Conjugate Calculator

About the Conjugate Calculator

The conjugate of a number is formed by changing the sign of one component while keeping the rest unchanged. For a complex number z = a + bi, the complex conjugate is z̄ = a − bi — the imaginary part flips sign. For a radical expression like a + √b, the radical conjugate is a − √b. Conjugates are indispensable in algebra and analysis: multiplying a complex number by its conjugate gives the real number |z|² = a² + b², and multiplying a radical expression by its conjugate eliminates the radical via the difference-of-squares identity. This calculator handles both types of conjugates, computing the product, sum, modulus, argument, and polar form. For complex conjugates, an interactive Argand diagram shows z and z̄ reflected across the real axis. For radical conjugates, it demonstrates rationalization — the technique used to clear radicals from denominators. Students encounter conjugates when dividing complex numbers, rationalizing denominators, and solving polynomial equations. Engineers use complex conjugates in signal processing, electrical impedance, and Fourier analysis, where the conjugate represents the mirror-image frequency component.

When This Page Helps

Conjugates are used in complex number division, denominator rationalization, and signal processing. This calculator handles both complex and radical conjugates in one tool, showing the conjugate, its product with the original (always real), the modulus, argument, and polar form. The Argand diagram visualization makes the reflection symmetry immediately clear, and radical mode demonstrates how conjugates eliminate radicals from denominators — essential for simplifying algebraic expressions.

How to Use the Inputs

1. Select the mode: Complex Conjugate (a + bi) or Radical Conjugate (a + c√b).
2. For complex mode, enter the Real Part and Imaginary Part of your complex number.
3. For radical mode, enter a (rational part), b (radicand), and optionally the coefficient c.
4. Click a preset like "3 + 4i" or "1 + √2" to load a common example.
5. Review the conjugate, modulus, argument, and the product z·z̄ in the output cards.
6. Examine the Argand diagram showing z and z̄ reflected across the real axis.
7. Adjust Precision to control decimal places in all outputs.

Example Calculation

Tips & Best Practices

• Check that all inputs use the same scale and assumptions before trusting the result.
• Compare the answer with the worked example or a rough estimate to catch entry mistakes.

Complex Conjugates and Division

To divide complex numbers, multiply both numerator and denominator by the conjugate of the denominator: (a + bi)/(c + di) = (a + bi)(c − di)/((c + di)(c − di)) = (a + bi)(c − di)/(c² + d²). The denominator becomes real because z·z̄ = |z|² always eliminates the imaginary part. This technique is the complex-number analog of rationalizing a denominator.

Radical Conjugates and Rationalization

For expressions with square roots, multiplying by the conjugate uses the difference of squares: (a + √b)(a − √b) = a² − b. This eliminates the radical, which is why we rationalize denominators by multiplying by the conjugate. For example, 1/(3 + √2) = (3 − √2)/((3 + √2)(3 − √2)) = (3 − √2)/7. This technique extends to cube roots and higher using sum/difference of cubes identities.

Conjugates in Physics and Engineering

In electrical engineering, the impedance of a circuit element is a complex number Z = R + jX. The complex conjugate Z* is used to compute average power: P = ½ Re(V·I*). In quantum mechanics, the probability of a state is |ψ|² = ψ·ψ*, the product of the wave function and its conjugate. In signal processing, the Fourier transform of a real signal has conjugate symmetry: F(−ω) = F(ω)*, which halves the computational work.

Frequently Asked Questions

• What is a complex conjugate?

  For z = a + bi, the complex conjugate is z̄ = a − bi. It reflects the number across the real axis in the complex plane.

• Why is z · z̄ always real?

  Because z · z̄ = (a + bi)(a − bi) = a² + b², which has no imaginary part. This product equals the squared modulus.

• What is a radical conjugate?

  For a + √b, the radical conjugate is a − √b. Multiplying the two eliminates the radical: (a + √b)(a − √b) = a² − b.

• How do you rationalize a denominator?

  Multiply both numerator and denominator by the conjugate of the denominator. This clears the radical or imaginary part from the denominator.

• What is the Argand diagram?

  A 2D plot where the horizontal axis represents the real part and the vertical axis the imaginary part. Each complex number is a point on this plane.

• Does every polynomial root have a conjugate root?

  For polynomials with real coefficients, complex roots always come in conjugate pairs. This is the Complex Conjugate Root Theorem.