to try to explain the idea, feel free to borrow this:

If something is real, then you can draw a picture of it, correct?

Then draw me a picture of heat or wind.  Most will be unable to, Some creative types will draw a picture of a frying pan with the "heat squiggles" or a bending tree with "wind squiggles".  From here just keep pressing the idea that you can't see wind or wind doesn't have squiggle lines in reality.  Eventually you prod them to say something along the lines of 'you have to imagine the heat/wind'.  You can see the effects of heat/wind on other objects, but not heat/wind itself.  Same with imaginary numbers.  We may not be able to "see" imaginary numbers, but we can see their impact on real numbers.

I didn't say imaginary numbers aren't real, but that the fact that you can't solve for them is a problem with the way we view mathematics and our numeral system, not that it's "unsolvable".  The square root of -1 really does exist, and we use it to great effect.  The "i" is just a way of us trying to shoehorn the number into our system, and sidestep the problem of sqrt(-1) being a mathematically paradox.

And yes, I will continue to talk about this sort of thing  philosophically.  Philosophy is applied logic, and logic is applied mathematics, but it should make sense from both directions.

/yeah, I probably suck at trying to explain this; I do like Hyjamon's explanation of imaginary numbers

Actually, we don't use "i" as a means of shoehorning it into our number system, its discovery led us to discover a richer field of "numbers" called the complex plane.  Much like integers are richer than whole numbers, rational numbers richer than integers, and irrational numbers added more density into our real number system.

"i" is just nice notation and more efficient than sqrt (-1) (think number of strokes to write the symbol).  when sqrt(-1) was first encountered in solving quadratics, people would leave them in the equations and they would work themselves out later on.  Imaginary numbers are not shoehorned into the real number system, but the reverse, we discovered real numbers to be a subset of the complex plane (a+bi).  real numbers are simply (a+0i) (a plus zero i).

/my favorite math book is "Basic Complex Analysis".  I never realized the oxymoronic title until my journalism roommate spotted it on my desk.

//Math is fun (sadly it really isn't)

dywed88: In this case they are the same result. Just with different notation. Try dividing by 3 (or 9):

0.99999.../3 = 0.33333...
1/3 = 0.33333...
Therefore:
0.99999.../3 = 1/3

I want to stress that I do actually accept that 0.999... and 1 are in fact the same number; what I question is whether this method is adequate proof of this. Division by one was probably not the best example to use. If we use a number which is not a divisor of 9, such as 2, we get the (apparently) different results of 0.4999... and 0.5. To say that "they are the same result. Just with different notation" is somewhat circular reasoning, since the fact that the different notations are the same result is the very thing that we are trying to prove.

Clockwork Kumquat: dywed88: In this case they are the same result. Just with different notation. Try dividing by 3 (or 9):

0.99999.../3 = 0.33333...
1/3 = 0.33333...
Therefore:
0.99999.../3 = 1/3

I want to stress that I do actually accept that 0.999... and 1 are in fact the same number; what I question is whether this method is adequate proof of this. Division by one was probably not the best example to use. If we use a number which is not a divisor of 9, such as 2, we get the (apparently) different results of 0.4999... and 0.5. To say that "they are the same result. Just with different notation" is somewhat circular reasoning, since the fact that the different notations are the same result is the very thing that we are trying to prove.

I'm not a mathematician, but if a hypothetical person who doubts equality isn't convinced based on the fact that they're interchangeable within so simple an arithmetic equation, I don't know what will convince them. It's not like one of the lines of threes will eventually end.

timharrod: Clockwork Kumquat: dywed88: In this case they are the same result. Just with different notation. Try dividing by 3 (or 9):

0.99999.../3 = 0.33333...
1/3 = 0.33333...
Therefore:
0.99999.../3 = 1/3

I want to stress that I do actually accept that 0.999... and 1 are in fact the same number; what I question is whether this method is adequate proof of this. Division by one was probably not the best example to use. If we use a number which is not a divisor of 9, such as 2, we get the (apparently) different results of 0.4999... and 0.5. To say that "they

are the same result. Just with different notation" is somewhat circular reasoning, since the fact that the different notations are the same result is the very thing that we are trying to prove.

I'm not a mathematician, but if a hypothetical person who doubts equality isn't convinced based on the fact that they're interchangeable within so simple an arithmetic equation, I don't know what will convince them. It's not like one of the lines of threes will eventually end.

The hypothetical doubter is likely to ask why interchangeable quotients do not arise consistently with all denominators, and not just a few, if the numerators are equal. I don't mean to be trollish; the equality is just highly counterintuitive, and I don't think this is a very solid means of illustrating it. I like the Digit Manipulation proof, though the part of the "Discussion" section immediately following it which reads, "But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all. William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation," was pretty much written with me in mind.

timharrod: Clockwork Kumquat: dywed88: In this case they are the same result. Just with different notation. Try dividing by 3 (or 9):

0.99999.../3 = 0.33333...
1/3 = 0.33333...
Therefore:
0.99999.../3 = 1/3

I want to stress that I do actually accept that 0.999... and 1 are in fact the same number; what I question is whether this method is adequate proof of this. Division by one was probably not the best example to use. If we use a number which is not a divisor of 9, such as 2, we get the (apparently) different results of 0.4999... and 0.5. To say that "they are the same result. Just with different notation" is somewhat circular reasoning, since the fact that the different notations are the same result is the very thing that we are trying to prove.

I'm not a mathematician, but if a hypothetical person who doubts equality isn't convinced based on the fact that they're interchangeable within so simple an arithmetic equation, I don't know what will convince them. It's not like one of the lines of threes will eventually end.

So to believe this proof you have to assume:

1. the obvious mapping f from rationals to their infinite decimal representation is invertible
2. f is like a field morphism, i.e. that it preserves operations on both sides, like f(a + b) = f(a) + f(b), f(a*b) = f(a) * f(b)
3. the specific definition of addition and multiplication for decimal expansions is the one you are thinking about (there are not others?)
4. oops, f doesn't work because there is some f(.99999) and f(1.00000) that should be equal but they are not.
5. To patch up the hole, let's make them equal. tada, problem solved!

all this without looking at the basic definition of decimal expansion. I think Clockwork is right, you are putting the cart before the horse..

murray208: So to believe this proof you have to assume:

1. the obvious mapping f from rationals to their infinite decimal representation is invertible
2. f is like a field morphism, i.e. that it preserves operations on both sides, like f(a + b) = f(a) + f(b), f(a*b) = f(a) * f(b)
3. the specific definition of addition and multiplication for decimal expansions is the one you are thinking about (there are not others?)
4. oops, f doesn't work because there is some f(.99999) and f(1.00000) that should be equal but they are not.
5. To patch up the hole, let's make them equal. tada, problem solved!

all this without looking at the basic definition of decimal expansion. I think Clockwork is right, you are putting the cart before the horse..

You've got the math down better than I, but it seems to me that to doubt equality is to suggest that one or both strings of threes comes to an end, and to deny equality is to claim it. I regret that that's the limit of my ability to argue.

Ther's an xkcd for every occasion.

timharrod: murray208: So to believe this proof you have to assume:

1. the obvious mapping f from rationals to their infinite decimal representation is invertible
2. f is like a field morphism, i.e. that it preserves operations on both sides, like f(a + b) = f(a) + f(b), f(a*b) = f(a) * f(b)
3. the specific definition of addition and multiplication for decimal expansions is the one you are thinking about (there are not others?)
4. oops, f doesn't work because there is some f(.99999) and f(1.00000) that should be equal but they are not.
5. To patch up the hole, let's make them equal. tada, problem solved!

all this without looking at the basic definition of decimal expansion. I think Clockwork is right, you are putting the cart before the horse..

You've got the math down better than I, but it seems to me that to doubt equality is to suggest that one or both strings of threes comes to an end, and to deny equality is to claim it. I regret that that's the limit of my ability to argue.

It has nothing to do with doubting the facts. The point is about how a mathematical proof supposed to start with some simpler axioms and indisputably prove something more complicated. That's a stronger statement than 'to give compelling evidence for', which how the word proof might be defined for common use.

Suppose for example you wanted to get the area that the area of a 2 unit square is 4. You could do that by integrating x between 0 and 2. You'll get 4, but this isn't really a proof in math. You've proved the area of a square assuming calculus, but don't you think that proof of calculus assumes the area of a square at some point? It not a proof if it doesn't expand the body of knowledge in some way.

Some gaps in my understanding:

Are 0.999... and 1 equal because the difference between them is an infinitesimal quantity?

If so, can this difference be represented by the fraction "one over infinity?"

If so, is that fraction equal to zero?

How would it be written as a decimal? I took SineSwiper's answer to that as a jest, but now I'm wondering.

Clockwork Kumquat: Some gaps in my understanding:

Are 0.999... and 1 equal because the difference between them is an infinitesimal quantity?

If so, can this difference be represented by the fraction "one over infinity?"

If so, is that fraction equal to zero?

How would it be written as a decimal? I took SineSwiper's answer to that as a jest, but now I'm wondering.

You need to appeal to the definition:  Rationals are numbers that can be written as fractions, and Reals are numbers that can be written as limits of Cauchy sequences of Rationals.

1 and 0.9999999. These are two limits (the first one being trivial). We define two limits to be equal if the limit of the difference is zero. Note that this definition is reflective, symmetric and transient, so it 'behaves' like we would expect it to.

murray208: The point is about how a mathematical proof supposed to start with some simpler axioms and indisputably prove something more complicated.

Absolutely incorrect.  The proof that gives you simple addition and multiplication sits on top of concepts FAR more complicated than arithmetic.  In fact, most simple proofs, even modern ones, sit on top of a far more complicated foundation and sets of assumptions.  While your statement is generally true for philosophical debating structures, it is not for mathematical or physics principles.

Do not mistake a mathematical learning pyramid structure (learning simple things as tools to do more complicated things, as you learn in school) for the proof structure that holds it up.

Clockwork Kumquat: How would it be written as a decimal? I took SineSwiper's answer to that as a jest, but now I'm wondering.

That's the reason that I first came to understand that 0.99999.... = 1.  I think it was someone in a Fark thread who explained that there's no such thing as a 1 coming after an infinite string of 0's. So, yes, you can write some shorthand for it, but it's not a legitimate number.

Khellendros: murray208: The point is about how a mathematical proof supposed to start with some simpler axioms and indisputably prove something more complicated.

Absolutely incorrect.  The proof that gives you simple addition and multiplication sits on top of concepts FAR more complicated than arithmetic.  In fact, most simple proofs, even modern ones, sit on top of a far more complicated foundation and sets of assumptions.  While your statement is generally true for philosophical debating structures, it is not for mathematical or physics principles.

Do not mistake a mathematical learning pyramid structure (learning simple things as tools to do more complicated things, as you learn in school) for the proof structure that holds it up.

Yes yes, I know about principa mathematica and all that. Obviously. Look, if you look at the context of what i said I was using simple to mean 'axiom' and complex to mean 'not following from axioms'. I was also talking someone to someone who didn't understand the concept of proof, so was trying to ease him in without more jargon.

0.9999 = 1

that is all

murray208: 1 and 0.9999999. These are two limits (the first one being trivial). We define two limits to be equal if the limit of the difference is zero. Note that this definition is reflective, symmetric and transient, so it 'behaves' like we would expect it to.

Huh. That triggered a faint glimmer of memory somewhere deep in my mind. I'm thinking now that I actually might have learned about a lot of this way back in math class, too many decades ago. Anyway that's probably as far as I need to go here. Thanks.

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