The devil is in the details............b/c the trickiest thing is if u have two......................and took its value...........as in two to the i..................using i as an exponent..........u would generally think that if u did make a triangle.....................the side would be two...........but between two and the origin......the distance is two........one more unit to negative one.  Put another way.......from 2 to negative one.........along the X axis......is a distance of 3........................3.....................3 across.............one up..................just how pi begins...........3.1.....

Different parts of the math cannon that seem to have nothing to do with each other.........I read in book about the R hyp..............that it would be shame.......if to solve it involved a trick..........pretty thick language..........but it is as if it were a trick.............the trick being to find the correct way to look at it.................

Complex plane

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Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z.

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.^[1]
The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
The complex plane is sometimes known as the Argand plane, and geometric plots in the plane as Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818).^[2] Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

Contents

 [hide] 

• 1 Notational conventions
• 2 Stereographic projections
• 3 Cutting the plane
  • 3.1 Multi-valued relationships and branch points
  • 3.2 Restricting the domain of meromorphic functions
  • 3.3 Specifying convergence regions
• 4 Gluing the cut plane back together
• 5 Use of the complex plane in control theory
• 6 Quadratic spaces
• 7 Other meanings of "complex plane"
• 8 Terminology
• 9 See also
• 10 Notes
• 11 References
• 12 External links

Notational conventions[edit]

In complex analysis, the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts:

${\displaystyle z=x+iy}$

for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.
In the Cartesian plane the point (x, y) can also be represented in polar coordinates as

${\displaystyle (x,y)=(r\cos \theta ,r\sin \theta )\qquad (r,\theta )=\left({\sqrt {x^{2}+y^{2}}},\quad \arctan {\frac {y}{x}}\right).}$

In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2 (in radians), and some care must be taken to define the real arctangent function for points (x, y) when x ≤ 0.^[3] In the complex plane these polar coordinates take the form

${\displaystyle z=x+iy=|z|\left(\cos \theta +i\sin \theta \right)=|z|e^{i\theta }}$

where

${\displaystyle |z|={\sqrt {x^{2}+y^{2}}};\quad \theta =\arg(z)={\frac {1}{i}}\ln {\frac {z}{|z|}}=-i\ln {\frac {z}{|z|}}.\,}$^[4]

Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π; and the last equality (to |z|e^iθ) is taken from Euler's formula. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the complex exponential function is periodic, with period 2π i. Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any integer ≠ 0.^[5]
While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers w and z is given by ${\displaystyle \Re (w{\overline {z}})}$; then for a complex number z its absolute value |z| coincides with its Euclidean norm, and its argument arg(z) with the angle turning from 1 to z.
The theory of contour integration comprises a major part of complex analysis. In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started.
Almost all of complex analysis is concerned with complex functions – that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as a set of points in the w-plane. In symbols we write

${\displaystyle z=x+iy;\qquad f(z)=w=u+iv}$

and often think of the function f as a transformation from the z-plane (with coordinates (x, y)) into the w-plane (with coordinates (u, v)).

Stereographic projections[edit]

Main article: Stereographic projection

Riemann sphere which maps all points on a sphere except one to all points on the complex plane

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane.
We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped onto the equator, and the exterior of the unit circle (|z| > 1) will be mapped onto the northern hemisphere, minus the north pole. Clearly this procedure is reversible – given any point on the surface of the sphere that is not the north pole, we can draw a straight line connecting that point to the north pole and intersecting the flat plane in exactly one point.
Under this stereographic projection the north pole itself is not associated with any point in the complex plane. We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane. We speak of a single "point at infinity" when discussing complex analysis. There are two points at infinity (positive, and negative) on the real number line, but there is only one point at infinity (the north pole) in the extended complex plane.^[6]
Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphere onto the flat plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin z = 0. And the lines of longitude will become straight lines passing through the origin (and also through the "point at infinity", since they pass through both the north and south poles on the sphere).
This is not the only possible yet plausible stereographic situation of the projection of a sphere onto a plane consisting of two or more values. For instance, the north pole of the sphere might be placed on top of the origin z = −1 in a plane that is tangent to the circle. The details don't really matter. Any stereographic projection of a sphere onto a plane will produce one "point at infinity", and it will map the lines of latitude and longitude on the sphere into circles and straight lines, respectively, in the plane.