Complex numbers can be expressed in many forms:
- Rectangular form: a + jb , where a and b are real numbers and j is an imaginary number given by sqrt(-1). Here, j is used as an operator to indicate that the real number which it precedes is measured along the imaginary axis.
- Polar form: r /_ theta , where r is a positive real number which is the modulus or the magnitude of the complex number and theta is the angle of the complex number measured counter-clockwise, about the origin, from the positive real axis. Its unit can be expressed in degrees or radians.
- Exponential form: r . e^j(theta) , here, the theta has to be in radians
- Trigonometric form: r . cos(theta) + r . j . sin(theta)
If z is a complex number, then:
= |z| /_ theta
= |z| . [ cos(theta) + j. sin(theta) ]
Since the complex number is not given, we'll write the rectangular form as:
z=x + y*i
where x is the real part,
x= Re(z) and
y is the imaginary part,
i is the imaginary unit, i^2=-1.
The trigonometric (polar) form is:
z=r(cos t + i sin t), where r = |z|
r = sqrt(x^2 + y^2)
r is the modulus of the complex number z.
If x and y are the coordinates of a point M(x,y), r is the position vector of the point M.
The formula for r = sqrt(x^2 + y^2) results from Pythagorean theorem, where r is hypotenuse.
cos t = x/r=>x=r cos t
sin t = y/r=>y=r sin t
z = r cos t + ir sin t
tan t = y/x => t = arctan y/x
z = r( cos t+i sin t) - trigonometric form of the complex number z.