Complex Reciprocal Calculator
The reciprocal of 3 + 4i is 0.12 − 0.16i, because its squared modulus is 25: enter the real and imaginary parts and get 1/z in rectangular form, ready for complex division. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Real
- 3
- Imaginary
- 4
- Real
- 0.12
- Imaginary
- -0.16
- Modulus squared
- 25
This complex reciprocal calculator finds the multiplicative inverse of a complex number written as a real part and an imaginary part. Enter the two components of a + bi, and it returns the real and imaginary components of 1/(a + bi), along with the squared modulus used as the denominator. The calculation is deterministic, keeps the result in rectangular form, and rejects zero because zero has no reciprocal. It is useful for checking algebra, preparing complex division, and building repeatable calculations into software.
Turn a complex number into its reciprocal
A reciprocal is the number that produces one when multiplied by the original value. For a real number, finding it means dividing one by that number. A complex denominator needs one extra algebraic step because it contains both a real and an imaginary component.
Supply the real part and imaginary part separately; for example, 3 + 4i is entered as real 3 and imaginary 4. The calculator returns the reciprocal as the same two-part structure, making the answer easy to use in code or further calculations. It also returns the squared modulus, a² + b², so you can inspect the denominator used in the operation.
No formatted expression parser is involved, which avoids ambiguity around signs, spacing, or the symbol used for the imaginary unit. Both inputs must be finite numbers. The single excluded complex value is 0 + 0i, whose reciprocal is undefined because no number multiplied by zero can equal one.
Why the conjugate makes division possible
For z = a + bi, the conjugate is a - bi. Multiplying z by its conjugate cancels the imaginary cross terms and gives a² + b², which is the squared modulus and is always positive for a nonzero complex number. Therefore 1/z equals (a - bi)/(a² + b²).
The returned real component is a divided by that squared modulus, while the returned imaginary component is negative b divided by it. Consider 3 + 4i: its squared modulus is 9 + 16 = 25, so the reciprocal is 3/25 - 4/25i, or 0.12 - 0.16i. Multiplying those two complex numbers confirms the result: the real component becomes one and the imaginary component becomes zero.
Showing the squared modulus alongside the answer makes the transformation auditable and helps students connect the geometric magnitude of a complex number with the algebraic technique used to remove an imaginary denominator.
Use the result accurately in later work
The output is designed for direct reuse rather than display-only mathematics. A reciprocal is especially helpful when dividing complex numbers: multiplying a numerator by the reciprocal of its denominator gives the quotient. It also appears in impedance and admittance relationships, transfer functions, signal analysis, iterative numerical methods, and textbook exercises involving rational expressions over complex values.
For routine values, the operation uses a fixed sequence of additions, multiplications, sign changes, and divisions, with no network calls, random choices, hidden rounding option, or dependence on the current time.
What you can do with it
Check complex-number homework
Verify the conjugate-over-squared-modulus method and compare the returned components with a hand calculation.
Prepare complex division
Compute the inverse of a nonzero denominator before multiplying it by another complex number.
Automate engineering formulas
Reuse numeric real and imaginary fields in impedance, admittance, signal, or control calculations.
FAQ
What does the calculator return?
It returns the real and imaginary components of the reciprocal and the squared modulus used as their denominator.
What formula does it use?
For a + bi, it computes (a - bi)/(a² + b²), so the components are a/(a² + b²) and -b/(a² + b²).
Why is zero rejected?
Zero has no multiplicative inverse: multiplying any finite complex number by zero still gives zero, never one.
Can I enter a formatted value such as 3+4i?
No. Enter 3 in the real field and 4 in the imaginary field so the input remains unambiguous and machine-readable.
How does the reciprocal help divide complex numbers?
Dividing by a complex number is the same as multiplying by its reciprocal. Find 1/z for the denominator here, then multiply the numerator by it to get the quotient without handling an imaginary denominator.
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/algebra/complex-reciprocal \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"real":3,"imaginary":4}'const res = await fetch("https://api.kit.forhosting.com/algebra/complex-reciprocal", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"real": 3,
"imaginary": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/complex-reciprocal",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"real": 3,
"imaginary": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/complex-reciprocal", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"real":3,"imaginary":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"real":3,"imaginary":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/complex-reciprocal", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"real": 3,
"imaginary": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.complex_reciprocal",
"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
Use the result accurately in later work
Its real and imaginary fields can feed another calculation without parsing a string, while modulus_squared records the common denominator. Results use ordinary finite JavaScript numbers, so they follow IEEE-754 floating-point precision. Very large components can make a² + b² overflow, and extremely tiny nonzero components can make that value underflow to zero; these cases are rejected instead of returning infinity or a misleading result that JSON cannot represent faithfully.
Technical questions
How much does the API request cost?
Each API request costs $0.002. The browser calculator is free to use on this page.