A Complex Number Division Calculator is used to calculate the division operation performed between two complex numbers. Complex numbers are unlike real numbers as they contain both Real and Imaginary parts.
To solve division for such numbers is therefore a computationally taxing job, and that is where this Calculator comes in to save you the trouble of going through all that computing.
What Is a Complex Number Division Calculator?
A Complex Number Division Calculator is an online tool designed to solve your complex number division problems in your browser in real-time.
This Calculator is equipped with a lot of computational power, and division is only one of the five different Mathematical Operations it can perform on a pair of complex numbers.
It is very easy to use, you just place your complex number inputs into the input boxes, and you can get your results.
How to Use the Complex Number Division Calculator?
To use the Complex Number Division Calculator, one must first have a pair of complex numbers to divide one against the other. Following that, the calculator requires to be set into the Correct Mode, which in this case would be Division. And finally, to get the result, one may enter the two complex numbers in their appropriate input boxes.
Now, a step-by-step procedure to use this calculator is given as follows:
Step 1
Go to the “Operation” drop-down option to select the one labeled, “Division (z1/z2)”. This is done for the setup of the Complex Number Division Calculator.
Step 2
Now, you may enter both your numerator complex number as well as your denominator complex number in the input boxes.
Step 3
Finally, you may press the button labeled “Submit” to get the solution to your problem. In case, you want to solve similar problems you can change the values in the input boxes and proceed.
It may be important to note that, when using this calculator, you must keep in mind the Format in which you enter your complex numbers. Keeping the mathematical rules for Precedence in check is very much advised.
How Does the Complex Number Division Calculator Work?
A Complex Number Division Calculator works by solving the denominator of a complex number division, and therefore solving the division altogether. The solution to a complex number in the denominator of said division is defined as the Transformation of this complex number into a real number.
Now, before we move on to understand Complex Number Divisions, let’s first understand Complex Numbers themselves.
Complex Number
A Complex Number is described as a combination of a real number and an imaginary number, linked to each other forming a whole new entity in the process. The Imaginary Part which contains the value $i$ referred to as “iota”. Where Iota has the following property:
\[i = \sqrt{-1}, i^2 = -1\]
Complex Number Division
Dividing Complex Numbers is indeed a complex process, whereas multiplication, subtraction, and addition are a bit more easily computed for them. This is because of the Imaginary Part in the complex number, as it is challenging to compute the behavior of such a number against traditional methods.
So, to cater to this problem, we intend to remove the Imaginary Part of the complex number in the denominator by using some mathematical operation. This Mathematical Operation includes identifying and multiplying a particular value that can, as mentioned above, rid the denominator of its imaginary part.
So, in general, to carry out Complex Number Division, we have to convert or transform the denominator of our division into a real number.
Complex Conjugate
The magical entity that we intend to use for transforming our complex number in the denominator of the division is also known as the Complex Conjugate of the denominator.
A Complex Conjugate of a complex number is referred to as the process of Rationalization for a said complex number. It is used to find the Amplitude of a function’s polar form, and in Quantum Mechanics it is used to find probabilities of physical events.
This Complex Conjugate of a complex number is thus calculated as follows.
Let there be a complex number of the form:
y = a + bi
The complex conjugate of this complex number can be found by inverting the sign of the coefficient associated with the imaginary part of this number. This means inverting the sign of the value corresponding to i.
It can be seen here:
y’ = (a + bi)’ = a – bi
Solve for Complex Number Division
So, we have come to learn above that to solve a Complex Number Division problem, we must first find the Complex Conjugate of the denominator term. This is therefore generally done as follows:
\[y = \frac{a + bi}{c + di}\]
y-denominator = c + di
y’-denominator = (c + di)’ = c – di
Once we have the Complex Conjugate of the denominator term, then we can simply multiply it to both the numerator and the denominator of our original fraction. This is done on the general division we have been using, as follows:
\[y = \frac{a + bi}{c + di} = \frac{a + bi}{c + di} \times \frac{c – di}{c – di}\]
And solving this leads to:
\[y = \frac{a + bi}{c + di} \times \frac{c – di}{c – di} = \frac{(a + bi)(c – di)}{c^2 + d^2}\]
Thus, finally, the denominator is free of Imaginary Terms and is completely real, as we initially intended it to be. This way, a Complex Number Division problem can be solved, and a computable solution is extracted from the fraction.
Solved Examples
Example 1
Now take a ratio of two complex numbers given as:
\[\frac{1 – 3i}{1 + 2i}\]
Solve this complex number division to get a resultant number.
Solution
We start by first taking the complex conjugate of the complex number in the denominator.
This is done as follows:
(1 + 2i)’ = 1 – 2i
Now that we have the complex conjugate of the denominator term, we move forward by multiplying this expression by both the numerator and the denominator of the original fraction.
We proceed here:
\[\frac{1 – 3i}{1 + 2i} = \frac{1 – 3i}{1 + 2i} \times \frac{1 – 2i}{1 – 2i} \]
\[\frac{1 – 3i}{1 + 2i} \times \frac{1 – 2i}{1 – 2i} = \frac{(1 – 3i)(1 – 2i)}{(1 + 2i)(1 – 2i)} = \frac{1 – 2i – 3i + (-3i)(-2i)}{1 – 2i + 2i + (-2i)(2i)} \]
\[\frac{1 – 2i – 3i + (-3i)(-2i)}{1 – 2i + 2i + (-2i)(2i)} = \frac{1 – 6 – 5i}{1 + 4} = \frac{-5}{5} – \frac{5i}{5} = -1 – i\]
And we have a result to our complex number division found as -1-i.
Example 2
Consider the given complex numbers’ ratio:
\[\frac{7 + 4i}{-3 – i}\]
Find the solution to this problem using the Complex Number Division.
Solution
We begin by first calculating the complex conjugate for the denominator term of this ratio. This is done as follows:
(-3 – i)’ = -3 + i
Now that we have the complex conjugate for the denominator complex number, we must move forward by multiplying and dividing the original fraction by this conjugate. This is carried forward below to calculate the solution to our problem:
\[\frac{7 + 4i}{-3 – i} = \frac{7 + 4i}{-3 – i} \times \frac{-3 + i}{-3 + i} \]
\[\frac{7 + 4i}{-3 – i} \times \frac{-3 + i}{-3 + i} = \frac{(7 + 4i)(-3 + i)}{(-3 – i)(-3 + i)} = \frac{-21 + 7i – 12i + (4i)(i)}{9 – 3i + 3i + (-i)(i)} \]
\[\frac{-21 + 7i – 12i + (4i)(i)}{9 – 3i + 3i + (-i)(i)} = \frac{-21 – 4 – 5i}{9 + 1} = \frac{-25}{10} – \frac{5i}{10} = -\frac{5}{2} – \frac{i}{2}\]
Hence, using the Complex Number Division, we were able to calculate the solution to our division problem. And the solution came out to be $-\frac{5}{2} – \frac{i}{2}$.
Example 3
Consider the given fraction of complex numbers:
\[\frac{-5 – 5i}{-5 + 5i}\]
Solve this division using the Complex Number Division method.
Solution
We start solving this problem by finding the denominator term’s complex conjugate. This is carried forth mathematically as follows:
(-5 + 5i)’ = -5 – 5i
Once we have acquired the complex conjugate of the denominator for this division, we move forward by multiplying the resulting conjugate to the original fraction’s numerator and denominator. Therefore, we solve to find the resulting complex number of this division here:
\[\frac{-5 – 5i}{-5 + 5i} = \frac{-5 – 5i}{-5 + 5i} \times \frac{-5 – 5i}{-5 – 5i} \]
\[\frac{-5 – 5i}{-5 + 5i} \times \frac{-5 – 5i}{-5 – 5i} = \frac{(-5 – 5i)(-5 – 5i)}{(-5 + 5i)(-5 – 5i)} = \frac{25 + 25i + 25i + (-5i)(-5i)}{25 + 25i – 25i + (+5i)(-5i)} \]
\[\frac{25 + 25i + 25i + (-5i)(-5i)}{25 + 25i – 25i + (+5i)(-5i)} = \frac{25 – 25 + 50i}{25 + 25} = \frac{50i}{50} = i\]
Finally, the Complex Number Division method delivers us a solution to the given fraction. The answer of which was found to be equal to the mathematical value known as Iota, i.
