Complex Numbers

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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