The plane takes off, 0.999... = 1, but what about...

[thesurge]

Originally posted by: fitzov

Originally posted by: khas

Hyperreal numbers are real numbers. Newton and Leibniz used infinitesimals, some mathematicians prefer their use to limits in the Calculus, and they often provide a simpler analysis to Calc problems. But even if you don't use them, the result does not make 0.999... equal to 1.

Here is a quote from a common Calculus book:

"The limit of an infinite series is sometimes called its 'sum at infinity,' but of course this is not a sum in the usual arithmetical sense when the number of terms is finite. You can't obtain the "sum" of an infinite series by adding because the number of terms to be added is infinite. When we speak of the "sum" of an infinite series, this is just a short way of naming its limit...

So what you're really saying when you say 0.999... = 1, is that the limit of the infinite series (denoted by '...')0.999 is 1. This is, strictly speaking, not the same thing as equality.

I believe that saying the Limit of ___ is 1 is the same as saying the Limit of ___ = 1. ('is' == 'equals')

Correct me if I'm wrong, but what you just quoted there indeed says that .999... is equal to 1.

I believe that book is saying that the notation .999...(repeating) is not a number but a representation of a Limit. This limit, as we've all read about to death, equals 1.

Or perhaps what you're saying is that the Limit does not equal 1, but converges at 1 (is that the proper term?). And that convergence is not the same as equality. I can't vouch for that one... might have to ask a mathematician.

It's the limit of the function that equals 1, not the function itself.

In other words 'f(x) = y' is not the same as 'lim f(x) as x approaches infinity =y'

Read this:

http://www.math.wisc.edu/~keisler/chapter_9.pdf

Right but my "argument" is, isn't .999... already inferred as the limit of a series (9/10^k) or is it just the sum of the infinite sequence (9/10^k)?

 

[potato28]

Would it be 2mL of product, or 2mL of liquid? Because as many ppl have stated, it could precipitate or evapourate.

 

[SilthDraeth]

For some reason I wasn't thinking along the lines of infinitely reoccurring decimals when talking about .999...

 

[Rastus]

No.

 

[khas]

It's the limit of the function that equals 1, not the function itself.

In other words 'f(x) = y' is not the same as 'lim f(x) as x approaches infinity =y'

Read this:

http://www.math.wisc.edu/~keisler/chapter_9.pdf

Right but my "argument" is, isn't .999... already inferred as the limit of a series (9/10^k) or is it just the sum of the infinite sequence (9/10^k)?

I understand the concepts of infinite sums/series, limits, converging/diverging functions, etc, but thanks for the link anyways. I confess it was a bit too long to read in depth at the moment... especially since i'd already gone through that stage of math long ago... and now that i'm out of college, don't feel the need to revisit this part of math in textbook form.

But really, I'm on the same field as thesurge here... and that chapter (as far as I could see by skimming it) didn't discuss the definition of what the notation .999... is defined as. (just let me know the page it mentions that if i missed it)

It discussed infinite sums... and we know that .999...'s equivalent is 9/10+9/100+9/1000+...

but is .999... defined as the function of the sums of 9/10^n n=1-to-infinity... or the limit of that function... or is it literally just a mathematically equivalent number to 1. I confess this may be more a question of symmantics, and is rather irrelavent to the earlier discussion in this part of the thread in which someone did not believe that .999... is always equivalent to 1 and can always be considered 1 as long as it's truly infinitely repeating. which it can be.

 

[jagec]

Originally posted by: Injury

The only way .999 repeating can't equal 1 is if you truncate the number of 9's

fixed.

 

[Injury]

Originally posted by: jagec

Originally posted by: Injury

The only way .999 repeating can't equal 1 is if you truncate the number of 9's

fixed.

That just makes you even more wrong.

 

[oCxTiTaN]

People who don't understand math shouldn't try to use it...

 

[GoodRevrnd]

--

 

[exdeath]

Originally posted by: JohnCU
If you have 2 liquids (you don't know what they are), 1 mL of each and you add them together, can you be 100% sure that the resulting solution is 2 mL?

No not at all. Try this with water and alcohol. The density will increase because the molecules intermingle and the volume of the sum will be less than the two parts, even if there isn't a reaction.

 

[fitzov]

Originally posted by: thesurge

Originally posted by: fitzov

Originally posted by: khas

Hyperreal numbers are real numbers. Newton and Leibniz used infinitesimals, some mathematicians prefer their use to limits in the Calculus, and they often provide a simpler analysis to Calc problems. But even if you don't use them, the result does not make 0.999... equal to 1.

Here is a quote from a common Calculus book:

"The limit of an infinite series is sometimes called its 'sum at infinity,' but of course this is not a sum in the usual arithmetical sense when the number of terms is finite. You can't obtain the "sum" of an infinite series by adding because the number of terms to be added is infinite. When we speak of the "sum" of an infinite series, this is just a short way of naming its limit...

So what you're really saying when you say 0.999... = 1, is that the limit of the infinite series (denoted by '...')0.999 is 1. This is, strictly speaking, not the same thing as equality.

I believe that saying the Limit of ___ is 1 is the same as saying the Limit of ___ = 1. ('is' == 'equals')

Correct me if I'm wrong, but what you just quoted there indeed says that .999... is equal to 1.

I believe that book is saying that the notation .999...(repeating) is not a number but a representation of a Limit. This limit, as we've all read about to death, equals 1.

Or perhaps what you're saying is that the Limit does not equal 1, but converges at 1 (is that the proper term?). And that convergence is not the same as equality. I can't vouch for that one... might have to ask a mathematician.

It's the limit of the function that equals 1, not the function itself.

In other words 'f(x) = y' is not the same as 'lim f(x) as x approaches infinity =y'

Read this:

http://www.math.wisc.edu/~keisler/chapter_9.pdf

Right but my "argument" is, isn't .999... already inferred as the limit of a series (9/10^k) or is it just the sum of the infinite sequence (9/10^k)?

'0.999...' is a function of 0.999
'...' denotes a function.

 

[compuwiz1]

Originally posted by: Syringer

Originally posted by: BrownTown
.99999 DOES EQUAL 1 (this is not up for debate, if you don't understand why this is true thats your own fault, but this is a fact in mathematics)

No it doesn't

Mixing ethanol and water results in a considerbly smaller volume since they hydrogen bond tightly and become more dense, there are hundreds of other examples, but ethanol + water is most famous.

Whats this plane taking off thing, never heard that b4?

If a plane is on a conveyor belt, will it take off?

Plane's do not take off from conveyors, but from catapults.

 

[smack Down]

Assuming the plane can take off from the treadmill and we replace the plane with a car will it move forward?

 

[DrPizza]

Originally posted by: smack Down
Assuming the plane can take off from the treadmill and we replace the plane with a car will it move forward?

That situation makes no sense. Some people may be thinking "well, if the car is moving forward, then the treadmill matches the speed of the car, keeping the car from moving." Well, if that's the case, then the car ISN'T moving forward, making the car's speed equal to zero. Thus, the treadmill wouldn't be moving either.

If the question were rephrased to "if the treadmill moved in the opposite direction at a speed equal to what the car's speedometer said..." then the car would be stationary. In fact, this situation actually is used in labs (well, I wouldn't necessarily call the thing the tires sit on a "treadmill")

 

[smack Down]

Originally posted by: DrPizza

Originally posted by: smack Down
Assuming the plane can take off from the treadmill and we replace the plane with a car will it move forward?

That situation makes no sense. Some people may be thinking "well, if the car is moving forward, then the treadmill matches the speed of the car, keeping the car from moving." Well, if that's the case, then the car ISN'T moving forward, making the car's speed equal to zero. Thus, the treadmill wouldn't be moving either.

If the question were rephrased to "if the treadmill moved in the opposite direction at a speed equal to what the car's speedometer said..." then the car would be stationary. In fact, this situation actually is used in labs (well, I wouldn't necessarily call the thing the tires sit on a "treadmill")

Umm how is that different then the orginal question? The fact that it is a plane has no effect on the question.

 

[DrPizza]

Originally posted by: smack Down

Originally posted by: DrPizza

Originally posted by: smack Down
Assuming the plane can take off from the treadmill and we replace the plane with a car will it move forward?

That situation makes no sense. Some people may be thinking "well, if the car is moving forward, then the treadmill matches the speed of the car, keeping the car from moving." Well, if that's the case, then the car ISN'T moving forward, making the car's speed equal to zero. Thus, the treadmill wouldn't be moving either.

If the question were rephrased to "if the treadmill moved in the opposite direction at a speed equal to what the car's speedometer said..." then the car would be stationary. In fact, this situation actually is used in labs (well, I wouldn't necessarily call the thing the tires sit on a "treadmill")

Umm how is that different then the orginal question? The fact that it is a plane has no effect on the question.

The plane accelerates forward because there is a net force on the plane in the forward direction. The treadmill provides a very small amount of force backwards on the plane (equal to the rolling resistance of the wheels) while the air provides a forward force on the plane per Newton's 3rd law (equal and opposite; plane is pushing backwards against the air.) The propeller (or jet engines) are what make the plane move forward.

In the case of the car, it moves forward by pushing backwards against the treadmill. The tires pushing against the ground (and the ground pushing back; again, Newton's 3rd law) are what makes it move forward (if it moved forward.)


If it helps, think of it this way: Imagine you have a toy plane with two motors in it. One spins the tires, and the other spins the propellers. Place that toy onto a super super slippery surface, more slippery than ice. Turn the motor on for the wheels, and the plane will just sit there with its wheels spinning in place. Turn on the motor for the propeller, and the plane will move forward. Leave the motor on for the propeller, and turn off the motor for the wheels; the plane will still move forward. Rewire the motor for the wheels so that it spins backwards. The plane will *still* move forwards, even if the wheels are spinning backwards.

In the problem with the real plane, the wheels are nearly irrelevant; the only thing they account for is some amount of force preventing the plane moving, equal to the frictional force of rolling friction which is quite low compared to the force due to the propellers pushing the plane forward.

 

[smack Down]

Originally posted by: DrPizza

Originally posted by: smack Down

Originally posted by: DrPizza

Originally posted by: smack Down
Assuming the plane can take off from the treadmill and we replace the plane with a car will it move forward?

That situation makes no sense. Some people may be thinking "well, if the car is moving forward, then the treadmill matches the speed of the car, keeping the car from moving." Well, if that's the case, then the car ISN'T moving forward, making the car's speed equal to zero. Thus, the treadmill wouldn't be moving either.

If the question were rephrased to "if the treadmill moved in the opposite direction at a speed equal to what the car's speedometer said..." then the car would be stationary. In fact, this situation actually is used in labs (well, I wouldn't necessarily call the thing the tires sit on a "treadmill")

Umm how is that different then the orginal question? The fact that it is a plane has no effect on the question.

The plane accelerates forward because there is a net force on the plane in the forward direction. The treadmill provides a very small amount of force backwards on the plane (equal to the rolling resistance of the wheels) while the air provides a forward force on the plane per Newton's 3rd law (equal and opposite; plane is pushing backwards against the air.) The propeller (or jet engines) are what make the plane move forward.

In the case of the car, it moves forward by pushing backwards against the treadmill. The tires pushing against the ground (and the ground pushing back; again, Newton's 3rd law) are what makes it move forward (if it moved forward.)


If it helps, think of it this way: Imagine you have a toy plane with two motors in it. One spins the tires, and the other spins the propellers. Place that toy onto a super super slippery surface, more slippery than ice. Turn the motor on for the wheels, and the plane will just sit there with its wheels spinning in place. Turn on the motor for the propeller, and the plane will move forward. Leave the motor on for the propeller, and turn off the motor for the wheels; the plane will still move forward. Rewire the motor for the wheels so that it spins backwards. The plane will *still* move forwards, even if the wheels are spinning backwards.

In the problem with the real plane, the wheels are nearly irrelevant; the only thing they account for is some amount of force preventing the plane moving, equal to the frictional force of rolling friction which is quite low compared to the force due to the propellers pushing the plane forward.

I really don't see where you are going with the slippery surface at all. I have never seen a treadmill made out of ice. I think your think off something more like a pair of free rolling rollers like in a dyno so that the car can't really apply any force. But that isn't what the question is asking about.

How you apply the force to the body doesn't matter. if the plane gets 10 pounds of force form the wheels or jets it still gets 10 pounds of force applyed.

 

[DrPizza]

Originally posted by: smack Down
I really don't see where you are going with the slippery surface at all. I have never seen a treadmill made out of ice. I think your think off something more like a pair of free rolling rollers like in a dyno so that the car can't really apply any force. But that isn't what the question is asking about.

How you apply the force to the body doesn't matter. if the plane gets 10 pounds of force form the wheels or jets it still gets 10 pounds of force applyed.

That's the point... if it's from the wheels, it is NOT getting 10 pounds of force applied if the treadmill is running backwards, matching the tire's speed.

 

[Vich]

Originally posted by: randay
No, the plane does not take off.

ahaha

 

[smack Down]

Originally posted by: DrPizza

Originally posted by: smack Down
I really don't see where you are going with the slippery surface at all. I have never seen a treadmill made out of ice. I think your think off something more like a pair of free rolling rollers like in a dyno so that the car can't really apply any force. But that isn't what the question is asking about.

How you apply the force to the body doesn't matter. if the plane gets 10 pounds of force form the wheels or jets it still gets 10 pounds of force applyed.

That's the point... if it's from the wheels, it is NOT getting 10 pounds of force applied if the treadmill is running backwards, matching the tire's speed.

You would have the same effect weather the force is applied by the tires or jets. The only difference would be the treadmill speed. It would take alot more revolutions of the wheel to counter act the 10 pounds of force from the jet because the wheels are free spinning but it would happen because as you said it has to match the tires speed.

 

[Toastedlightly]

Alright, if anyone on my floor has a damn digital video camera, I will put my RC plane on a treadmill and SHOW you that it can go forward and that the damn wheel speed doesn't have a huge effect (although it would have greater as it doesn't have bearings and such).

anyone at the University of Minnesota willing to lend me a camera for a few hours? Let me know.

 

[Horus]

THE PLANE DOES NOT TAKE OFF. Aircraft wheels are not designed to roll at 400+ MPH that you'd be getting with this scenario.

If you were using NASCAR tires ont he plane, ok. But it won't take off otherwise.

 

[mercanucaribe]

Originally posted by: Horus
THE PLANE DOES NOT TAKE OFF. Aircraft wheels are not designed to roll at 400+ MPH that you'd be getting with this scenario.

If you were using NASCAR tires ont he plane, ok. But it won't take off otherwise.

You're missing the point of the riddle. If you are going to say "the bearings fail" then you might as well say "there is no such thing as a treadmill the size of a runway".

 

[Howard]

not sure yet

 

[ryan256]

Originally posted by: JohnCU
If you have 2 liquids (you don't know what they are), 1 mL of each and you add them together, can you be 100% sure that the resulting solution is 2 mL?

Well.... if it happened to be 1mL of liquid oxygen and 1mL of liquid hydrogen..... you certainly wouldn't have 2mL of solution left after the reaction

Would be hard to find 2mL of YOU left after that one