Compute the conjugate and modulus of any complex number z = a + bi step-by-step.
z = a + bi
This calculator helps you compute the complex conjugate and absolute value (modulus) of any complex number z = a + bi instantly. Simply enter the real part a and the imaginary part b, and the calculator displays step-by-step results.
Provide the two components of your complex number:
The calculator instantly updates the results as you type.
The calculator computes:
Results are displayed clearly, with the sign of the imaginary part adjusted automatically.
Ensures reliable calculations:
Ideal for learning and practice:
1. Enter the real part (a) and imaginary part (b) of your complex number.
2. The calculator computes the complex conjugate automatically using: z̄ = a − bi.
3. The absolute value (modulus) is calculated with: |z| = √(a² + b²).
4. Results are displayed instantly in the result section.
5. If either input is missing or invalid, an error message is shown and results are hidden.
This real-time functionality helps students, beginners, and learners explore complex numbers easily.
A complex number is expressed as z = a + bi, where a is the real part and b is the imaginary part. This calculator computes two important properties of any complex number: the complex conjugate and the absolute value (modulus). Understanding these formulas is essential for algebra, engineering, and advanced mathematics.
Complex Conjugate: z̄ = a − bi
The complex conjugate flips the sign of the imaginary part while keeping the real part unchanged. This is used in dividing complex numbers, simplifying expressions, and in many electrical engineering calculations.
Absolute Value (Modulus): |z| = √(a² + b²)
The absolute value or modulus represents the distance of the complex number from the origin on the complex plane. It is calculated using the Pythagorean theorem, combining the squares of the real and imaginary parts, then taking the square root. This is widely used in signal processing, physics, and mathematics for measuring magnitude.
By using these formulas, you can perform operations such as addition, subtraction, division, and finding magnitudes of complex numbers accurately. They form the foundation of complex number arithmetic and are essential for students, engineers, and scientists.
Suppose you have a complex number:
z = 3 + 4i
The calculator computes:
Conjugate: z̄ = 3 − 4i
Absolute Value: |z| = 5
All results are displayed automatically as you enter any new values for a or b.
Everything you need to know about computing complex conjugates and absolute values accurately and instantly.
This calculator computes the complex conjugate (z̄ = a − bi) and the absolute value |z| = √(a² + b²) of any complex number z = a + bi, showing results instantly and step-by-step.
Yes! It is completely free with no registration, ads, or hidden charges.
You only need to enter the real part (a) and the imaginary part (b) of the complex number. The calculator will automatically compute the conjugate and absolute value.
Yes, both negative and zero values are supported for real and imaginary parts.
If either input is missing or not a number, the calculator will not display a result and may show an error message until valid numbers are entered.
Yes. Use the “Clear All” button to reset the input fields or the “Reload Calculator” button to reload the page completely.
The conjugate is displayed as z̄ = a − bi, and the absolute value is shown as |z| = √(a² + b²), rounded to two decimal places if necessary.
Yes, the calculator is fully responsive and works smoothly on smartphones, tablets, laptops, and desktops.
Students, teachers, or anyone learning complex numbers can use this tool to compute conjugates, absolute values, and better understand complex number operations.
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