Spiral Numbers

The idea is simplest when thinking in terms of polar coordinates.

For a,c > 0 and b,d real , the complex numbers satisfy

(a,b) (c,d) = (ac , b + d mod 2 pi)

The idea of spiral numbers is

(a,b)(c,d) = (ac , b + d)

So far for products.

The sum for spiral numbers is defined by

X + Y = ln( exp(X) exp(Y) ).

So it comes down to finding a good ln and exp.

My guess is exp(a,b) =

( exp(a + ab) , e b)

Where |*| is the absolute value.

And the ln is just the inverse.

For X^Y we use exp( ln X * Y ).

I wonder how the algebra works out.

Is this a good idea ?

I wonder what you think.

Regards

Tommy1729

The idea is simplest when thinking in terms of polar coordinates.

For a,c > 0 and b,d real , the complex numbers satisfy

(a,b) (c,d) = (ac , b + d mod 2 pi)

The idea of spiral numbers is

(a,b)(c,d) = (ac , b + d)

So far for products.

The sum for spiral numbers is defined by

X + Y = ln( exp(X) exp(Y) ).

So it comes down to finding a good ln and exp.

My guess is exp(a,b) =

( exp(a + ab) , e b)

Where |*| is the absolute value.

And the ln is just the inverse.

For X^Y we use exp( ln X * Y ).

I wonder how the algebra works out.

Is this a good idea ?

I wonder what you think.

Regards

Tommy1729