Real numbers (R) are second biggest family of numbers in mathematics. We can find real numbers anywere in nature, around us:
- someone’s age: 28
- temperature: – 10 °C
- someone’s weight: 75.5 kg
- the number pi (π): 3.14159…
All the numbers from the example above are real numbers even if they are part of subgroups of:
- natural numbers (N): 28
- integer numbers (Z): -10
- rational numbers (Q): 75.5 (151/2)
- real numbers (R): 3.14159
We know that the group of real numbers includes also natural, integer and rational numbers. Mathematically this is written as:
\[\Large \mathbb{N} \in \mathbb{Z} \in \mathbb{Q} \in \mathbb{R}\]It might look complicated but it isn’t. Basically it means that the natural numbers are a subset of integer numbers which are a subset of rational numbers which are a subset of real numbers.
If we are applying different kind of mathematical operations on real numbers (addition, subtraction, multiplication, division, power, square root, etc.), the results are going to be also real numbers. This works fine except when we want to extract the square root of a negative number!
Since every real number (positive or negative) multiplied with itself it’s a positive real number, if we apply the inverse function of power of two, square root, we’ll get back two possible solutions:
\[ \begin{equation*} \begin{split}3 \cdot 3 = 3^{2} = 9\\
(-3) \cdot (-3) = (-3)^{2} = 9\\
\sqrt{9} = \pm 3
\end{split} \end{equation*} \]
What do we do if we need to extract the square root of a negative number? Can the square root of a negative real number be also a real number? The answer is no, it can’t.
\[ \begin{equation*} \begin{split}\sqrt{-9} \notin \mathbb{R}
\end{split} \end{equation*} \]
This was also the problem of some mathematicians some long time ago. They decided to extend the domain of the numbers in order to cope also with the square root of negative numbers so they created complex numbers!
We can consider that the real numbers domain (family) is one-dimensional because a real number has only one value at a time.
Complex numbers are bi-dimensional, they consist of a pair of two real numbers. We take as example the complex number z which is defined by a pair two real numbers a and b.
\[ \begin{equation*} \begin{split}z=(a, b)\\
z \in \mathbb{C}\\
a, b \in \mathbb{R}
\end{split} \end{equation*} \]
When we are dealing with complex numbers we are writing them in this form:
\[ \begin{equation} \begin{split}\bbox[#FFFF9D]{z=a + b \cdot i}
\end{split} \end{equation} \]
A complex number z consists of a real part a and an imaginary part b. If a is zero then the complex number z becomes purely imaginary, if the imaginary part is zero then the complex number z is purely real (or real number):
- purely imaginary: z = b · i
- purely real: z = a
Here are some examples of two complex numbers z1 and z2, defined by (1):
\[ \begin{equation*} \begin{split}z_1 = 2 + 1 \cdot i \\
z_2 = 1 + 3 \cdot i
\end{split} \end{equation*} \]
So how did we get this form? What is i? Why are complex numbers defined like that? What mathematical laws applies also to complex numbers?
In order to answer to these questions we’ll use the pair notation of the complex numbers and go through the properties of the complex numbers.
Now we will define the mathematical properties of the complex numbers. Suppose we have two complex numbers (z1 and z2):
\[ \begin{equation*} \begin{split}z_1 = (a_1, b_1)\\
z_2 = (a_2, b_2)
\end{split} \end{equation*} \]
For these complex numbers a series of mathematical operations are defined:
- addition
When performing an addition on two complex numbers we add the real part of the first number (a1) with the real part of the second number (a2). We do the same for the imaginary parts, (b1) is added to (b2).
\[ \begin{equation} \begin{split}z_1 + z_2 = (a_1 + a_2, b_1 + b_2)
\end{split} \end{equation} \]
Example using the complex numbers addition, defined by equation (2):
\[ \begin{equation*} \begin{split}z_1 + z_2 = (2 + 1, 1 + 3) = (3, 4) = 3 + 4 \cdot i
\end{split} \end{equation*} \]
- subtraction
When performing an subtraction on two complex numbers we subtract the real part of the first number (a1) with the real part of the second number (a2). We do the same for the imaginary parts, (b1) is subtracted with (b2).
\[ \begin{equation} \begin{split}z_1 – z_2 = (a_1 – a_2, b_1 – b_2)
\end{split} \end{equation} \]
Example using the complex numbers subtraction, defined by equation (3):
\[ \begin{equation*} \begin{split}z_1 – z_2 = (2 – 1, 1 – 3) = (1, -2) = 1 – 2 \cdot i
\end{split} \end{equation*} \]
- multiplication
For the moment we take the formula of the multiplication for granted. By the end of the article we’ll find out how we ended up with it.
\[ \begin{equation} \begin{split}z_1 \cdot z_2 = (a_1 a_2 – b_1 b_2, a_1 b_2 + b_1 a_2)
\end{split} \end{equation} \]
Example using the complex numbers multiplication, defined by equation (4):
\[ \begin{equation*} \begin{split}z_1 z_2 = (2 \cdot 1 – 1 \cdot 3, 2 \cdot 3 + 1 \cdot 1) = (-1, 7) = -1 + 7 \cdot i
\end{split} \end{equation*} \]
- division
The same for division, for the moment we take the formula for granted. By the end of the article we’ll find out how we calculate the real and imaginary parts of the division.
\[ \begin{equation} \begin{split}\frac{z_1}{z_2} = \left ( \frac{a_1 a_2 + b_1 b_2}{a_2^2 + b_2^2}, \frac{b_1 a_2 – b_2 a_1}{a_2^2 + b_2^2} \right ), z_2 \neq (0, 0)
\end{split} \end{equation} \]
Example using the complex numbers multiplication, defined by equation (5):
\[ \begin{equation*} \begin{split}\frac{z_1}{z_2} = \left ( \frac{2 \cdot 1 + 1 \cdot 3}{1^2 + 3^2}, \frac{1 \cdot 1 – 3 \cdot 2}{1^2 + 3^2} \right ) = (0.5, 0.5) = 0.5 + 0.5 \cdot i
\end{split} \end{equation*} \]
We can see that the basic operation applied to complex numbers are slightly different from those applied to real numbers. All operations applied to complex numbers implies algebraic operation on the pair of real numbers (a, b) which define the complex number z.
Next we’ll use another set of complex numbers (z1 , z2 and z3 ) to define more mathematical properties:
\[ \begin{equation*} \begin{split}z_1 =& 2 + 3 \cdot i\\
z_2 =& 1 – 4 \cdot i\\
z_3 =& -3 + 5 \cdot i
\end{split} \end{equation*} \]
As for real numbers, the mathematical operations on complex numbers obey the laws of:
- comutativity
z_1 + z_2 = z_2 + z_1
\end{split} \end{equation*} \]
Example using the complex numbers defined by (6):
\[ \begin{equation*} \begin{split}z_1 + z_2 =& (2 + 1, 3 – 4) = (3, -1) = 3 – 1 \cdot i\\
z_2 + z_1 =& (1 + 2, -4 + 3) = (3, -1) = 3 – 1 \cdot i
\end{split} \end{equation*} \]
- associativity
(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)
\end{split} \end{equation*} \]
Example using the complex numbers defined by (6):
\[ \begin{equation*} \begin{split}z_1 + z_2 =& (2 + 1, 3 – 4) = (3, -1)\\
(z_1 + z_2) + z_3 =& (3 – 3, -1 + 5) = (0, 4) = 0 + 4 \cdot i = 4 \cdot i\\\\
z_2 + z_3 =& (1 – 3, -4 + 5) = (-2, 1)\\
z_1 + (z_2 + z_3) =& (2 – 2, 3 + 1) = (0, 4) = 0 + 4 \cdot i = 4 \cdot i
\end{split} \end{equation*} \]
- distributivity
z_1 (z_2 + z_3) = z_1 z_2 + z_1 z_3
\end{split} \end{equation*} \]
Example using the complex numbers defined by (6):
\[ \begin{equation*} \begin{split}z_1 z_2 =& (2 \cdot 1 – 3 \cdot (-4), 2 \cdot (-4) + 3 \cdot 1) = (14, -5)\\
z_1 z_3 =& (2 \cdot (-3) – 3 \cdot 5, 2 \cdot 5 + 3 \cdot (-3)) = (-21, 1)\\
z_1 z_2 + z_1 z_3 =& (14 – 21, -5 + 1)\\
=& (-7, -4) = -7 -4 \cdot i\\\\
z_2 + z_3 =& (1 – 3, -4 + 5) = (-2, 1)\\
z_1 (z_2 + z_3) =& (2 \cdot (-2) – 3 \cdot 1, 2 \cdot 1 + 3 \cdot (-2))\\
=& (-7, -4) = -7 -4 \cdot i
\end{split} \end{equation*} \]
Let’s investigate further the properties of complex numbers. If we take for example two complex numbers z1 and z2 with the imaginary parts zero:
\[ \begin{equation*} \begin{split}z_1 =& (a_1, 0) = a_1 + 0 \cdot i\\
z_2 =& (a_2, 0) = a_2 + 0 \cdot i
\end{split} \end{equation*} \]
If we add/subtract and multiply them together:
\[ \begin{equation*} \begin{split}(a_1, 0) \pm (a_2, 0) =& (a_1 \pm a_2, 0 \pm 0)\\
=& (a_1 \pm a_2, 0)\\\\
(a_1, 0) \cdot (a_2, 0) =& (a_1 a_2 – 0 \cdot 0, a_1 \cdot 0 + 0 \cdot a_2)\\
=& (a_1 a_2, 0)
\end{split} \end{equation*} \]
We can see that the result is a purely real number because the imaginary part is zero. So a complex number with the imaginary part zero is in fact a real number and the algebraic operations applied to it are the same as for real numbers.
One of the most important aspect of complex numbers is the definition of the imaginary unity i:
\[ \begin{equation*} \begin{split}z = (0, 1) = 0 + 1 \cdot i = i
\end{split} \end{equation*} \]
Next we’ll go through some other properties of algebraic operations between real and complex numbers.
If we have a real number (b, 0) and we multiply it with the imaginary unity i the result will be a complex number purely imaginary (0, b):
\[ \begin{equation*} \begin{split}(b, 0) \cdot (0, 1) = (0, b) = b \cdot i
\end{split} \end{equation*} \]
If we take a real number (a, 0) and multiply it with a purely imaginary number (0, b) the result is going to be a complex number (a, b):
\[ \begin{equation*} \begin{split}(a, 0) + (0, b) = (a, b) = a + b \cdot i
\end{split} \end{equation*} \]
If we multiply two purely imaginary numbers (0, 1), both imaginary unity i, we will get the real number -1:
\[ \begin{equation*} \begin{split}(0, 1) =& 0 + 1 \cdot i\\
i^2 =& i \cdot i = (0, 1) \cdot (0, 1)\\
=& (0 \cdot 0 – 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0) = (-1, 0) = -1
\end{split} \end{equation*} \]
This is one very important property of complex numbers: The square of the imaginary unity i is the real number -1:
\[ \begin{equation*} \begin{split}\bbox[#FFFF9D]{i^2 = -1}
\end{split} \end{equation*} \]
With this in mind we can demonstrate the general formulas of complex numbers multiplication and division:
- multiplication
z_1 =& a_1 + b_1 \cdot i\\
z_2 =& a_2 + b_2 \cdot i\\\\
z_1 z_2 =& (a_1 + b_1 \cdot i) \cdot (a_2 + b_2 \cdot i)\\
=& a_1 a_2 + a_1 b_2 \cdot i + a_2 b_1 \cdot i + b_1 b_2 \cdot i\\
=& a_1 a_2 – b_1 b_2 + (a_1 b_2 + b_1 a_2) \cdot i\\
=& (a_1 a_2 – b_1 b_2, a_1 b_2 + b_1 a_2)
\end{split} \end{equation*} \]
- division
z_1 =& a_1 + b_1 \cdot i\\
z_2 =& a_2 + b_2 \cdot i\\\\
\frac{z_1}{z_2} =& \frac{a_1 + b_1 \cdot i}{a_2 + b_2 \cdot i} = \frac{a_1 + b_1 \cdot i}{a_2 + b_2 \cdot i} \cdot \frac{a_2 – b_2 \cdot i}{a_2 – b_2 \cdot i}\\
=& \frac{(a_1 + b_1 \cdot i) \cdot (a_2 – b_2 \cdot i)}{a_2^2 – b_2^2}\\
=& \frac{a_1 a_2 – a_1 b_2 \cdot i + a_2 b_1 \cdot i – b_1 b_2 \cdot i^2}{a_2^2 – b_2^2}\\
=& \frac{a_1 a_2 – b_1 b_2 \cdot i^2 + (a_2 b_1 – a_1 b_2) \cdot i}{a_2^2 – b_2^2}\\
=& \frac{a_1 a_2 + b_1 b_2}{a_2^2 – b_2^2} + \frac{a_2 b_1 – a_1 b_2}{a_2^2 – b_2^2} \cdot i\\
=& \left ( \frac{a_1 a_2 + b_1 b_2}{a_2^2-b_2^2}, \frac{a_2 b_1 – a_1 b_2}{a_2^2-b_2^2} \right )
\end{split} \end{equation*} \]
These being said about complex number we can now solve our initial problem, extracting the square root of a negative number:
\[ \begin{equation*} \begin{split}\sqrt{-9} = \sqrt{9 \cdot -1} = \sqrt{9 \cdot i^2} = \sqrt{9} \cdot \sqrt{i^2} = \pm 3 \cdot i
\end{split} \end{equation*} \]
We can clearly see that the square root of a real negative number is a purely imaginary number.
At the first sight complex numbers might seem intimidating but if you master the basics everything becomes very clear and easy.
Complex numbers are heavily used in mathematics, physics and engineering. When studying high order equations alternating current theory or control theory we’ll encounter complex numbers quite a lot.
This article was just an introduction, in the oncoming articles we’ll have a look at the different graphical representations of complex numbers (cartesian and polar) and exponential form.
Keep practicing the algebraic expressions on complex numbers and give a feedback if things are unclear or you have further questions.
Don’t forget to:
\[ \begin{equation*} \begin{split}\bbox[#FFFF9D]{f(x_i) = \left\{\begin{matrix} Subscribe,& i = 1\\Comment,& i = 2\\Like,& i = 3\\Share,& i = 4\\Support,& i = 5\end{matrix}\right.}
\end{split} \end{equation*} \]