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Can Power of a Number Indicates Dimensions ? 
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#1
Jun1804, 07:05 AM

P: 1,443

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. [itex]2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, ...[/itex]. 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. 


#2
Jun1804, 07:18 AM

Sci Advisor
PF Gold
P: 2,226

what is (√2)^{2}, what is 4^{2}?
The indice of a unit often tells you something about the 'dimensions' of a quantity, but the vale says nothing. 


#3
Jun1804, 09:31 AM

P: 1,443

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 multidimensional, 2 is 1 dim, 3 is 1dim, 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. 


#4
Jun1804, 09:38 AM

Sci Advisor
PF Gold
P: 2,226

Can Power of a Number Indicates Dimensions ?
It really doesn't mean anything at all thoug, quantities such as area can take on any real values.



#5
Jun1804, 09:44 AM

PF Gold
P: 2,901

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. 


#6
Jun1804, 09:52 AM

P: 1,443




#7
Jun1804, 09:52 AM

PF Gold
P: 2,901

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 


#8
Jun1804, 09:57 AM

P: 1,443

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


#9
Jun1904, 06:14 AM

P: 1,443

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 onedimensional number: 1,2,5,7. There are 2 twodimensional number: 4, 9. There is only one threedimensional 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 onedimensional. 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. 


#10
Jun1904, 06:44 AM

PF Gold
P: 2,901

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. 


#11
Jun1904, 09:01 AM

P: 1,443




#12
Jun1904, 09:13 AM

Emeritus
Sci Advisor
PF Gold
P: 16,091

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



#13
Jun1904, 09:25 AM

P: 1,443

Hurkyl,
You lost me there. 


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