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@BilboGrabbins Hi there!🙋I have a Query, Especially for You.Interested?A_______BA simple line.Start point A.End point B.I suppose i can keep dividing it into Halves.But considering Planck's Length, Can i Divide it until Infinity?If i Know the Line's point of Origin(A) & am Aware of the Line's point of Cessation(B)...Does that sound like Infinity?Or if i didn't know the Origin, n was clueless bout the End.Infinity!Ps - Thanks E_S for the suggestions.I've Realized my capacity to understand & learn is Not infinite.Hence i do not bother myself, & also spare Others of goin thru the troubles of explaining me things which i Firmly Believe i shall never understand.(Reason i Request for short & brief answers, i Hate it when Teachers waste their Precious Time on Futile Things): )
I've Realized my capacity to understand & learn is Not infinite.
No, to divide through the Planck length infinitely many times would be equivalent to a singularity. That is itself equivelent to a breakdown in physics.
...The Planck length is expected to be the shortest measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would inevitably result in black hole production.... [from Wikipedia]
What prevents us from expanding a 1 meter rod to 1 meter plus half Planck's length, eg. by heating it up?
You can't localise a particle to one point in space unless it's momentum → ∞
It's sort of the point here. Physics breaks down below this length.
https://twitter.com/pickover/status/1541962088926560256?t=ml0TBOGlhgCZIKDaLXserg&s=03There are many hypercomplex numbers known by modern math. Do they have the same size as real number?
Apparently, the complex numbers have the same cardinality as the reals and I'm tempted to assume that the hypercomplex ones do too.
Quote from: Bored chemist on 01/07/2022 09:59:31Apparently, the complex numbers have the same cardinality as the reals and I'm tempted to assume that the hypercomplex ones do too.There are as many real numbers in a set of complex numbers where imaginary parts is 0. Like wise, there are as many real numbers in a set of complex numbers where imaginary parts is 1. You can change the imaginary part with any magnitude. Each of them contains a whole set of real numbers. How can we say that they have one to one relationship?
So why would there be a problem with the complex numbers having the same cardinality as the reals?
To show this (a bijection exists between C and R ) you need to show every real number maps to a unique complex number and that every complex number maps to a unique real number. The first part is easy -- every real number already is a unique complex number. So let's concentrate on the second part, mapping the complex numbers to the reals.Each complex number a+bi, a and b real, may be mapped to a unique real number as follows: Expand a and b as decimals, taking care not to end either in repeating 9s for uniqueness. Then you can interleave the digits of the two decimal expansions so, e.g, the even numbered digits are from a and the odd from b. Start at the decimal points and work out in both directions. Call the result of this interleaving c.So you now have a real number c from which you can recover a and b, and thus the original complex number a+bi.Since you did this mapping both ways, by the Cantor-Bernstein theorem (or common sense) the cardinality of the complex numbers and real numbers are the same.
There are as many real numbers in a set of complex numbers where imaginary parts is 0. Like wise, there are as many real numbers in a set of complex numbers where imaginary parts is 1. You can change the imaginary part with any magnitude. Each of them contains a whole set of real numbers. How can we say that they have one to one relationship?
There isn't a Mathematics section in this forum. It seemed that the OP was asking something about Mathematics without any reference to some application in Physics.
To be clear,
I didn't read the whole thread, but perhaps you could quote the place where somebody asserted that this would be problematic.
How can we say that they have one to one relationship?
Since you did this mapping both ways, by the Cantor-Bernstein theorem (or common sense) the cardinality of the complex numbers and real numbers are the same.
We can represent any rational numbers by combining two integers as numerator and denominator. Some irrational numbers can be stated as a rational number powered by another rational number. Let's call them power numbers. How many more expansion procedures like that are required to cover the whole real numbers?