Divide complex numbers
(3 + 4i) divided by (1 + 2i) is 2.2 − 0.4i. Enter the real and imaginary parts of both numbers and get the quotient's two parts, worked out through the conjugate of the divisor; a zero divisor is refused. Free in your browser.
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Worked example
- Real1
- 3
- Imag1
- 4
- Real2
- 1
- Imag2
- 2
- Complex1
- 3+4i
- Complex2
- 1+2i
- Quotient real
- 2.2
- Quotient imag
- -0.4
Dividing one complex number by another is the rectangular-form operation (a plus b i) divided by (c plus d i). The algebraic method multiplies the dividend by the conjugate of the divisor and divides by the squared modulus of the divisor, so the quotient's real part is (a c plus b d) over (c squared plus d squared) and the imaginary part is (b c minus a d) over (c squared plus d squared). Students use this identity when simplifying fractions of complex numbers, converting polar-form division back to rectangular parts, or checking homework without a computer-algebra system. Engineers apply the same identity to impedance ratios, transfer functions, and phasor quotients. This divide-complex-numbers calculator accepts the real and imaginary parts of both numbers, rejects a zero divisor, and returns the quotient's real and imaginary parts together with compact display strings.
What dividing complex numbers means algebraically and geometrically
A complex number lives in the plane with its real part on the horizontal axis and its imaginary part on the vertical axis. Dividing z1 by z2 is multiplication by the reciprocal of z2: geometrically you scale lengths by one over the modulus of z2 and rotate by the negative of the argument of z2. In rectangular coordinates you never need that polar conversion.
Write the dividend as a plus b i and the divisor as c plus d i. The reciprocal of the divisor is the conjugate c minus d i divided by c squared plus d squared, because a number times its conjugate equals the squared modulus. Multiplying the dividend by that reciprocal produces the single fraction whose numerator is (a c plus b d) plus (b c minus a d) i and whose denominator is c squared plus d squared.
The real part of the quotient is therefore (a c plus b d) divided by (c squared plus d squared) and the imaginary part is (b c minus a d) divided by (c squared plus d squared). Division is undefined when the divisor is the zero complex number, that is when both c and d are zero, because the squared modulus in the denominator vanishes. When the divisor is a pure real number the formula reduces to ordinary division of each part by c.
When the divisor is pure imaginary the quotient is a ninety-degree rotation scaled by one over the absolute value of d. Classic check: three plus four i divided by one plus two i equals 2.2 minus 0.4 i. Another check: one plus i divided by one minus i equals i.
Where dividing complex numbers comes up
Classrooms and tutoring platforms use a divide-complex-numbers calculator so learners can verify (a plus b i) over (c plus d i) without sign mistakes when expanding the conjugate by hand. Exam generators and LMS plugins need a deterministic oracle that rejects a zero divisor, returns stable rounded parts, and never invents a quotient when the denominator vanishes. Electrical engineering notebooks reduce impedance ratios, voltage-over-current phasors, and transfer-function samples to a single rectangular division when a full computer-algebra system would be overkill.
Signal-processing and control scripts apply the same identity to complex gains and residuals without standing up heavy numeric libraries on the edge. Failed validation, including division by zero, does not consume balance.
What you can do with it
Classroom and homework checks
Let students submit real and imag parts for (a+bi)/(c+di) and verify the quotient against the conjugate method from the textbook.
Engineering notebooks and phasors
Compute impedance ratios or voltage-over-current phasors in rectangular form without pulling a full computer-algebra library.
FAQ
How do I divide two complex numbers in rectangular form?
Send finite numbers real1 and imag1 for the dividend and real2 and imag2 for the divisor.
What happens if the divisor is zero?
Division by the zero complex number 0+0i is undefined, and failed validation is not billed.
Can I send textbook names a, b, c, and d instead of real1 and imag1?
Canonical names in the response remain real1, imag1, real2, and imag2.
Does this calculator return polar form (modulus and argument)?
No. This capability returns the rectangular quotient only. Use a complex modulus or argument tool if you need polar coordinates of the result.
What does dividing by a complex number do geometrically?
It scales lengths by one over the modulus of the divisor and rotates by minus its argument. You do not need that polar view to get the answer, though: the rectangular formula gives both parts directly.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/divide-complex-numbers \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"real1":3,"imag1":4,"real2":1,"imag2":2}'const res = await fetch("https://api.kit.forhosting.com/math/divide-complex-numbers", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"real1": 3,
"imag1": 4,
"real2": 1,
"imag2": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/divide-complex-numbers",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"real1": 3,
"imag1": 4,
"real2": 1,
"imag2": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/divide-complex-numbers", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"real1":3,"imag1":4,"real2":1,"imag2":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"real1":3,"imag1":4,"real2":1,"imag2":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/divide-complex-numbers", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"real1": 3,
"imag1": 4,
"real2": 1,
"imag2": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.divide_complex_numbers",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
What dividing complex numbers means algebraically and geometrically
This endpoint returns those rectangular parts and does not emit polar magnitude or argument; pair it with a modulus or argument calculator if you need the polar pair after the quotient is known.
Input contract, response shape, and validation behaviour
Classic check: real1 equals three, imag1 equals four, real2 equals one, imag2 equals two yields quotient real 2.2 and quotient imag negative 0.4 with divisor squared modulus five. Use the two quotient fields as the primary answer; the display string is a convenience for worksheets and logs. POST a JSON body with four required finite numbers: real1 and imag1 for the dividend z1, and real2 and imag2 for the divisor z2. Non-finite values, missing fields, empty strings, and non-numeric text fail with invalid_input naming the field, and those failures are not billed. A divisor whose real and imaginary parts are both zero also fails with invalid_input, because division by the zero complex number is undefined. On success the body echoes real1, imag1, real2, and imag2 with stable twelve-decimal rounding; compact complex1 and complex2 display strings such as three plus four i or one minus two i; quotient_real and quotient_imag for the rectangular parts of z1 divided by z2; a quotient display string for that result; divisor_squared_modulus equal to real2 squared plus imag2 squared, which is the denominator of the conjugate formula; and a formula string restating (a plus b i) over (c plus d i) equals ((a c plus b d) plus (b c minus a d) i) over (c squared plus d squared).
Where a complex-division API earns its keep
Keep requests small, pin the rounded quotient_real and quotient_imag in tests, and you get reproducible rectangular quotients for homework, LMS plugins, and edge workers without importing symbolic math infrastructure of your own.
Integration use cases
Exam and LMS auto-grading
Mark complex-division items with a deterministic oracle that returns quotient_real and quotient_imag and rejects a zero divisor.
Technical questions
What does the divide complex numbers calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input, including a zero divisor, is not billed.