by Michael Kirsch complex modulus.
"the schema of complex numbers is a system of equidistant points which are laid on equidistant straight lines such that the infinite plane is decomposed into infinitely many squares. Every number which is divisible by a complex number a + bi = m will likewise form infinitely many squares, whose sides are equal to Öa^{2} + b^{2} or whose area is equal to a^{2} + b^{2}; the latter squares will have an inclined position to the former if neither of the two numbers a, b are equal to zero."^{1}
"Every number which is not divisible with respect to modulus m will correspond to a point, which lies either within such a square, or on the boundary line of two squares; the latter case, however, can only occur if a, b have a common divisor..."
^{2} The line drawn from our number divisible by the modulus, 9+2i, is -3.
Since we are looking at the same point, the two numbers should be congruent relative to this number. Well, is 6+2i º -3 mod9+2i? Our arithmetic seems to be following along nicely.
Here the least residue is -1+2i. All other numbers are congruent to -1+2i relative to a modulus which is either one of the 4 associates of 1+4i, or some multiple of it. How are all of these similar? powers of complex numbers. Likewise any complex number taken to any power will be congruent to one of these least residues inside the square, zero, or the sides of the square in the case mentioned by Gauss where a and b have a common divisor.
An example of this would be (1+2i)^{2} º -i mod1+4i"
norm,of 1+4i and 1-4i, its conjugate. We will refer to the norm as p . Its general form is thus a^{2}+b^{2}, the sum of two squares. Is every norm, the sum of two squares? Also, what would the area of the square be? If our two sides are 1+4i and 4-i, what is their product? Wait a second, how can an area be 8+15i? Hmmmm. That doesn't seem to work. Rather, if we want to describe the length of the sides of these squares, they would be the Ö{a^{2} + b^{2}}, and the area would thus be a^{2} + b^{2}. What does this tell us about the nature of these complex numbers, that they cannot represent areas?## Footnotes:^{1}The geometric representation given here is one way to represent the arithmetic of complex numbers. There are other physical forms which complex numbers can assume.
^{2} Arising as a corollary from this, we here have a way to determine whether uneven complex numbers like 2+9i are divisible by any number and thus not prime. It is not hard to demonstrate that any complex number which connects two points on this grid will not be prime. |