Matter /CAN/ be created!

[Makhiel]

Looking at the example of 1/3, in base 10, we need a zero followed by an infinity of 3s to asymptotically approach the value of 1/3.

Which lucky for us, we can do. We just don't write it out and instead use a special symbol for it. Just like we can represent pi up to the "last" digit by simply writing the greek letter.

 

[ElPatron]

Let's say that your asymptote boundaries are X=1 and y=1. That means that no matter how close to the boundary the line will get i.e. .999... it will never touch it because that will mean that 1 would be a solution to the equation of the graph, which for equations with asymptotes would not be possible because it would cause a 0 to appear in the denominator of the original equation. (equations that have asymptote have a variable in the denominator i.e. Y= 1/(X-1).)

Therefore .999... cannot equal 1.

I was thinking that OP never set foot in college, then you posted highschool maths.

I safely assume OP never set foot inside highschool, thank you.

 

[Tac0qvy]

Matter actually CAN be created. It takes energy.

E=MC^2

This Q&A answers it pretty well.


http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970724a.html

 

[Bara_no_Hime]

**snip**

Um... of course you can create mater. You do so by converting the required amount of energy into matter. E=MC^2 works both ways.

I don't see how your math has anything to do with the creation of matter. All you proved is that there is no practical difference between 0.999_ and 1. And there isn't any practical difference between the two. However, that has nothing to do with matter, the creation of it or otherwise.

Edit: Ninjaed.

 

[oktalist]

To convince yourself of the cardinality problem, start with a finite number of digits and then extend the process by adding more digits. Keep going till you get to an infinite number of digits. You'll see that no matter how many digits you have, you always end up with an extra 0 after multiply by 10 and a remaining 1 after doing the subtraction in the last digit.

What.

Did you honestly write that with a straight face?

Yes, I did. You do it abstractly. It's called a limit process. You do it all the time in Calculus to compute derivatives and integrals.

f(x) = 1 - 10-x
f(0) = 0
f(1) = 0.9
f(2) = 0.99
f(3) = 0.999
limx→∞ f(x) = 1

And the hidden zero thing is rubbish.

I've seen this trick a couple times, and it is incorrect because it breaks cardinality.

It's not a trick. 0.999... = 1.

The key thing to realize is that infinity is not the same size in all cases.

Infinity doesn't really have a size, as such.

When you multiply x=0.9999rep by 10, you have increased the cardinality by 1 with a hidden zero all the way at the end.

There is no "at the end". Recurring decimals are endless.

Thus, 10*x and x don't have the same cardinality.

So 1 ≠ 1.0 because they have a different number of digits? ("hidden zero")

As a result, 10*x - x =/= 9.0000rep; there is a hidden 1 all the way at the last digit.

There is no "last digit".

EDIT: I should have pointed out that cardinality is the size of a set, and it can be used to deal with infinite sets like 0.999rep.

0.999... is not a set. It's a number.

 

[The_root_of_all_evil]

I'm surprised at you, Root. You are talking nonsense.

You shouldn't be surprised. I talk nonsense some of the time.

None of the time with Maths though. You're getting rational/irrational numbers mixed up with finite and infinitely recurring decimals.

Any infinitely recurring decimal is, by definition, an approximation because there is no way to represent infinity, or an infinite recurrence, on a number line, except for an approximation.

If you want to disprove me, solve 1/0. Or x^3+y^3=z^3(xyz). Or the square root of -1.

And an equivalency, if that were proper mathematical terminology,

http://en.wikipedia.org/wiki/%E2%89%83
Asymptotic analysis. The terminology may have changed, the definition hasn't.
http://en.wikipedia.org/wiki/Congruence_relation
http://en.wikipedia.org/wiki/Approximation_theory
http://en.wikipedia.org/wiki/Diophantine_approximation

Please read up. This issue has been around for a lot longer than the 'net and has been "solved" many times.

 

[De_Roll_Le]

Oh god, this thread again. It's such an easy way to get a hundred pages of back and forth between those who understand complicated math and those who don't. I refuse to get sucked into another debate about this. It simply never ends.

Yes, .999... is exactly equal to 1. No, it does not involve rounding. They are exactly the same number. No, I'm not going to spend time explaining all of the different proofs and ideas about infinities to people. Search for one of the other threads about this if you really care. Every point I could make has been repeated over and over again in them.

Also, to the OP: If you really wanted to get lots of views, you could have just made the title "FREE TITTIES INSIDE CLICK NAO!!1". It would have had about the same level of relevance to the content of the thread.

Rabid Toilet is awesome.

 

[Axolotl]

What the fuck guys? Seriously what the fuck?

We've had this disscusion. It was long enough and stupid enough that it was used in a Cracked article. Seriously why the fuck are we doing this again. Nothing will be said that hasn't been said before.

It would be one thing if this were even something you could have an opnion on but no, it has a right answer and if you disagree with it then all the arguements and posts you can make won't stop you from being wrong. And it's even worse because the people being wrong pepper their nonsense with phrases along the lines of "Maths Fail!" and other self-congratulating pieces of inspid idiocy as if being wrong wasn't enough they have to be obnoxious about it as well. At last creationists generally admit their total ignorance.

Lok the OP is a either a troll (very likely)or by now has made it plenty clear they don't know what they're talking about. So please just walk away. If you really are deperate to know about 0.999 recurring equaling 1 then just go ask a maths professor I'm sure they'll be happy to explain and give proof.

 

[Vigormortis]

If you thought that was cool, here's some Astrophysics 101.

There is a gravitational attration between EVERY object. The equation to calculate that attration is:

Fg=G((M1 * M2)/D^2)

Where
Fg= The force of gravity
G= Universal gravity constant (don't worry about it, it's a really small number)
M1 and M2= Mass of object 1 and 2
D= distance

Now that we have that established...
Black holes have such huge masses that not even light can escape it.
You're probably putting two and two together now...
If you look at the equation, for that statement to be true both objects must have a mass.
Light has mass.

EDIT:

Oh yeah, and according to wikianswer matter can be created when gamma rays collide, or something like that.

I can explain all of this quite easily. Even using "layman's" terms.

Gravitation isn't just some magical attractive force. It's the curving of space/time. I'm sure I don't need to crack out the old bowling ball/rubber sheet analogy.

As such, regardless of whether something has mass, if it's in the presence of a gravity well it will inevitably fall towards the object creating the well. Ergo, light "bends" in the presence of gravity. This has nothing to do with light having or not having mass.

The equation you've listed is used to calculate the gravitation pull between two objects. Since light technically has no mass, it exudes no gravitation on the black hole. However, that does not stop it from reacting TO the black hole's own gravitation.

Oh, and as for the wikianswers thing....
It's a wiki-site. No matter what, take ANYTHING you find on those sites with a hefty dose of salt.

And gamma rays "colliding", or whatever the page was talking about, wouldn't "create" matter. What they were likely saying was that when the rays collide, the energy contained within the waves could be transformed into matter.

There is a difference.

I was going to type up a more specific, refined explanation, but Berenzen beat me to it.

Any mathematician will tell you that .999... is equal to 1. This is because 0.000...1 does not exist. A mathematician could explain it better than I can.

The closest way that we can create matter is by converting energy into it. However, Mass and Energy must be conserved unless you want to get into insane multidimensional physics that is no more than conjecture.

The equation to Mass-Energy equivalence is E^2=m^2c^4 + p^2c^2.

Light has mass-energy equivalence depending on the energy of the photon. Measured in MeV (Mega electronVolts), energy is affected by gravity because of warping in space-time, as defined through general relativity.

A black hole has a definite mass, and in fact it can be calculated by the size of it's schwartzchild radius- radius of the event horizon. Equation is r=2GM/c^2. All matter has a schwartzchild radius, but most of them are contained within the actual particles itself.

F=GMM/r^2 cannot equal infinite. Even as you approach the center of a black hole it does not equal infinite. However, it approaches it asymptotically. In order for an object to have infinite gravitational attraction to another object, it must exist at the exact same location of the other object, a physical impossibility.

People, before you start spewing out physics, you should probably take a physics/astronomy course beforehand, instead of just looking at equations and thinking you know it.

 

[Vigormortis]

Clegane]

SNIP

I don't know why you bothered posting that long wall of text. People have known that matter can be created for over 60 years and theres no need for all that garbage, it can be summed up in one line. Ready?

E=MC^2

There, that's all it takes. E is energy, M is matter. It's been extensively proven (for 1000s of years) that we can make energy from matter and therefore it must be possible to recreate matter from energy. Which Einstein conclusively proved in the 40s or 50s or whatever.

I think you're wording is a bit off.

The 'creation' of matter is impossible. Queue long winded explanation of the law of thermodynamics. However, energy can be 'converted' into matter. And vice-versa. That's the difference.

Also, it hasn't been thousands of years. It's barely been over a century since we've understood this fact.

 

[Lukeje]

there is no way to represent infinity, or an infinite recurrence, on a number line, except for an approximation.

That's why you use extensions of the number line.

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

 

[ACman]

^ That's where you made your error, if you multiply X and the value of X in this equation by 10, that means there's 1 less '9' than in just X, infinity - 1 '9's, if that helps; so 10X-X (9X) would equal 0.000...1 in the infinity place, basically.

You're dealing with infinitesimals here, and of course by definition infinitesimals can't be measured, and so will lead to errors if you try to use them in equations.

In short, .999... does NOT = 1, OP is wrong.

Ummmmmmmmmmmmm.

1/3 = 0.3 rep

2/3 = 0.6 rep

3/3 = ?

 

[The_root_of_all_evil]

there is no way to represent infinity, or an infinite recurrence, on a number line, except for an approximation.

That's why you use extensions of the number line.

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

But that confounds the problem because the only way of creating a recurring decimal is the decimal approximation of a fraction. (Pi or e don't count as they don't recur precisely - that's why they have a name)

Therefore, everytime you see a .1rep or .4rep, you KNOW that it's actually talking about a fraction; because there's no other way to create one. We don't have the technology to measure anything that could have one.

So, .9rep is already 1 (which is what people think), because .9rep is a decimal approximation of a fraction.

Start with x = 0.999rep

means x=1.

Multiply .1rep by 9 and you get 1. That's how it works. Riemann sphere's aren't necessary as we already know where the recurrence came from - approximations.

The bit where the whole argument breaks down is that x=0.9rep, right at the start. It's a mathematic bait/switch just like sawing the lady in half. Or tenths.

 

[geizr]

f(x) = 1 - 10-x
f(0) = 0
f(1) = 0.9
f(2) = 0.99
f(3) = 0.999
limx→∞ f(x) = 1

And the hidden zero thing is rubbish.

Your function only approaches 1 in the limit, but it never actually gets there. This is because n/(infinity) only approaches zero in the limit; it never actually gets there. However, for purposes of convenience, we often ignore that subtlety because it is below our error tolerance, but, when we need to be more mathematically precise, we can not make that assertion.

It's not a trick. 0.999... = 1.

Yes, it is a trick because it is ignoring the underlying cardinality.

Infinity doesn't really have a size, as such.

Infinity very much has a size, and that size can be different in different cases. That is why you can take the limit of a numerator and denominator both going to infinity and, yet, obtain a finite ratio. It's because they are different sized infinities.

There is no "at the end". Recurring decimals are endless.

It doesn't matter how endless they are, you can always add one more digit to make a larger infinity. That's what happens when you multiply by 10.

So 1 ≠ 1.0 because they have a different number of digits? ("hidden zero")

Numerically, yes, that is precisely true. The number 1 does not have the same precision as 1.0. The representation 1.0 is, at best, an approximation of the number 1. As I said above, for most practical purpose, it is below our error tolerance, so we don't care. But, when we need to be more mathematically precise, we do care.

As a result, 10*x - x =/= 9.0000rep; there is a hidden 1 all the way at the last digit.

There is no "last digit".

EDIT: I should have pointed out that cardinality is the size of a set, and it can be used to deal with infinite sets like 0.999rep.

0.999... is not a set. It's a number.

The digits that we use to represent numbers are sets. Each element in the set represents a particular fraction multiplied by some factor. The arithmetic operations with which we are familiar perform transformations on those elements, which can themselves be sets. When you multiply x = 0.999rep by 10, you shift all the elements upward and then have to add an extra empty set element at the very end to represent the digit that was vacated as a result of the multiplication. Now, when you do the subtraction, the very last element in 0.999rep, is matched against an empty set element rather than another 9, hence why you don't get 9, exactly, but something infinitesimally smaller than 9.

This is necessarily nonsense. It's something that happens precisely because infinity is not the same size everywhere. It doesn't always have the same meaning. Just like I can not label all the real numbers with integers because there is a greater infinity of real numbers than there is the infinity of integers. They are not the same size infinities. That's the cardinality difference.

EDIT: Correction. You don't get something infinitesimally larger than 9; you get something infinitesimally smaller than 9.

 

[OrokuSaki]

If you thought that was cool, here's some Astrophysics 101.

There is a gravitational attration between EVERY object. The equation to calculate that attration is:

Fg=G((M1 * M2)/D^2)

Where
Fg= The force of gravity
G= Universal gravity constant (don't worry about it, it's a really small number)
M1 and M2= Mass of object 1 and 2
D= distance

Now that we have that established...
Black holes have such huge masses that not even light can escape it.
You're probably putting two and two together now...
If you look at the equation, for that statement to be true both objects must have a mass.
Light has mass.

According to a documentary on black holes that I watched, physicists hate'em because once you get to a certain point near the singularity, that (or some other physics gravity equation) comes out to a big-fat infinity sign staring up at you, suggesting that a singularity has infinite mass and infinite gravity.

I'm definitely not a physicist, but I do know the meaning of the words "Infinite" and "Gravity". Wouldn't an object with infinite gravity consume EVERYTHING regardless of distance? Because if it's truly infinite then it doesn't decrease, 1/2 of infinity is still infinity because infinity goes on forever and cannot be decreased. So if a black hole has infinite gravity wouldn't we all be dead?

 

[Makhiel]

It doesn't matter how endless they are, you can always add one more digit to make a larger infinity. That's what happens when you multiply by 10.

That's not what happens, one more item added to a set that has the cardinality of infinity will result in a set the cardinality of which is still infinity (infinity of the same "size"). You need to do more than adding items to reach higher cardinality.

 

[randomsix]

If you thought that was cool, here's some Astrophysics 101.

There is a gravitational attration between EVERY object. The equation to calculate that attration is:

Fg=G((M1 * M2)/D^2)

Where
Fg= The force of gravity
G= Universal gravity constant (don't worry about it, it's a really small number)
M1 and M2= Mass of object 1 and 2
D= distance

Now that we have that established...
Black holes have such huge masses that not even light can escape it.
You're probably putting two and two together now...
If you look at the equation, for that statement to be true both objects must have a mass.
Light has mass.

According to a documentary on black holes that I watched, physicists hate'em because once you get to a certain point near the singularity, that (or some other physics gravity equation) comes out to a big-fat infinity sign staring up at you, suggesting that a singularity has infinite mass and infinite gravity.

I'm definitely not a physicist, but I do know the meaning of the words "Infinite" and "Gravity". Wouldn't an object with infinite gravity consume EVERYTHING regardless of distance? Because if it's truly infinite then it doesn't decrease, 1/2 of infinity is still infinity because infinity goes on forever and cannot be decreased. So if a black hole has infinite gravity wouldn't we all be dead?

The force gravity of a BH is only infinite arbitrarily close to the singularity.

Because the D in the above equation is zero, and you're dividing by it.

Make D a more reasonable number, and the force is finite.

And photons do not have mass. The reason that light cannot escape black holes is that the gravity of the BH bends space to the extent that local space around the BH is falling into the hole at a speed faster than c.

 

[Lukeje]

there is no way to represent infinity, or an infinite recurrence, on a number line, except for an approximation.

That's why you use extensions of the number line.

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

But that confounds the problem because the only way of creating a recurring decimal is the decimal approximation of a fraction. (Pi or e don't count as they don't recur precisely - that's why they have a name)

Therefore, everytime you see a .1rep or .4rep, you KNOW that it's actually talking about a fraction; because there's no other way to create one. We don't have the technology to measure anything that could have one.

So, .9rep is already 1 (which is what people think), because .9rep is a decimal approximation of a fraction.

Start with x = 0.999rep

means x=1.

Multiply .1rep by 9 and you get 1. That's how it works. Riemann sphere's aren't necessary as we already know where the recurrence came from - approximations.

The bit where the whole argument breaks down is that x=0.9rep, right at the start. It's a mathematic bait/switch just like sawing the lady in half. Or tenths.

We're talking about completely separate issues here. Representing infinity as a number requires an extension of the `number line' (by which, to be clear, I mean the set of real numbers). So does representation of the number generated by the square root of -1.

A completely separate issue is that of the representation of repeating decimals as fractions. This does not require any extension of the real numbers. Every repeating decimal can be mapped into at least one fractional representation. This may be proved to varying degrees of rigour if one has the time or inclination. And you're right; 0.(9) is equivalent to 1 by definition. (The definition is that 0.(0)1 is equivalent to zero, if you were wondering).

I also don't understand why you think that the only non-repeating decimals are those that have names. One can easily prove that there is a set of size aleph[sub]1[/sub] that don't.

(I've also noticed that you seem confused by the notion of mathematical equivalence. Saying that a is equivalent to b means that a=b if and only if b=a. Equivalence thus implies equality).

 

[dobahci]

This is hilarious. It's funny how there's another thread where the vast majority of Escapist readers believe themselves to be of above average intelligence, and then you have this thread, where a sizable quantity of readers prove themselves unable to comprehend what is essentially grade school mathematics.