Complex Number Calculator
Equations involving square roots of negative values require precise tracking of real and imaginary components. The complex number calculator processes addition, subtraction, multiplication, division, modulus extraction, and polar conversion instantly. Enter your numerical inputs to receive exact outputs with verified step-by-step breakdowns.
Polar Conversion
Complex number calculator operations and formulas
Every input follows the standard algebraic notation $a + bi$. The variable $a$ represents the real coordinate, while $b$ scales the imaginary unit $i$ where $i^2 = -1$. The calculator applies these core arithmetic rules automatically:
| Operation | Formula | Result Format | ||
|---|---|---|---|---|
| Addition | $(a + bi) + (c + di)$ | $(a + c) + (b + d)i$ | ||
| Subtraction | $(a + bi) - (c + di)$ | $(a - c) + (b - d)i$ | ||
| Multiplication | $(a + bi) \times (c + di)$ | $(ac - bd) + (ad + bc)i$ | ||
| Division | $(a + bi) / (c + di)$ | $\frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$ | ||
| Modulus | $ | z | = \sqrt{a^2 + b^2}$ | Non-negative real number |
| Conjugate | $\overline{z} = a - bi$ | Flipped imaginary sign |
Division requires multiplying the numerator and denominator by the conjugate of the denominator. This process eliminates the imaginary component from the divisor, leaving a real denominator equal to $c^2 + d^2$. The tool applies this rationalization step before extracting final coordinates.
How do you convert between rectangular and polar coordinates?
Standard algebraic notation positions numbers on a two-dimensional plane with orthogonal axes. Polar representation describes the same point using radial distance and angular direction. Conversion relies on trigonometric relationships derived from the Pythagorean theorem.
Rectangular to Polar
- Calculate modulus: $r = \sqrt{a^2 + b^2}$
- Calculate argument: $\theta = \arctan(b / a)$
- Adjust $\theta$ for quadrants II and III by adding 180° or π radians
- Format result: $r(\cos\theta + i\sin\theta)$ or $re^{i\theta}$
Polar to Rectangular
- Extract real part: $a = r \cos\theta$
- Extract imaginary part: $b = r \sin\theta$
- Format result: $a + bi$
The calculator accepts both degrees and radians for angular inputs. Negative moduli are automatically converted to positive form with a corresponding 180° phase shift, preserving mathematical equivalence.
Step-by-step calculation example
Multiplication demonstrates how real and imaginary terms redistribute across four separate products. Consider $(4 + 3i) \times (2 - 5i)$:
- Apply the distributive property: $4(2) + 4(-5i) + 3i(2) + 3i(-5i)$
- Simplify individual terms: $8 - 20i + 6i - 15i^2$
- Replace $i^2$ with $-1$: $8 - 14i - 15(-1)$
- Combine real components: $8 + 15 = 23$
- Final output: $23 - 14i$
The modulus of this result equals $\sqrt{23^2 + (-14)^2} = 26.93$. The argument calculates to $-31.4^\circ$. You can verify intermediate steps using the tool above to ensure algebraic accuracy.
Engineering and scientific applications
Impedance modeling forms the foundation of alternating current circuit design. Resistors contribute real values, while capacitors and inductors introduce imaginary reactance. Engineers combine these elements using series and parallel rules that require exact complex arithmetic.
Signal processing relies on Fourier transforms to decompose waveforms into frequency components. Each harmonic carries amplitude and phase information stored in polar coordinates. Time-domain multiplication translates to frequency-domain convolution, demanding rapid calculation of magnitudes and angles.
Quantum mechanics uses complex probability amplitudes to determine particle behavior. Wave functions evolve according to the Schrödinger equation, which operates entirely within complex Hilbert spaces. State vectors normalize to unit modulus before extracting measurable probabilities.
Control system stability depends on pole placement in the complex left-half plane. Transfer functions map input frequencies to output responses using $s$-domain algebra. Designers calculate damping ratios and natural frequencies directly from real and imaginary root components.
Consult authoritative mathematical references like NIST Digital Library of Mathematical Functions for advanced properties and domain definitions.
This content provides mathematical reference material. Verify critical design calculations with licensed engineering standards before implementation.