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The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. [1]
Note: solving for ′ returns the resultant angle in the first quadrant (< <). To find , one must refer to the original Cartesian coordinate, determine the quadrant in which lies (for example, (3,−3) [Cartesian] lies in QIV), then use the following to solve for :
For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of are the vertical axis. A complex number can also be defined by its geometric polar coordinates : the radius is called the absolute value of the complex number, while the angle from the positive real axis is ...
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal.
In polar form, if and are real numbers then the conjugate of is . This can be shown using Euler's formula . The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates ).
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
Generally, if = ( + ) (in polar form) and w are arbitrary complex numbers, then the set of possible values is = ( + ) = { (+) + (+) |}. (Note that if w is a rational number that equals p / q in lowest terms then this set will have exactly q distinct values rather than infinitely many.
x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, e ix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.