Introduction to complex numbers
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Motivation behind complex numbers
Suppose we wanted to solve the equation:
Clearly, squaring any two real numbers will give us a positive value, which means that this equation is not solvable using real numbers. However, instead of leaving the equation unsolved, mathematicians have come up with a way to express the solution using so-called complex numbers.
Let's go ahead and take the square root of both sides of
Again, it does not make sense to take the square root of a negative number. At the heart of complex numbers is the imaginary number
The solution
The reason we somewhat forcefully solve these equations using complex numbers is to understand more about the nature of the equation instead of marking it as unsolvable. As we shall explore in the latter chapters, complex numbers are required to prove important theorems such as the fundamental theorem of algebra and the central limit theorem.
Complex numbers
If
we often denote a complex number as
.some textbooks define the imaginary number as
.
Some complex number
The following are all examples of complex numbers:
Notice how
Set of all complex numbers
Just like how
Real and imaginary part of complex numbers
If
is known as the real part of , denoted by . is known as the imaginary part of , denoted by .
Finding the real and imaginary part of a complex number
Consider the following complex numbers:
. .
Find the real parts and imaginary parts of
Solution. Let's first tackle
Next, the real and imaginary parts of
Purely real and purely imaginary complex numbers
Suppose we have a complex number
if
and , then the complex number is called purely real.if
and , then the complex number is called purely imaginary.
Arithmetic operations of complex numbers
Just like for real numbers, we can also add, subtract, multiply and divide between complex numbers.
Addition
Adding two complex numbers:
It is always a good idea to explicitly split the complex number into its real and imaginary parts as we did here.
Subtraction
Subtracting a complex number from another complex number:
Multiplication
Multiplying two complex numbers:
Here, we used the fact that
Division
Dividing two complex numbers is slightly tricky. We multiply both the numerator and denominator by
Performing arithmetic operations on complex numbers
Consider the following complex numbers:
Compute the following:
. . . .
Solution. 1. Addition is:
2. Subtraction is:
3. Multiplication is:
4. Division is:
Properties of arithmetic operations of complex numbers
Commutative property of multiplication of complex numbers
If
This means that we can perform the multiplication in either order.
Proof. Let
This completes the proof.
Associativity property of multiplication of complex numbers
If
Proof. We define complex numbers
We start from the left-hand side:
This completes the proof.
Visualizing complex numbers using the Argand plane
We can plot complex numbers on a so-called complex plane or the Argand plane. For instance, the complex number

The key difference between the traditional Cartesian plane and the Argand plane is what the axis lines represent. In Cartesian plane:
the horizontal axis represents the
-axis.the vertical axis represents the
-axis.
In contrast, for an Argand plane:
the horizontal axis represents the real part of the complex number.
the vertical axis represents the imaginary part of the complex number.
In fact, there is a one-to-map mapping between the Cartesian plane and the Argand plane. For instance, the point
Modulus of a complex number
Let
Note that the modulus is always a real number since
Discussion. The reason we use the absolute value symbol to indicate the modulus is that if
Here, the right-hand side represents the absolute value of the real number
Finding the modulus of a complex number
Let
Solution. The modulus of
Let's also visualize

Here, the modulus