There is no Such a Thing as a Next Number

[AlexB23]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

In math, there are multiple types of numbers, with the main types listed below. The next whole number after 1 is 2. But, there are an infinite amount of rational numbers between 1 and 2, or say, between 2 and 3, etc...

Numbers
↳Real Numbers
Includes integers (positive and negative numbers that have nothing after their decimal point) and natural numbers (all numbers that are whole and non negative, including zero, used for counting). Real numbers also include rational numbers, such as 3/4 and 2.57. Finally, real numbers include irrational numbers such as pi, and √2, which have digits that go on forever.

↳Complex numbers
These numbers are square roots of negative numbers, such as √(-1) = i, -5i + 3.2, and etc. These numbers have some uses in electrical engineering.

Complex numbers can be plotted on graphs such as the one below.

 

[sjastro]

In math, there are multiple types of numbers, with the main types listed below. The next whole number after 1 is 2. But, there are an infinite amount of rational numbers between 1 and 2, or say, between 2 and 3, etc...

Numbers
↳Real Numbers
Includes integers (positive and negative numbers that have nothing after their decimal point) and natural numbers (all numbers that are whole and non negative, including zero, used for counting). Real numbers also include rational numbers, such as 3/4 and 2.57. Finally, real numbers include irrational numbers such as pi, and √2, which have digits that go on forever.

↳Complex numbers
These numbers are square roots of negative numbers, such as √(-1) = i, -5i + 3.2, and etc. These numbers have some uses in electrical engineering.

Complex numbers can be plotted on graphs such as the one below. View attachment 336642

Quantum mechanics is built around the use of complex numbers which describes wavefunctions and also the processes of addition, subtraction and multiplication of wavefunctions as well as their constructive and destructive interference and in the calculation of probability amplitudes.
While it is possible to construct a mathematical theory of quantum mechanics using only real numbers which turns out to being more complicated, it appears nature prefers the complex number version of quantum mechanics.

Complex numbers are essential in quantum theory, experiments reveal – Physics World

Imaginary numbers are more that just a mathematical convenience

physicsworld.com

 

[AlexB23]

Quantum mechanics is built around the use of complex numbers which describes wavefunctions and also the processes of addition, subtraction and multiplication of wavefunctions as well as their constructive and destructive interference and in the calculation of probability amplitudes.
While it is possible to construct a mathematical theory of quantum mechanics using only real numbers which turns out to being more complicated, it appears nature prefers the complex number version of quantum mechanics.

Complex numbers are essential in quantum theory, experiments reveal – Physics World

Imaginary numbers are more that just a mathematical convenience

physicsworld.com

That is pretty cool. Quantum physics is quite complicated, but fascinating stuff. So cool that a particle can be in two places at once, or quantum computers could operate faster than classical silicon based computers. I guess complex numbers have more applications than just electricity. What do you think of De Broglie, and how anything with mass has a wavelength?

 

[sjastro]

That is pretty cool. Quantum physics is quite complicated, but fascinating stuff. So cool that a particle can be in two places at once, or quantum computers could operate faster than classical silicon based computers. I guess complex numbers have more applications than just electricity. What do you think of De Broglie, and how anything with mass has a wavelength?

De Broglie got his idea from optics, a light ray travelling perpendicular to the wave fronts which are equally spaced planes in the media is refracted at the interface of the two homogenous media.

Instead of a pathway of a light ray de Broglie thought of it as the trajectory of a particle namely a photon.
He put forward this idea as a hypothesis in 1923, by 1927 electrons like x-rays were found to be diffracted by crystals indicating electrons have a wave like nature.
In more recent times much larger particles such as buckyballs undergo diffraction when passed through slits and undergoes interference like a wave.

Wave-particle duality seen in carbon-60 molecules – Physics World

The Vienna team sent a collimated beam of carbon-60 molecules through a slit made of silicon nitride and detected the interference pattern by ionizing the

physicsworld.com

 

[AlexB23]

De Broglie got his idea from optics, a light ray travelling perpendicular to the wave fronts which are equally spaced planes in the media is refracted at the interface of the two homogenous media.

Instead of a pathway of a light ray de Broglie thought of it as the trajectory of a particle namely a photon.
He put forward this idea as a hypothesis in 1923, by 1927 electrons like x-rays were found to be diffracted by crystals indicating electrons have a wave like nature.
In more recent times much larger particles such as buckyballs undergo diffraction when passed through slits and undergoes interference like a wave.

Wave-particle duality seen in carbon-60 molecules – Physics World

The Vienna team sent a collimated beam of carbon-60 molecules through a slit made of silicon nitride and detected the interference pattern by ionizing the

physicsworld.com

Wave-particle duality is fascinating stuff, as photons can be both waves and particles, electrons and buckyballs as you mentioned. Some people say macroscopically large objects such as humans could have a wavelength. But do think quantum physics could scale up? There is somewhat of a debate going on in physics which is trying to uncover the effects of quantum physics on a larger scale.

Quantum Effects size: Why do quantum effects only happen on the atomic scale?

Large scale superposition: Heaviest Macroscopic System Yet Put In Quantum Superposition

 

[sjastro]

Wave-particle duality is fascinating stuff, as photons can be both waves and particles, electrons and buckyballs as you mentioned. Some people say macroscopically large objects such as humans could have a wavelength. But do think quantum physics could scale up? There is somewhat of a debate going on in physics which is trying to uncover the effects of quantum physics on a larger scale.

Quantum Effects size: Why do quantum effects only happen on the atomic scale?

Large scale superposition: Heaviest Macroscopic System Yet Put In Quantum Superposition

The mass of the object plays an important role in determining if the object behaves quantum mechanically or by using classical physics.
Electrons behave quantum mechanically because their rest energy (= mass) is small enough to be scattered by X-ray photons of the same energy.

The h/mₒc term in the equation is the Compton wavelength λ of the electron and equals 2.43 x 10⁻¹²m.

The Heisenberg uncertainty principle applies for quantum mechanics where ΔxΔp ≥ h/4π and Δx and Δp are the uncertainties in the position x and momentum p of the particle.
If the energy of the incident photon exceeds the electron, there may be sufficient energy remaining to be converted into an electron/positron pair which along with the scattered electron produces an uncertainty in the position x of the electron as one cannot differentiate between the scattered electron and the electron produced in the particle/antiparticle pair.

Humans being considerably more massive than electrons have Compton wavelengths of around of around 3-4 x 10⁻⁴³m and are not scattered even by the most energetic photons and can therefore be described using classical physics.

 

[AlexB23]

The mass of the object plays an important role in determining if the object behaves quantum mechanically or by using classical physics.
Electrons behave quantum mechanically because their rest energy (= mass) is small enough to be scattered by X-ray photons of the same energy.

View attachment 336740​

The h/mₒc term in the equation is the Compton wavelength λ of the electron and equals 2.43 x 10⁻¹²m.

The Heisenberg uncertainty principle applies for quantum mechanics where ΔxΔp ≥ h/4π and Δx and Δp are the uncertainties in the position x and momentum p of the particle.
If the energy of the incident photon exceeds the electron, there may be sufficient energy remaining to be converted into an electron/positron pair which along with the scattered electron produces an uncertainty in the position x of the electron as one cannot differentiate between the scattered electron and the electron produced in the particle/antiparticle pair.

Humans being considerably more massive than electrons have Compton wavelengths of around of around 3-4 x 10⁻⁴³m and are not scattered even by the most energetic photons and can therefore be described using classical physics.

Wow, that is a good way of looking at it. The Compton wavelength of an electron is equivalent to the wavelength of an X-ray photon, but the Compton wavelengths of a human is shorter than any photon. So, where does temperature play into this? Current generation quantum computers need extremely cold operating temperatures to work, for their atoms to have superposition.

By the way, during the very late 20th century, NASA had a space based observatory named the Compton Gamma Ray Observatory which analyzed gamma ray emissions from cosmic rays and gamma ray bursts (extremely powerful supernovas/supernovae). The space telescope slowly fell and burned up in our atmosphere in the year 2000, as the 9 year mission was completed.

 

[sjastro]

Wow, that is a good way of looking at it. The Compton wavelength of an electron is equivalent to the wavelength of an X-ray photon, but the Compton wavelengths of a human is shorter than any photon. So, where does temperature play into this? Current generation quantum computers need extremely cold operating temperatures to work, for their atoms to have superposition.

By the way, during the very late 20th century, NASA had a space based observatory named the Compton Gamma Ray Observatory which analyzed gamma ray emissions from cosmic rays and gamma ray bursts (extremely powerful supernovas/supernovae). The space telescope slowly fell and burned up in our atmosphere in the year 2000, as the 9 year mission was completed.

The explanation I gave was based on a classical physics perspective (top diagram), much like de Broglie using optics to explain the wave nature of particles.
Explaining quantum mechanics from a quantum mechanical perspective is somewhat more complicated (bottom diagram).

Instead of a particle being scattered, the superimposed state is being scattered or exposed to the surrounding environment.
The superimposed state is a coherent state coupled to the environment which becomes decoherent as information is lost to the environment.
A superimposed state can remain coherent for longer periods of time by reducing the temperature of the environment.
The objective of quantum computing is not to completely isolate the superimposed state which would remain permanent with no information transfer but maintain the state for as long as possible in order for quantum calculations to occur.

 

[ViaCrucis]

There are, in fact, an infinite number of next numbers; not just one infinite either, there are infinites of next numbers.

-CryptoLutheran

 

[AlexB23]

The explanation I gave was based on a classical physics perspective (top diagram), much like de Broglie using optics to explain the wave nature of particles.
Explaining quantum mechanics from a quantum mechanical perspective is somewhat more complicated (bottom diagram).

View attachment 336945​

Instead of a particle being scattered, the superimposed state is being scattered or exposed to the surrounding environment.
The superimposed state is a coherent state coupled to the environment which becomes decoherent as information is lost to the environment.
A superimposed state can remain coherent for longer periods of time by reducing the temperature of the environment.
The objective of quantum computing is not to completely isolate the superimposed state which would remain permanent with no information transfer but maintain the state for as long as possible in order for quantum calculations to occur.

Quantum computing is pretty fascinating. So, trying to hold the superimposed state for a certain time duration, and then return the particle(s) back to normal state, which return the calculation's result? Some companies such as D-Wave and Google are working on quantum computers, and adding more q-bits each year, basically more particles to compute with. Quantum physics was created by God, like all physics, and is extremely complex stuff for most humans to grasp, so this goes to show how awesome our Creator is, for developing a universe with complex nuances and physics that make our reality here.

So, could one use a quantum computer to operate a device* similar to one in Star Trek, such as a replicator (molecular scale 3D printer)? A regular classical computer would take ages to calculate the positions where proteins would need to be in order to make scrambled eggs that the Star Trek replicator could make in a fraction of a second.

*I also need this to be realistic for a sci-fi plot where a group of aliens named the Xanadu use molecular assemblers on their version of an airplane with a quantum computer as an autopilot and on spaceships, set in the 2040s-2070s. The Xanadu have airplanes and spacecraft just like humans do, but a little more advanced. Their tech in the late 1990s was similar to our tech in 2023 for sci-fi's sake, such as internal combustion engines running on biodiesel and hybrid cars.

 

[Lost4words]

And yet, there is a sequence created that goes 1 2 3 4 5...

So what if there's actually an infinite number of half numbers between each whole number? There is still a sequence of sequential numbers created by Arabic scholars and mathematicians in the 10th century AD. Why? Because it's easy and simple.

So, i am 59 now, but next birthday i wont be 60....lol...

 

[The IbanezerScrooge]

Sagan made it up. In fact, I reckon he might have thought of the idea that then built a story around it.

I imagine Sagan thought of it while thinking of evidence that might convince him that our existence was designed. At the time that came out I think was on the cusp of when creationism was beginning it's foray into attempting to sound scientific to justify itself.

 

[The Barbarian]

There is no such a thing as a next number. What do I mean by that? Let's use 1 as an example. 1 does not have a number that is next to it. You might think 2 is the next number, but 1.5 is actually in between them. You might think 1.5 is the next number, but 1.25 is between them. This goes on infinitely for every number.

You've discovered Zeno's paradox. It was answered by the development of the concept of limits and calculus.

 

[sjastro]

You can show Zeno's paradox is not a paradox by showing the series S converges to a finite value.

Multiply S by 2.

Subtract the original series from the series that has been multiplied by 2.
2*S - S = S.

Hence S converges to one.

 

[Farid7]

You can show Zeno's paradox is not a paradox by showing the series S converges to a finite value.

View attachment 337592

Multiply S by 2.

View attachment 337593

Subtract the original series from the series that has been multiplied by 2.
2*S - S = S.

View attachment 337594

Hence S converges to one.

View attachment 337595

Yes, it's true that adding every part of a number together adds to a finite value. But, that doesn't mean there is such a thing as a next number in numbers.

There are, in fact, an infinite number of next numbers; not just one infinite either, there are infinites of next numbers.

-CryptoLutheran

Yes, as it was pointed out in one of the posts. But, that is true only with a certain set of numbers called integers. So, if you pick certain numbers, it can have next numbers, but not if you include all numbers.

 

[sjastro]

Yes, it's true that adding every part of a number together adds to a finite value. But, that doesn't mean there is such a thing as a next number in numbers.

This was simply to show Zeno's paradox expressed as a mathematical series is not a paradox as the series converges.
Your thread was specifically addressed in post#9.

 

[Farid7]

This was simply to show Zeno's paradox expressed as a mathematical series is not a paradox as the series converges.
Your thread was specifically addressed in post#9.

Zeno has never claimed the mathematical series does not converge. He only claimed that time is infinitely divisible.

 

[Tinker Grey]

Of course there is a next number. Numbers serve us. If I have an apple and someone hands me another, then I have two. I need not go through infinite numbers in between. Those are irrelevant.

Math is cool; infinities are neat. But if reality shows I can traverse space, then Zeno's paradoxes are just an interesting puzzle -- and fundamentally wrong.

 

[Farid7]

Of course there is a next number. Numbers serve us. If I have an apple and someone hands me another, then I have two. I need not go through infinite numbers in between. Those are irrelevant.

Math is cool; infinities are neat. But if reality shows I can traverse space, then Zeno's paradoxes are just an interesting puzzle -- and fundamentally wrong.

There is no next number when it comes to numbers. If there is, then please tell me the next number to 0.

 

[Farid7]

So, i am 59 now, but next birthday i wont be 60....lol...

As pointed out in the previous posts, a set of integer numbers have next numbers. Suppose, there is a set of integer that contains 1,2,3. Obviously, in that set, 2 is next to 1. But, that doesn't mean that 2 is the next number to 1. There are infinite numbers in between them.

Let me give another example. Suppose, a set contains the letters A, C, Z. The next letter to A in the set is C, but C is not actually next to A, B is.