Wednesday, August 1, 2012

Remarkable Relationships (4)

Once again any root of 1 can be expressed in terms of both (real) cos and (imaginary) sin components.

Now if we consider the n roots of unity (where n is an odd values integer ≠ 1), the absolute value of the product of all the cos values = 1/{2^(n - 1)}.

The absolute value of the product of all the sin values in turn = n/{2^(n - 1)}. (In the case where the root = n/n, the sin value = o; so this value is ignored in obtaining the product of sin values!)

For example the where n = 3, the 3 roots of unity are

- .5 + .866i and
- .5 - .866i

Therefore the absolute value of the product of the 3 cos values = .25 = 1/(2^2)

Likewise the absolute value of the product of the 2 (significant) sin values = .75 = 3/(2^2).

However when n is even this relationship does not hold. For example when n = 2, the 2 roots are

+ 1 and
- 1

So the absolute value of the product of the cos values = 1. However no product of sin values exists in this case!

Then when n = 4, the 4 roots are

+ 1
- 1
+ i and
- i

So the absolute value of the product of (nonzero) cos values = 1 while the absolute value of (nonzero) sin values also = 1.

So in the behaviour of these values we see a pattern that is akin to the Riemann Zeta Function where the nature of the Function for odd integer values of s is uniquely distinct from corresponding even values!

As all prime numbers other than 2 are odd, this implies that where n = p (with p a prime number ≠ 2), the the absolute value of the product of of the real (cos) part of prime roots of 1 = 1/{2^(p - 1)} whereas the corresponding absolute value of the imaginary (sin) part of prime roots of 1 = p/{2^(p - 1)}

There is also an unexpected connection (with respect to the product of the real part with number partitions (where each arrangement represents a new partition).

For example in standard terms (without rearrangement) 3 has 3 partitions i.e. 3, 1 + 2 and 1 + 1 + 1. However when we allow each permutation to represent a fresh partition, then 3 has 4 partitions i.e. 3, 1 + 2, 2 + 1 and 1 + 1 + 1.

So the number of partitions here of 3 (allowing rearrangement) = 2^2.

More generally the number of partitions of n (allowing rearrangement) = 2^(n - 1).

Where n is a prime number (= p) the number of partitions (allowing rearrangement) = 2^(p - 1).

And this result is the inverse of the absolute product of the real part of the prime roots of 1.

So in this way we can perhaps see an intimate connection as between partitions and the prime roots of 1!

We have seen before that whereas the cardinal nature of number corresponds to linear logic, the ordinal nature (pertaining to the relationship between numbers) strictly corresponds to an alternative circular logic.

Therefore just as the roots of 1 represent the most basic example of such linear/circular behaviour, not surprisingly partitions (pertaining to a fundamental relationship between numbers) are characterised by the same linear/circular behaviour.