4
3
5
25
8
0
0
-6
0.8
-0.6
4
3
5
25
8
0
0
-6
0.8
-0.6
The Complex Number Conjugate Calculator computes the complex conjugate of any complex number and displays key related properties including the magnitude and the product with its conjugate. The complex conjugate is one of the most important operations in complex analysis, formed by negating the imaginary part of a complex number while keeping the real part unchanged.
For a complex number $$z = a + bi$$, its complex conjugate is denoted $$\bar{z} = a - bi$$. Geometrically, the conjugate is the reflection of the complex number across the real axis in the complex plane. This reflection operation preserves the magnitude (distance from the origin) while reversing the sign of the argument (angle with the positive real axis).
The conjugate satisfies several fundamental algebraic properties that make it indispensable in mathematics. The product of a complex number with its conjugate yields a non-negative real number: $$z \cdot \bar{z} = a^2 + b^2 = |z|^2$$. This property is the foundation for complex division, as multiplying by the conjugate rationalizes complex denominators. The sum $$z + \bar{z} = 2a$$ extracts twice the real part, while $$z - \bar{z} = 2bi$$ extracts twice the imaginary part.
Complex conjugation distributes over all arithmetic operations: $$\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$$, $$\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}$$, and $$\overline{z_1 / z_2} = \bar{z_1} / \bar{z_2}$$. These properties make conjugation an automorphism of the complex number field — a structure-preserving operation that is its own inverse ($$\overline{\bar{z}} = z$$).
In quantum mechanics, the conjugate appears in the computation of probability amplitudes and inner products of state vectors. In signal processing, the conjugate is used in matched filtering and correlation operations. The Hermitian (conjugate) transpose of matrices generalizes the conjugate to linear algebra, forming the basis of unitary transformations and spectral theory. In polynomial theory, the Complex Conjugate Root Theorem states that if a polynomial with real coefficients has a complex root $$z$$, then $$\bar{z}$$ is also a root.
This calculator not only computes the conjugate but also displays the magnitude $$|z|$$, the squared magnitude $$|z|^2 = z \cdot \bar{z}$$, and the sum $$z + \bar{z}$$, providing a comprehensive view of the conjugate's relationship to the original number.
The complex conjugate is obtained by negating the imaginary part of the complex number.
Given a complex number:
$$z = a + bi$$
The conjugate is:
$$\bar{z} = a - bi$$
Key properties computed by this calculator:
Magnitude:
$$|z| = |\bar{z}| = \sqrt{a^2 + b^2}$$
Product with conjugate:
$$z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2 = |z|^2$$
Sum with conjugate:
$$z + \bar{z} = (a + bi) + (a - bi) = 2a = 2\,\text{Re}(z)$$
Difference with conjugate:
$$z - \bar{z} = (a + bi) - (a - bi) = 2bi = 2i\,\text{Im}(z)$$
The conjugate reflects the original number across the real axis. If the original number is in the upper half-plane (positive imaginary part), the conjugate lies in the lower half-plane, and vice versa. The magnitude is identical for both the original and conjugate. The product $$z \cdot \bar{z}$$ is always a non-negative real number equal to the square of the distance from the origin. A real number is its own conjugate, while a purely imaginary number's conjugate is its negative.
Inputs
Results
The conjugate of 4-3i is 4+3i. Magnitude = √(16+9) = 5. Product z×z̄ = 25. Sum = 8.
Inputs
Results
The conjugate of 7i is -7i. For purely imaginary numbers, the conjugate is the negative. Magnitude = 7, z×z̄ = 49.
The complex conjugate of $$z = a + bi$$ is $$\bar{z} = a - bi$$. It is formed by changing the sign of the imaginary part while keeping the real part the same. Geometrically, it reflects the number across the real axis in the complex plane. The conjugate operation is its own inverse: the conjugate of the conjugate returns the original number.
The conjugate is essential because multiplying a complex number by its conjugate produces a real number: $$z\bar{z} = |z|^2$$. This property enables complex division (rationalizing denominators), defines the modulus of complex numbers, and underpins Hermitian inner products in quantum mechanics and linear algebra. It also appears in the Schwarz reflection principle and analytic continuation.
A real number $$r$$ can be written as $$r + 0i$$, so its conjugate is $$r - 0i = r$$. Real numbers are their own conjugates. This means a complex number equals its conjugate if and only if it is real, providing a useful test: $$z = \bar{z} \iff z \in \mathbb{R}$$.
No, the conjugate has the same magnitude as the original: $$|\bar{z}| = |z|$$. Since $$|z| = \sqrt{a^2+b^2}$$ and $$|\bar{z}| = \sqrt{a^2+(-b)^2} = \sqrt{a^2+b^2}$$, both distances from the origin are identical. Only the angle changes: $$\arg(\bar{z}) = -\arg(z)$$.
The Complex Conjugate Root Theorem states that if a polynomial has all real coefficients and $$z = a+bi$$ is a root, then $$\bar{z} = a-bi$$ is also a root. This means complex roots of real polynomials always come in conjugate pairs. For example, if $$2+3i$$ is a root, then $$2-3i$$ must also be a root.
To divide $$z_1/z_2$$, we multiply numerator and denominator by $$\bar{z_2}$$: $$\frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{z_2 \bar{z_2}} = \frac{z_1 \bar{z_2}}{|z_2|^2}$$. Since the denominator becomes a real number, the result is a standard complex number. This rationalization technique is the standard method for complex division.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Complex Number Calculator
Complex Number Calculators
Complex Number Addition Calculator
Complex Number Calculators
Complex Number Multiplication Calculator
Complex Number Calculators
Complex Number Division Calculator
Complex Number Calculators
Complex Number Modulus Calculator
Complex Number Calculators
Complex Number Polar Form Calculator
Complex Number Calculators
