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Divide Complex Numbers Calculator

Created by Anna Szczepanek, PhD
Reviewed by Komal Rafay
Last updated: Jan 18, 2024

Omni's divide complex numbers calculator is here to help when you need to quickly work out the quotient of two complex numbers, be they in the rectangular or in the polar form! The calculator displays the answer in both forms as well, so that you can pick the one which is more convenient for you.

In what follows, we explain how to divide complex numbers and provide formulas in case you need to compute it by hand. In particular, we'll see how nice it is to divide complex numbers in polar form (in contrast to the rectangular form).

Dividing complex numbers in rectangular form

To divide two complex numbers in rectangular form (a+ib form), use the formula:

z1/z2=a+ibc+id=ac+bd+i(bcad)c²+d²\small \begin{align*} z_1 / z_2 & = \frac{a + \mathrm{i}b}{c + \mathrm{i}d} \\[0.75em] & = \frac{a c + b d + \mathrm{i}(b c - a d) } {c² + d²} \end{align*}

Of course, the division is possible only if z20z_2 \neq 0, which is equivalent to z22=c2+d20|z_2|^2 = c^2+d^2 \neq 0.


As you can see, this formula is quite complicated. Fortunately, you don't need to learn it by heart. It is much easier to derive it! There's only one trick you need to remember: start by multiplying both the numerator and denominator by the conjugate of the latter:

z1/z2=a+ibc+id=(a+ib)(cid)(c+id)(cid)=aciad+ibci2bdc²(id)²=ac+bd+i(bcad)c²+d²\small \begin{align*} z_1 / z_2 & = \frac{a + \mathrm{i}b}{c + \mathrm{i}d} \\[0.75em]& = \frac{(a + \mathrm{i}b) \cdot (c - \mathrm{i}d)}{(c + \mathrm{i}d) \cdot (c - \mathrm{i}d)} \\[0.75em]& = \frac{a c - \mathrm{i}a d + \mathrm{i} b c - \mathrm{i}^2 b d }{c² - (\mathrm{i}d)²} \\[0.75em]& = \frac{a c + b d + \mathrm{i}(b c - a d) } {c² + d²} \end{align*}

Dividing complex numbers in polar form

Division of complex numbers in much easier in the polar form (r×exp(iφ) form):

z1z2=z1exp(iφ1)z2exp(iφ2)=z1z2exp(iφ1)exp(iφ2)=z1z2exp[i(φ1φ2))]\small \begin{align*} \frac{z_1}{z_2} &= \frac{|z_1|\exp(\mathrm{i}\varphi_1)} {|z_2|\exp(\mathrm{i}\varphi_2)} \\ &= \left|\frac{z_1}{z_2}\right| \exp(\mathrm{i}\varphi_1) \exp( - \mathrm{i}\varphi_2) \\ &= \left|\frac{z_1}{z_2}\right| \exp[\mathrm{i}(\varphi_1 - \varphi_2))] \end{align*}

We can see that:

  • z1/z2=z1/z2|z_1 / z_2| = |z_1| / |z_2|; and
  • arg(z1/z2)=φ1φ2\arg( z_1/ z_2) = \varphi_1 - \varphi_2.

How to use this dividing complex number calculator?

To most efficiently use our divide complex numbers calculator, follow these steps:

  1. Enter the numbers z1z_1 and z2z_2 in the respective boxes. The calculator performs the division z1/z2z_1/z_2.
  2. For each number you can choose if you want to input it in the rectangular form or in the polar form:
    • For the rectangular form, enter the real and imaginary part.
    • For the polar form, enter the magnitude and phase.
  3. For the second complex number you can use the same form as for the first number or the other form - our divide complex numbers calculator will know what to do!
  4. The calculator performs the division and returns the quotient in both the rectangular and polar form.
  5. Click the advanced mode button below the calculator if you want to adjust the precision of computations (number of decimal places). The default is three and you can go up to ten.

Omni's calculators for complex numbers


How do I divide complex numbers in polar form?

To divide r×exp(iφ) by s×exp(iψ)), follow these steps:

  1. Divide the two magnitudes: r / s. This will be the magnitude of the result.
  2. Subtract the phase of the divisor from that of the dividend: φ - ψ . This will be the phase of the result.
  3. Write down the result as (r/s) × exp(i(φ - ψ)).
  4. That's it! Dividing complex numbers in polar form is sooo nice, isn't it? :)

How do I divide complex numbers in rectangular form?

To find the quotient (a+ib)/(c+id):

  1. Determine the magnitude of the divisor: c² + d².
  2. Compute ac + bd.
  3. Divide it by the number from Step 1.
  4. Compute bc - ad.
  5. Divide it by the number from Step 1.
  6. The real part of the result is the number obtained in Step 3.
  7. The imaginary part of the result is the number obtained in Step 5.

Can I divide by i?

Yes, you can divide by every non-zero complex number. In fact, dividing by i is equivalent to multiplying by -i. That is, we have 1/i = -i. To verify that this claim is true, it suffices to observe that i × (-i) = 1.

What is 1 divided by i?

The answer is -i. To get this result, expand both the numerator and denominator by the complex conjugate of i, which is equal to -i. We obtain 1 / i = 1 × (-i) / i × (-i) = -i / -i².
Now we recall that i² = -1, and so our computation continues as follows: -i / -i² = -i / -(-1) = -i, as claimed.

Anna Szczepanek, PhD
Related calculators
First number z₁:
a + bi
Real part (a₁)
Imaginary part (b₁)
Second number z₂:
a + bi
Real part (a₂)
Imaginary part (b₂)
Quotient z₁/z₂:
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