5
25
0.927295
rad
—
deg
5
5
25
0.927295
rad
—
deg
5
The Complex Number Modulus Calculator computes the absolute value (modulus) of a complex number, which represents its distance from the origin in the complex plane. The modulus is one of the two fundamental components of the polar representation of complex numbers, along with the argument (angle), both of which this calculator provides.
For a complex number $$z = a + bi$$, the modulus is defined as $$|z| = \sqrt{a^2 + b^2}$$. This formula is a direct application of the Pythagorean theorem: the real part $$a$$ and imaginary part $$b$$ form the legs of a right triangle, and the modulus is the hypotenuse. The modulus is always a non-negative real number, with $$|z| = 0$$ if and only if $$z = 0$$.
The modulus satisfies several important properties that make it a norm on the complex numbers. The triangle inequality $$|z_1 + z_2| \leq |z_1| + |z_2|$$ states that the distance of a sum is at most the sum of the distances. The multiplicative property $$|z_1 z_2| = |z_1| |z_2|$$ means that multiplication scales distances multiplicatively. The quotient property $$|z_1/z_2| = |z_1|/|z_2|$$ shows that division scales distances by the reciprocal. These properties establish the modulus as a multiplicative norm, making the complex numbers a normed algebra.
In physics and engineering, the modulus corresponds to the amplitude or magnitude of a signal. In AC circuit analysis, the modulus of an impedance gives the impedance magnitude, which determines the ratio of voltage to current amplitudes. In quantum mechanics, the modulus squared $$|z|^2$$ gives the probability associated with a quantum amplitude. The modulus is also central to the definition of convergence in complex analysis: a sequence of complex numbers converges if and only if the modulus of the differences converges to zero.
This calculator also computes the argument (angle) of the complex number in both radians and degrees using the two-argument arctangent function $$\text{atan2}(b, a)$$, which correctly handles all four quadrants. Together, the modulus and argument fully determine the complex number in polar form: $$z = |z| e^{i\arg(z)}$$.
The squared modulus $$|z|^2 = a^2 + b^2$$ is displayed separately because it appears frequently in applications — it equals the product $$z \cdot \bar{z}$$ and avoids the computational cost of a square root. In probability and signal processing, power quantities are proportional to the squared modulus rather than the modulus itself.
The modulus of a complex number is computed using the Pythagorean theorem in the complex plane.
Given:
$$z = a + bi$$
Modulus (absolute value):
$$|z| = \sqrt{a^2 + b^2}$$
Squared modulus:
$$|z|^2 = a^2 + b^2 = z \cdot \bar{z}$$
Argument (angle with positive real axis):
$$\arg(z) = \text{atan2}(b, a)$$
The argument is measured in radians from $$-\pi$$ to $$\pi$$, with conversion to degrees:
$$\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}$$
Key properties:
$$|z_1 z_2| = |z_1| \cdot |z_2|$$
$$|z_1 + z_2| \leq |z_1| + |z_2| \quad \text{(triangle inequality)}$$
$$|\bar{z}| = |z|$$
The modulus tells you how far the complex number is from the origin (0,0) in the complex plane. A larger modulus means the number is farther from zero. The argument tells you the direction: 0° points along the positive real axis, 90° along the positive imaginary axis, 180° along the negative real axis, and -90° along the negative imaginary axis. Together, modulus and argument give the complete polar description of the complex number.
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Results
|3+4i| = √(9+16) = √25 = 5. This is the classic 3-4-5 Pythagorean triple. The argument is arctan(4/3) ≈ 53.13°.
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Results
|-5-12i| = √(25+144) = √169 = 13. The number is in the third quadrant, so the argument is about -112.6° (or equivalently 247.4°).
The modulus (or absolute value) of $$z = a + bi$$ is $$|z| = \sqrt{a^2 + b^2}$$. It represents the distance from the origin to the point $$(a, b)$$ in the complex plane. The modulus is always non-negative and equals zero only when both the real and imaginary parts are zero.
The modulus $$|z|$$ is the distance from the origin (how far), while the argument $$\arg(z)$$ is the angle measured from the positive real axis (which direction). Together they form the polar representation: $$z = |z|(\cos\theta + i\sin\theta)$$. The modulus is a single non-negative real number, while the argument is an angle typically in $$(-\pi, \pi]$$.
The squared modulus $$|z|^2 = a^2 + b^2$$ appears frequently because it avoids computing a square root, is always rational when $$a$$ and $$b$$ are rational, equals the product $$z\bar{z}$$, and represents physical quantities like power or probability in applications. In quantum mechanics, $$|\psi|^2$$ gives the probability density.
The triangle inequality states $$|z_1 + z_2| \leq |z_1| + |z_2|$$ for any complex numbers $$z_1$$ and $$z_2$$. Geometrically, the length of one side of a triangle is at most the sum of the other two sides. Equality holds when $$z_1$$ and $$z_2$$ point in the same direction (have the same argument).
In polar form, a complex number is written as $$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$, where $$r = |z|$$ is the modulus and $$\theta = \arg(z)$$ is the argument. The modulus is the radial distance and the argument is the angular coordinate. Converting between rectangular and polar forms requires computing both the modulus and argument.
No, the modulus is always non-negative: $$|z| \geq 0$$ for all complex numbers $$z$$. Since $$|z| = \sqrt{a^2 + b^2}$$ and the square root function returns non-negative values, the modulus cannot be negative. It equals zero only when $$z = 0 + 0i$$, the origin of the complex plane.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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