The Algebra of complex numbers is a very interesting topic in Mathematics. The topic is one of the first in A-level mathematics. This makes it very important for college students.
While in most aspects of mathematics, the values and numbers used are real numbers, there are very few situations when some values will not fall under the real number category. The real number line generally includes Natural numbers, Integers, rational numbers, and irrational numbers. The real number generally includes any of the following -2, -1, 0, 1, 2, ¾, and 0.32.
Any of the numbers above is known as a real number. However, there might be instances where a value may, unfortunately, end up being
Clearly, the number above is not part of the real number system. This can be confirmed by inputting it into a calculator. You will get the syntax error because it does not exist. Also, even if you take the square root of the value, it will not be 7 0r –7. This is because
This article will focus on special complex numbers that is like
The Complex Number and The Power of i
A complex number is a number made up of real and imaginary numbers and is not in the real number line. The complex number is usually gotten in a situation where the square root of a negative number needs to be found. Mathematically, the square root of a negative number cannot be determined. Also, the only way a mathematician will end up finding a square root is when determining a quadratic equation using the general formula.
We will show an example.
Find the solution of the equation using the quadratic equation.
2x2 + 9x + 7 = 0
Note that a fully comprehensive explanation of finding solutions to a quadratic equation will not be given as we are trying to show how complex numbers are formed. Discussing how to use the quadratic equation to find the solutions of equations can be found here
The quadratic equation is given as
From the above equation 5x2 – 6x + 5 = 0, a = 5, b = – 6 and c = 5
Now it is impossible to find
and as such, the quadratic equation cannot have any real solution. This is where the complex number factor comes in. To find it though, it will be important to determine the i and what it means in the Complex number arrangement.
The Power of i
Now any number in the square root of a negative number can be converted to an imaginary number.
i is the symbol of representation for complex numbers and always means;
Therefore, every negative square root can always be simplified in such a way that it will now become a real number x i. Another example would be;
= 9 x i = 9i
As can be seen from the two simplifications of the negative squares, their solution had I to the real number. This particular situation will always be the case when finding the square root of a negative value. The two solutions 8i and 9i are known as the imaginary parts of complex numbers. As an extension, all real numbers multiplied by i will always be recognized as imaginary numbers of complex numbers because they cannot fit into the real number system. This also means that the pure real numbers like 9 cannot be added/subtracted to complex numbers like 9i. i.e 9 + 9i cannot give a singular value.
However, Multiplication and division operations work. 9 x 9i = 81i.
The power of i
As can be seen from the derived i, a complex number or complex equations must have values that are multiplied by i, which is the square root of a negative 1. All the derived i2, i3 and i4 will be used in the subsequent sections to determine the addition, subtraction, multiplication and division of complex numbers.
Finally, now recall that the definition of a complex number is a number that must be made of both real and imaginary parts.
Therefore, 9i is not a complex number but just an imaginary.
A complex number will have to be something like 10 + 9i, 5 +6i or 8 + 2i.
All these numbers have real parts, the parts before the signs, and the imaginary parts, which are the values with i after the signs. The symbol representing the complex number is z.
Therefore a complex number is actually;
Z = a + bi
Addition Subtraction of Complex Numbers
The addition and subtraction rule for complex numbers is simple. Real numbers can only work with real numbers, while imaginary numbers can only work with their likes.
Simplify the following complex number let z1 = 5 + 6i and z2 = 2 – 4i, Find z1+z2
z1+z2 means the addition of the two complex numbers. So;
z1+z2 = (5 + 6i)+(2 – 4i) = 5 + 6i + 2 – 4i
Rearrange so that real parts come closer together while imaginary parts go closer t themselves
= 5 + 2 + 6i – 4i
= 7 + 2i
(5 + 6i)+(2 – 4i) = 7 + 2i
Multiplication of Complex Numbers
Finding the multiplication of complex numbers is not in any way difficult. Every form of multiplication rule still holds. The only thing is that only like terms can be added or subtracted if needed. we will consider some examples
Multiply (5 + 6i) by (2 + 4i)
We just need to follow the normal multiplication process.
(5 + 6i)(2 + 4i) = 10 +20i + 12i + 24i2
= 10 + 32i + 24i2
Recall that i2 = – 1
This will therefore makes
(5 + 6i)(2 + 4i) = 10 + 32i + 24(-1)
= 10 + 32i – 24
= 10 + 32i + 24
= – 14 + 32i
It is clear that no further simplification can be carried out. Therefore;
(5 + 6i)(2 + 4i) = -14 + 32i or 32i – 14
Following the above multiplication process will always lead to the right answers.
Conjugate Complex Numbers
The conjugate of any complex number is itself, with the only difference being the change of the sign that separates its real and imaginary parts. For example, the complex number (4 + 3i) has a conjugate that is presented as (4 – 3i).
The conjugates are pretty much the same except for the change in the sign separating their imaginary parts from their real parts.
The multiplication of a complex number by its conjugate will give a real number. This particular feature of complex numbers is very important for successfully dividing complex numbers.
We will take an example
Multiply the complex (5 + 7i) by its conjugate.
The conjugate of (5 + 7i) is (5 – 7i)
Therefore, we will have to find (5 + 7i)(5 – 7i)
(5 + 7i)(5 – 7i) = 25 – 35i + 35i – 49i2
(5 + 7i)(5 – 7i) = 25 – 49 (-1) = 25 + 49
(5 + 7i)(5 – 7i) = 74
As can be clearly seen, the solution was indeed a real number.
Division of Complex Numbers
As stated in the above subheadings, Carrying out a Division of complex numbers may sometimes need the application of conjugates and sometimes not. We will consider certain examples o complex number division.
Find the solution for the complex numbers below
If you look at the equation above more properly, you will see that the denominator can be factored. Hence;
It is that simple to find the answer.
Divide the complex number (4 + 5i) by (3 + 2i)
There is obviously no way to factor in the values. So we will have
We can use the conjugate rule to actually get a solution. Now we could multiply the numerator and denominator of our division with the conjugate of the denominator. This will ensure that the initial supposed answer is not altered while making calculations seamless. This is because multiplying the denominator with its conjugate will lead to a real number.
So we have;
No further simplification can be performed, so this is the solution to the division.
Complex number as a topic is extremely vast. However, its foundation starts from here, where its philosophy s established. Like the real number system, the complex number has its own vast branch of mathematics, which will be discussed in other topics. What has been done in this article is establish its basics and the foundation on which other work will be considered. Several examples were considered in this article to provide users with total complete information