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Complex Numbers

Terms and Formulae

Complex Numbers

Introduction to Complex Numbers

Terms

Argument  -  The angle created by the positive real axis and the segment connecting the origin to the plot of a complex number in the complex plane.
Complex Conjugate  -  The complex conjugate of a given complex number a + bı is a - bı .
Complex Number  -  The set of all numbers of the form a + bı , where a and b are real numbers. The real numbers are all complex numbers.
Complex Plane  -  A plane with two perpendicular axes, the real axis and the imaginary axis, on which a complex number a + bı is plotted at the coordinate (a, b) . It is customary for the real axis to coincide with the x -axis of the rectangular coordinate system, and for the imaginary axis to coincide with the y -axis of the rectangular coordinate system.
Imaginary Axis  -  The axis in the complex plane that customarily coincides with the y -axis of the rectangular coordinate system, and on which the imaginary part bı of the complex number a + bı is plotted.
Imaginary Number  -  A number that can be expressed in the form bı , where b is a real number.
Imaginary Part  -  The bı term in every complex number a + bı .
Imaginary Unit  -  The imaginary unit is ı . ı = .
Modulus  -  for a complex number a + bı . In the complex plane, it is the distance between the plot of a complex number and the origin.
Polar Form of a Complex Number  -  The polar form of a complex number z = a + bı is this: z = r(cos(θ) + ısin(θ)) , where r = | z| and θ is the argument of z .
Real Axis  -  The axis in the complex plane that typically coincides with the x -axis of the rectangular coordinate system, and on which the real part a of a complex numbers a + bı is plotted.
Real Part  -  In a complex number a + bı , a .
Standard Form  -  For a complex number, a + bı .

Formulae

 
De Moivre's Theorem Let z = r(cos(θ) + ısin(θ).Thenz n = [r(cos(θ) + ısin(θ)]n = r n(cos() + ısin() , where n is any positive integer.
 
Roots of a Complex Number A complex number z = r(cos(θ) + ısin(θ) has exactly n n th roots given by the equation [cos() + ısin()] , where n is a positive integer, and k = 0, 1, 2,..., n - 2, n - 1 .

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