Cartesian Form
Trigonometry Complex Numbers Geometric Series Integrals of Complex Functions and Integration by parts
Cartesian Form Polar or Exponential Form Euler's Identities Conjugate Operations on two complex numbers nth power and nth roots of a complex number Functions of a complex variable Complex Functions of a real variable Magnitude and Phase Plot Examples and References

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Definition

We will use the letter $j$ to refer to the imaginary number $\sqrt{-1}$. Even though $j$ is not a real number, we can perform all arithmetic operations such as addition, subtraction, multiplication, division with $j$ using the algebra of real numbers.

A complex number $z$ is any number of the form $z = x+jy$, where $x$ is called the real part of $z$ and $y$ is called the imaginary part of $z$.Note: The imaginary part is not $jy$, rather it is only $y$. It is important to stick to this terminology, otherwise computations can go wrong.Often, it is useful to think of a complex number $z = x + j y$ as a vector in a two-dimensional plane as shown in the figure below, where $x$ is the $X$-coordinate and $y$ is $Y$-coordinate of the vector. Due to this relationship between a complex number and the corresponding vector, we will abuse the terminology and use the terms complex number and vector interchangeably, if the context should resolve any possible ambiguity. For example, a complex number is said to lie in the first quadrant (or, second quadrant etc) if the corresponding vector lies in the first quadrant (or, second quadrant etc).

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