Complex Number Division Calculator + Online Solver With Free Steps (2024)

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.

Complex Number Division Calculator + Online Solver With Free Steps (1)

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.


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.


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.


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.

Stoichiometry Calculator < Math Calculators List > L’Hopital’S Rule Calculator

Complex Number Division Calculator + Online Solver With Free Steps (2024)


How to calculate complex number division? ›

To divide a complex number a+ib by c+id, multiply the numerator and denominator of the fraction a+ib/c+id by c−id and simplify.

How to compute complex numbers? ›

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a+bi where a is the real part and bi is the imaginary part. For example, 5+2i is a complex number. So, too, is 3+4√3i.

How do you multiply complex numbers? ›

What is Multiplication Of Complex Numbers Formula? The formula for multiplying complex numbers is: (a + ib) (c + id) = (ac - bd) + i(ad + bc).

What is the basic formula of a complex number? ›

The complex number is basically the combination of a real number and an imaginary number. The complex number is in the form of a+ib, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1.

What is the formula for division problems? ›

Division Math Formula

The formula to calculate the division of two numbers is: Dividend ÷ Divisor = Quotient + Remainder. The remainder is the leftover number in the division operation. If 46 is divided by 5, we get the quotient as 9 and the remainder 1.

How do you solve complex numbers easily? ›

Add or subtract the real parts and then the imaginary parts. Example 2: Add: ( 3 − 4 i ) + ( 2 + 5 i ) . Solution: Add the real parts and then add the imaginary parts. To subtract complex numbers, subtract the real parts and subtract the imaginary parts.

What is the world's most complex calculator? ›

What Is the Most Advanced Calculator in the World?
  • Texas Instruments TI-Nspire CX II CAS. ...
  • Wolfram Alpha. ...
  • GeoGebra. ...
  • Maple. ...
  • Casio ClassPad II (fx-CP400) ...
  • MATLAB. ...
  • HP Prime. ...
  • Symbolab. An online calculator that provides step-by-step solutions for algebra, calculus, and other math problems.

What is the most complex math calculation? ›

It's called a Diophantine Equation, and it's sometimes known as the “summing of three cubes”: Find x, y, and z such that x³+y³+z³=k, for each k from one to 100.

What is a complex number for dummies? ›

A complex number has a term with a multiple of i, and i is the imaginary number equal to the square root of –1. Many of the algebraic rules that apply to real numbers also apply to complex numbers, but you have to be careful because many rules are different for these numbers.

What are the rules for calculating with complex numbers? ›

Equality of Complex Number Formula
  • Addition of Complex Numbers: (a+bi)+(c+di) = (a+c) + (b+d)i.
  • Subtraction of Complex Numbers: (a+bi)−(c+di) = (a−c) + (b−d)i.
  • Multiplication of Complex Numbers: (a+bi)×(c+di) = (ac−bd) + (ad+bc)i.
  • Multiplication Conjugates: (a+bi) × (a+bi) = a2+b2

Why is 17 a complex number? ›

There is no imaginary part. In other words, the imaginary part is 0. We can think of 17 as 17 + 0i. In fact all real numbers can be thought of as complex numbers which have zero imaginary part.

How to do division of complex numbers? ›

Formula for Dividing Complex Numbers

The quotient a + ib/c + id represents the division of two complex numbers z1 = a + ib and z2 = c + id. This is determined using the z1/z2 = ac + bd/c2 + d2 + i (bc – ad / c2 + d2) formula for division of complex numbers.

What are the four powers of i? ›

The powers of i is always equal to either one of these 4 numbers: 1, i , -1,-i.

How do you solve complex number operations? ›

To add two complex numbers , add the real part to the real part and the imaginary part to the imaginary part. To subtract two complex numbers, subtract the real part from the real part and the imaginary part from the imaginary part. To multiply two complex numbers, use the FOIL method and combine like terms .

How to find the quotient of complex numbers? ›

Correct answer:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator. To find the conjugate, just change the sign in the denominator.

How do you solve complex division fractions? ›

Remember the main fraction bar means "divide".

Rewrite the fraction using the ÷ symbol. Follow the normal rules for dividing fractions: invert the second term (the denominator of the complex fraction) and multiply by the numerator of the complex fraction. Simplify whenever possible.

How to divide two complex numbers in polar form? ›

We can divide two complex numbers in polar form by dividing their moduli and subtracting their arguments.


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