Product of complex conjugates calculator
(3 + 4i) times its conjugate (3 − 4i) is 25: enter the real and imaginary parts and get a² + b², the squared modulus of the number, always real and never negative. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Real
- 3
- Imag
- 4
- Conjugate real
- 3
- Conjugate imag
- -4
- Product
- 25
- Modulus squared
- 25
Every complex number z written as a plus bi has a conjugate z bar formed by flipping the sign of the imaginary part. The product of a complex number and its conjugate is always a real, nonnegative scalar: (a plus bi) times (a minus bi) expands to a squared plus b squared, which is exactly the squared modulus of z. That identity underpins normalization of phasors, energy calculations in circuits, magnitude checks in signal processing, and every algebra course that introduces |z| before polar form. This product of complex conjugates calculator accepts the two finite real coefficients of a single rectangular operand, multiplies z by z bar with stable rounding, and returns the product, the echoed conjugate parts, rectangular display strings, and a formula string you can show next to the answer.
Why z times conjugate equals the squared modulus
A complex number in rectangular form is a pair of real numbers written a plus bi, where a is the real part, b is the imaginary part, and i denotes the imaginary unit with i squared equal to negative one. The complex conjugate of z is obtained by reflecting across the real axis: z bar equals a minus bi. Algebraically the product expands with the difference of squares identity: (a plus bi) times (a minus bi) equals a squared minus (bi) squared.
Because i squared is negative one, minus (bi) squared becomes plus b squared, so the whole product collapses to a squared plus b squared with no leftover imaginary part. Geometrically that quantity is the squared Euclidean length of the vector from the origin to the point (a, b) in the Argand plane, which is why textbooks write z times z bar equals |z| squared. The result is always real and always greater than or equal to zero; it is zero only when both a and b are zero.
The formula field always restates the identity so tutorials can place the rule beside the numeric answer without maintaining a separate content table.
What you can do with it
Classroom and homework checks
Let students submit real and imaginary parts of z and verify that z times conjugate equals a squared plus b squared with no residual imaginary part.
Exam and LMS auto-grading
Mark conjugate-product and modulus-squared items with a deterministic oracle that rejects non-finite coefficients and never invents polar intermediate steps.
Phasor power and magnitude scripts
Compute |z| squared for circuit power, energy density, or magnitude-squared filters in automation without importing a full computer-algebra library.
FAQ
How do I compute the product of a complex number and its conjugate?
Send two finite numbers: real (a) and imag (b) for z = a+bi.
Is the result always real and nonnegative?
Yes. The product z·z̄ equals a²+b², which is always a real number greater than or equal to zero. It is zero only when both real and imag are zero.
Does this return the modulus |z| or only |z| squared?
Take a square root yourself if you need the unsquared Euclidean length |z|.
What happens if a coefficient is missing or not a number?
Invalid input is not billed and does not return a partial product.
What is the product of a complex number and its conjugate used for?
It gives the squared magnitude without square roots. It appears when normalizing phasors, computing power or energy in circuits and checking magnitude-squared values in signal processing, as well as in algebra exercises before polar form.
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-conjugate-product \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"real":3,"imag":4}'const res = await fetch("https://api.kit.forhosting.com/algebra/complex-conjugate-product", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"real": 3,
"imag": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/complex-conjugate-product",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"real": 3,
"imag": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/complex-conjugate-product", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"real":3,"imag":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"real":3,"imag":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/complex-conjugate-product", 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,
"imag": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.complex_conjugate_product",
"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
Missing or non-finite coefficients fail with invalid_input and are not billed.
Why z times conjugate equals the squared modulus
This endpoint does not convert to polar form, does not return the unsquared modulus unless you take a square root yourself, and does not multiply two unrelated complex numbers—only a number by its own conjugate.
Input contract, response shape, and validation
Pure real numbers omit the imaginary unit, pure imaginary numbers omit a zero real part when appropriate, unit coefficients render as i or minus i rather than one i, and mixed signs use a single plus or minus between the real and imaginary pieces. Pair the response with your own Argand-plane plot if you need a diagram; this capability computes the conjugate product only. POST a JSON body with two required finite numbers: real for the real part a of z, and imag for the imaginary part b. Each field may arrive as a JSON number or as a numeric string; empty strings, missing keys, Infinity, NaN, and non-numeric text fail with invalid_input naming the field, and those failures do not consume prepaid balance. Optional aliases such as a, re, real_part, a_real for the real part and b, im, imag_part, a_imag, imaginary for the imaginary part map to the same canonical names so generated clients and hand-written scripts stay flexible. On success the body echoes the rounded input coefficients as real and imag, returns conjugate_real and conjugate_imag (equal to a and negative b), product and modulus_squared as the same nonnegative a squared plus b squared value rounded to twelve decimal places for stable goldens across runtimes, and supplies display strings z and conjugate in standard rectangular form plus product_display for the scalar result. Example: real three and imag four yield product twenty-five, z string three plus four i, conjugate three minus four i, and product_display twenty-five.
Where a conjugate-product API earns its keep
Classrooms and tutoring platforms use a product of complex conjugates calculator so learners can check that z times z bar is real and equals a squared plus b squared without sign errors when the imaginary part is negative. Engineering notebooks in circuits and signal processing often need |z| squared for power, energy density, or magnitude-squared filters before taking a square root for the modulus itself; calling a tiny API avoids pulling a full computer-algebra system for two multiplications and an addition. Because the implementation is pure parse, every successful call costs the published $0.002 base price.
Technical questions
What does the product of complex conjugates calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.