1+1 Dosen't Always Equal 2?! What?!

[The Jedi Master]

Ah, you mean ℤ₂? That's a so called group which only consists of the numbers 0 and 1. Whatever you do with these numbers, on the result you must calculate the remainder and use that as the result. And that will always be 0 or 1. You can't break out of it. That's a group axiom (closure).

Maybe it's because I'm not a mathematician, but that system sounds a little useless.

[andrewas]

ℤ₂ is not very useful, at least not to my knowledge. Its probably very useful in some areas, but not to the layman. ℤ₁₂, on the other hand, comes in handy any time you deal with a 12 hour clock. ℤ₂₄ when you use a proper clock. ℤ₇ when dealing with days of the week. And the concept is very important in cryptography - some operations that are easily reversible in normal arithmetic become almost impossible to reverse in modular arithmetic.

[*Aqua*]

ℤ₂ describes a bit. Although it is very simple and basic it is fundamental for computer science.

1 bit -> ℤ₂

2 bits -> ℤ₂ x ℤ₂

3 bits -> ℤ₂ x ℤ₂ x ℤ₂

etc.

That's how you can describe a binary number. Why should you do that? Using this construct you can explain why binary numbers work like they do. You can do that for any set of elements that fulfill the axioms of groups and rings. Every wondered why matrices have these weird calculation rules? Their group and ring properties cause that.

It is higher maths but if you understand groups and rings, half of mathematics won't be a problem anymore. Instead all the weirdness will start to make sense.

But I still hate math.

[KASASpace]

This is why we don't allow people to divide by zero.

Then explain to me why Einstein did so.

[andrewas]

Einstein didn't. Einstein, if I remember what you are referring to, discovered a formula which results in division by zero when particles with mass travel at lightspeed.

[KASASpace]

Einstein didn't. Einstein, if I remember what you are referring to, discovered a formula which results in division by zero when particles with mass travel at lightspeed.

"Discovered"

Well, considering we haven't accelerated anything macroscopic to near lightspeed, we have almost no proof of Einstein's energy equation.

You see, Quantum Physics can be what prevents us from measuring microscopic particles speed accurately. So, they could be going at light speed or faster, but because to measure the speed you need to know the position, you lose a lot of accuracy in measuring speed, such as within LHC.

[Ralathon]

"Discovered"

Well, considering we haven't accelerated anything macroscopic to near lightspeed, we have almost no proof of Einstein's energy equation.

You see, Quantum Physics can be what prevents us from measuring microscopic particles speed accurately. So, they could be going at light speed or faster, but because to measure the speed you need to know the position, you lose a lot of accuracy in measuring speed, such as within LHC.

Learn how math works. The lorentz factor is the only possible outcome if the speed of light is constant (which has been measured many times to be the case). Einstein just made the logical steps from A to Z.

If you know that A+B = C and that A = B = 3 then you don't need to actually measure C to know it is 6.

Yellow is a lightbeam bouncing between 2 mirrors on a train. Left picture is the PoV of someone on the train, right picture is someone on the station. To avoid paradoxes both lightbeams need to complete an equal amount of bounces in the same time period (Else the guy on the station and the guy on the train would start arguing after a few minutes). Only way to do this is to slow down time for the guy on the train by a factor of sqrt(1-v^2/c^2). Everything else, such as mass energy equivalence and the light speed barrier flows from adding this factor into newtonian physics.

As long as you take the assumptions that

1 - Lightspeed is constant from every reference frame

2 - The universe behaves according to mathematical laws that we can discover

all those other things MUST be the case. Since we've been measuring the speed of light as constant, measured time dilation and use nuclear energy there is plenty of proof for special relativity.

Edited March 30, 2014 by Ralathon

[KASASpace]

Learn how math works. The lorentz factor is the only possible outcome if the speed of light is constant (which has been measured many times to be the case). Einstein just made the logical steps from A to Z.

If you know that A+B = C and that A = B = 3 then you don't need to actually measure C to know it is 6.

http://imgur.com/3cs2eDS.png

Yellow is a lightbeam bouncing between 2 mirrors on a train. Left picture is the PoV of someone on the train, right picture is someone on the station. To avoid paradoxes both lightbeams need to complete an equal amount of bounces in the same time period (Else the guy on the station and the guy on the train would start arguing after a few minutes). Only way to do this is to slow down time for the guy on the train by a factor of sqrt(1-v^2/c^2). Everything else, such as mass energy equivalence and the light speed barrier flows from adding this factor into newtonian physics.

As long as you take the assumptions that

1 - Lightspeed is constant from every reference frame

2 - The universe behaves according to mathematical laws that we can discover

all those other things MUST be the case. Since we've been measuring the speed of light as constant, measured time dilation and use nuclear energy there is plenty of proof for special relativity.

You do realize this ONE simple thing:

Not every place in the universe has to follow our understanding of the laws of physics.

BTW, I would suggest learning how Quantum Physics works.

In layman's terms:

You can't take the speed accurately,

and take the position accurately.

[andrewas]

But you can take the speed accurately without taking the position accurately.

[Ralathon]

You do realize this ONE simple thing:

Not every place in the universe has to follow our understanding of the laws of physics.

Yes, and right behind you there could be a dancing pink elephant. If we want to talk about the universe in a meaningful way we need to make some assumptions. Else every discussion is going to boil down to "The only thing I can be certain of is that I am a thinking entity. Everything else could be an illusion" which isn't very conductive when trying to figure out the nature of the universe.

The speed of light flows directly from 2 fundamental constants: The permeability and permittivity of free space. If the universe behaves differently somewhere you either need to change the laws of physics or those constants. Changing either of those will create an area of space with a higher or lower vacuum energy. Meaning this area will either expand or collapse at the speed of light in a vacuum metastability event. The universe doesn't like inhomogeneous laws.

BTW, I would suggest learning how Quantum Physics works.

In layman's terms:

You can't take the speed accurately,

and take the position accurately.

Yes, this is the Heisenberg uncertainty principle. Congratulations at understanding the absolute basis of quantum mechanics. What you fail to realize is just how small the uncertainty is. It only really comes into play when you use a very small momentum or very accurate position. We also don't measure the velocity of small particles in the same way we do large objects. We don't take 2 snapshots and time how long it took the particles to move, instead we measure their energy or look how they deflect within a electric/magnetic field.

If you want particles that appear to exceed the speed of light you should look into quantum tunnelling. Zero spin particles can jump across a barrier instantaniously. This doesn't violate relativity though since these particles don't carry information and it is the result of the way the probability wave function works.

[*Aqua*]

Zero spin particles can jump across a barrier instantaniously. This doesn't violate relativity though since these particles don't carry information and it is the result of the way the probability wave function works.

I never understood this. Isn't the existence of a particle a piece of information?

[K^2]

BTW, I would suggest learning how Quantum Physics works.

That is an absolutely hilarious statement. Fortunately, we do have Relativistic Quantum Field Theory, which tells us exactly how particles behave near light speed, and we have used it to make computations, such as anomalous magnetic moment of electron, to confirm these equations to 12 orders of magnitude. Experiments with similar precision exist in General Relativity. Between the two, equations of Special Relativity, as a special case of the above, are some of the most precisely tested.

[The Jedi Master]

You do realize this ONE simple thing:

Not every place in the universe has to follow our understanding of the laws of physics.

Yes, they do. That's why they are laws, not suggestions.

[KASASpace]

It's quite simple, as all things we know come from this pocket of the universe, logically we must ASSUME that not all areas of the universe follow our understanding of physics.

In fact, we can only see about 14 billion lightyears (how we get the age) in the visible universe, so how can you say that it would immediately destroy itself?

Yes, this is the Heisenberg uncertainty principle. Congratulations at understanding the absolute basis of quantum mechanics.

Seriously?

Whatever.

It comes into play whenever a quantum particle is present, and it does not matter how you measure it.

Position is necessary to measure it, and, logically, you lose considerable accuracy.

- - - Updated - - -

Yes, they do. That's why they are laws, not suggestions.

You do realize that it's not a law if it can't be broken, right?

[*Aqua*]

it does not matter how you measure it.

Unfortunately it's the measurement which makes exact measures impossible. You'll either have to shoot a high frequency laser at a quantum and look what remains after it's fried or use a detector which will absorb the particle or its remains. In both cases the particle is gone afterwards and the only things you can measure are the effects of its death.

That's why Heisenberg said it's impossible to directly observe a particle.

[K^2]

Seriously?

Whatever.

It comes into play whenever a quantum particle is present, and it does not matter how you measure it.

Position is necessary to measure it, and, logically, you lose considerable accuracy.

Again, not how Quantum Mechanics works. More importantly, we don't need to measure velocity to verify these equations, as I've already specified.

[Dodgey]

Yes, they do. That's why they are laws, not suggestions.

Just a minor point, whenever we talk about the laws of the universe we treat them as descriptive laws, not proscriptive laws. That is to say the laws are derived from what we know about the universe, Newton derived the laws from what he saw in his expirments. This means that they are liable to change depending on the information we have present. That being said, untill there is good reason to we should operate under the presumption that they are consistent throughout the universe.

I really don't like the label laws, I personally think that a better term would be models. Though admittedly laws have a better ring to it. Silly English language, why do you have to be confusing .

[NKL]

you can set you own rule of addition and multiplication ...

like x+y = 2x*4y

so that 1+1 = 2*4 = 6

in that particular set.

[JumpsterG]

Wow, these answers are a lot smarter than what I first thought of.

In vector math, two vectors with magnitudes of 1 added together can equal a new vector of any magnitude between 0 and 2, depending on the directions of the original vectors.

1up + 1up = 2up

but

1up + 1down = 0up

[KASASpace]

Again, not how Quantum Mechanics works. More importantly, we don't need to measure velocity to verify these equations, as I've already specified.

Until I see a Phd I won't take your "talk backs" seriously.

And plus, your original comment never said that it wasn't how quantum mechanics worked, nor did it say how it did (according to you)

[KASASpace]

To actually contribute to the thread:

Binary: 1+1 = 10

Or, as I call it, monetary:

1+1= Infinity, as 1 is 10 and 10 is 100 and so on

And, it also depends on things like signs. Positive or negative.

You see, as it was never specified which EXACT sign it was.........

[Guest]

Until I see a Phd I won't take your "talk backs" seriously.

And plus, your original comment never said that it wasn't how quantum mechanics worked, nor did it say how it did (according to you)

I'm a first year student of physics and I can already see you don't know squat about quantum mechanics. Stop trying to explain things you don't understand, quantum physics are confusing enough as it is.

Your "layman's terms" explanation of Heisenberg's principle is wrong. It's impossible to *precisely* (meaning with absolute, 100% precision) measure both energy and momentum of a particle. You can measure one, but then you don't know anything about the other, or you can measure both with a reasonable accuracy. Also, your statement that "it doesn't matter how you measure it" is perhaps the biggest lie in this thread - it does matter, and that's what things like "wave function collapse" and "Heisenberg's uncertainty" are all about. In fact, the biggest problem with quantum physics is that any measurement has a non-negligible effect on the system measured. As for the "laws" being broken, you forgot (or didn't hear of) the correspondence principle: If large-scale data is inputted into quantum mechanical equations, they should produce results corresponding to predictions of the classical theory within a very close margin. That basically means classical theory must be a close approximation of any quantum mechanical theory in scales where the former would apply. You can derive Newton's equations of motion from any quantum theory worth it's salt, but it's a rather complicated process.

Now, don't worry if you don't understand this. Many of my fellow students don't. The point is, quantum mechanics won't let you perform miracles, nor "cheat" long established stuff like Newton's equations and thermodynamics. Quite the contrary, in fact, it'd be possible to derive both from a good quantum theory, in the exact same form we know them.

Edited April 1, 2014 by Guest

[KASASpace]

I'm a first year student of physics and I can already see you don't know squat about quantum mechanics. Stop trying to explain things you don't understand, quantum physics are confusing enough as it is.

Your "layman's terms" explanation of Heisenberg's principle is wrong. It's impossible to *precisely* (meaning with absolute, 100% precision) measure both energy and momentum of a particle. You can measure one, but then you don't know anything about the other, or you can measure both with a reasonable accuracy. Also, your statement that "it doesn't matter how you measure it" is perhaps the biggest lie in this thread - it does matter, and that's what things like "wave function collapse" and "Heisenberg's uncertainty" are all about. In fact, the biggest problem with quantum physics is that any measurement has a non-negligible effect on the system measured. As for the "laws" being broken, you forgot (or didn't hear of) the correspondence principle: If large-scale data is inputted into quantum mechanical equations, they should produce results corresponding to predictions of the classical theory within a very close margin. That basically means classical theory must be a close approximation of any quantum mechanical theory in scales where the former would apply. You can derive Newton's equations of motion from any quantum theory worth it's salt, but it's a rather complicated process.

Now, don't worry if you don't understand this. Many of my fellow students don't. The point is, quantum mechanics won't let you perform miracles, nor "cheat" long established stuff like Newton's equations and thermodynamics. Quite the contrary, in fact, it'd be possible to derive both from a good quantum theory, in the exact same form we know them.

I'm just saying, it IS the internet, after all. (and that's why I typically don't trust some of the fellow forum users...........)

About Layman's terms, it is an approximation as to what it is, not necessarily correct.

As a scientist, one has to take in to account all variables, and as such, because space isn't a complete vacuum (quantum particles popping in and out of existence) it might be possible to actually have a terminal velocity, even while in space. Not to mention that according to that, you will never be in free fall.

Remember, Science is pretty much all about expanding what we know, and as such people in the past might have been wrong, or they might not have been right, but close, etc.

Edited April 1, 2014 by KASASpace

[K^2]

Until I see a Phd I won't take your "talk backs" seriously.

ABD stands for "All but Dissertation," which I'm writing at the moment. There is nothing a typical recent Ph.D. knows that I don't. And my field is particle physics, which is all Quantum, naturally. If you want a link to my university page as verification of credentials, I'll be happy to PM it to you.

[Doozler]

You'll either have to shoot a high frequency laser at a quantum and look what remains after it's fried or use a detector which will absorb the particle or its remains. In both cases the particle is gone afterwards and the only things you can measure are the effects of its death.

Not quite right - its not that the particles are destroyed. The idea here is that to measure the position more accurately, you need a higher frequency laser. The higher the frequency, the more energy the photon have so the bigger the "knock". You now have an accurate measurement of the position, but have no idea how fast the particle is moving after its knock.

That's the basic idea. I don't particularly like this description as it implies that the uncertainty principle is just due to "poor experiments" and that some bright spark could come up with a clever idea to get around it. If you believe in wavefunctions as "real" then the point is that well-defined position and momentum do not exist at the same time, not just that they can't be measured.

(P.S. I have a PhD in physics and K^2 is correct)