Can Power of a Number Indicates Dimensions ?

[Antonio Lao]

Can Power of a Number Indicates Dimensions ?

When we raise a number to certain power, does the result indicates or tells about its dimension? So that for each integer value there is a dimension exclusively associated with each number.

Obviously, this will not work for the number 1. But for number 2, it works nicely.

$2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, ...$. This tells us that the number 1 is the only number that can be in any dimension while the number 2 is basically one dimensional. The number 4 is basically two dimensional. The number 8 is basically three dimensional, the number 16 is basically four dimensional, etc.

 

[jcsd]

what is (√2)², what is 4²?

The indice of a unit often tells you something about the 'dimensions' of a quantity, but the vale says nothing.

 

[Antonio Lao]

jcsd,

The notation square root of 2 to the power 2 is just another notation for the number 2 which is one dimensional. 4 to the power 2 is the same as 2 to the 4th power and the number 16 has dimension of 4.

The integer 1 is multi-dimensional, 2 is 1 dim, 3 is 1-dim, 4 is 2 dim, 5 is 1 dim, 6 is 1 dim, 7 is 1 dim, 8 is 3 dim, 9 is 2 dim, etc.

It seems that all prime numbers are one dimensional numbers.

 

[jcsd]

It really doesn't mean anything at all thoug, quantities such as area can take on any real values.

 

[arivero]

Antonio,

2^n is the number of subsets you can from a set of n elements. Possibly you are grasping an intuition of this fact, and mistakenly confusing it with the concept of dimension.

The 2^n idea was used to define transfinite numbers time ago by getting a generalization of the inequality 2^n > n . A pending question from early XXth century mathematics is if there is some infinite number between a0 and 2^a0, being a0 the countable infinite of natural numbers.

 

[Antonio Lao]

It really doesn't mean anything at all thoug, quantities such as area can take on any real values.

Can we just restrict the number to integer values? Or quantum values? Without decimals and hence not dealing with irrationals and transcendental numbers.

 

[arivero]

It seems that all prime numbers are one dimensional numbers.

Ah, this is a different concept. Are you calling d(n)="number of prime factors of n"? Or D(n)="number of divisors of n"? It does not matter a lot, because obviously d(n)=<D(n)=<2^d(n).

The collection of numbers having d(n)=2 is very important for cryptography, so I am pretty sure they have been studied deeply. The collection d(n)=1 is of course the set of prime numbers as you have said. I am not aware of interest for d(n)>2

 

[Antonio Lao]

Arivero,

I will get back with you with my reply. My warning is that I am not good at math.

 

[Antonio Lao]

arivero,

I think, what I'm saying is that I am defining dimension as the number of equal prime factors. Obviously, this would not work for the number 6. I guess, 6 is the product of two primes (2 and 3). So I really can't say anything about the dimension of the number 6. Between 1 and 9, with the exception of 6, there are 4 one-dimensional number: 1,2,5,7. There are 2 two-dimensional number: 4, 9. There is only one three-dimensional number: 8. Again, the number 10 would have the same problem as 6. Can't tell anything about the dimension of 10. But 11 is one-dimensional. 12 is also a problem. 13 is one dimensional. 14 is problem. 15 is problem. 16 is four dimensional. 17 is one dimensional. 18 is problem. 19 is one dimensional. 20 is problem. 21 is problem. The next three dimensional number would be the number 27.

 

[arivero]

Hmm. I do not forsee how to attach this concept to the usual ones of geometry.

Btw 72 is a hell of problem for your view.

 

[Antonio Lao]

Btw 72 is a hell of problem for your view.

I don't understand? Could you explain a little bit more.

 

[Hurkyl]

Normally we just call these things; 2, 3, 4, 5, 7, 8, 9, 11, ... "prime powers".

 

[Antonio Lao]

Hurkyl,

You lost me there.