Complex Numbers

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Complex number `z` can be represented in the form `z=a+bi`. This is called algebraic form of complex number. In this representation `i` is called imaginary unit, `a` is real part and `b` is imaginary part. If imaginary part of complex number not 0 then such number is called imaginary, for example `3+2i`. If `a=0` and `b!=0` then complex number is called purely imaginary. For example, `5i` is purely imaginary.

Except for `0`, any complex number can be represented in the trigonometric form or in polar coordinates: `z=r(cos alpha+i sin alpha)` or `z=r.cis(alpha)`. Each way has their advantages when doing specific problems. Knowing how to convert from one representation to another gives the student flexibility in being able to work with many different problems that use complex numbers.

Created with GeoGebra by Vitor Nunes


Move the point `z_1` in the first quadrant and observe its representation in algebraic and trigonometric form..

Uses of complex numbers

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Complex numbers can be used to solve quadratics for zeros. The quadratic formula solves `ax^2 + bx + c = 0` for the values of `x`. If the formula provides a negative in the square root, complex numbers can be used to simplify the zero.

Complex numbers are used in electronics and electromagnetism. A single complex number puts together two real quantities, making the numbers easier to work with. For example, in electronics, the state of a circuit element is defined by the voltage (V) and the current (I). The laws of electricity can then be expressed using the addition and multiplication of complex numbers. As mentioned before, this can also be applied to electromagnetism. Instead of being described as electric field strength and magnetic field strength, you can create a complex number where the electric and magnetic components are the real and imaginary numbers.