What is a complex number?

A complex number is a number that can be written in the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit satisfying $i^2 = -1$.

What is the imaginary unit i?

The imaginary unit $i$ is defined as the mathematical constant whose square is -1: $i^2 = -1$.

How do you add or subtract complex numbers?

Add or subtract their real parts and imaginary parts separately: $(a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i$.

How do you multiply complex numbers?

Multiply them using the distributive law (FOIL) and substitute $i^2 = -1$: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$.

What is a complex conjugate?

The complex conjugate of $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate always yields a real number: $a^2 + b^2$.

How do you convert from rectangular to polar form?

Calculate the magnitude $r = \sqrt{a^2 + b^2}$ and the phase angle $\theta = \text{atan2}(b, a)$.

What is polar form of a complex number?

Polar form represents a complex number by its distance from the origin (magnitude $r$) and its angle from the positive real axis (argument $\theta$), written as $r \angle \theta$ or $r e^{i\theta}$.

A phasor is a complex number used in engineering to represent a sinusoidal signal, capturing its amplitude and phase angle.

What is De Moivre's Theorem?

De Moivre's Theorem states that raising a complex number to a power $n$ corresponds to raising its magnitude to $n$ and multiplying its angle by $n$: $Z^n = r^n \angle (n\theta)$.

How many n-th roots does a complex number have?

A complex number has exactly $n$ distinct $n$-th roots, which are spaced symmetrically on a circle of radius $r^{1/n}$ in the complex plane.

Can you divide complex numbers in polar form?

Yes, division is easy in polar form: divide the magnitudes and subtract the angles: $(r_1 \angle \theta_1) / (r_2 \angle \theta_2) = (r_1/r_2) \angle (\theta_1 - \theta_2)$.

What is an Argand diagram?

An Argand diagram is a geometric plot of complex numbers as points or vectors in a two-dimensional plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

math calculator

Complex Number & Phasor Calculator

Perform rectangular and polar complex arithmetic, phasor conversions, and find complex roots.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Complex Operator

Complex Number Z₁

Real Part (a)

Imaginary Part (b)

Complex Number Z₂

Real Part (c)

Imaginary Part (d)

Results

Result Z_res (Rectangular)

4.0000 +2.0000i

Magnitude |Z|4.4721

Phase Angle θ26.57°

Argand Plane Plotter

Z₁vector | Z₂vector | Result phasor

Step-by-Step Mathematical Solution

Given $Z_1 = 3 + 4i$ and $Z_2 = 1 -2i$:

Polar conversion:

\[Z_1 = 5.000 \angle 53.1^\circ, \quad Z_2 = 2.236 \angle -63.4^\circ\]

Add real parts and imaginary parts separately:

\[Z_1 + Z_2 = (3 + 1) + (4 + -2)i = 4 +2i\]

Formula

Complex Number formulas

Z_1 \cdot Z_2 = (ac-bd) + (ad+bc)i

Polar coordinate math utilizes Euler's formula to simplify multiplication, division, and complex exponentiation.

Step by step

How the calculation works

1Enter coordinates for Z₁ and Z₂. Choose Rectangular input formats.
2Select the math operation (Add, Subtract, Multiply, Divide, or Roots).
3View vector representations plotted on the Argand plane coordinate grid.

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math calculator

Complex Number & Phasor Calculator

Perform rectangular and polar complex arithmetic, phasor conversions, and find complex roots.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Complex Operator

Complex Number Z₁

Real Part (a)

Imaginary Part (b)

Complex Number Z₂

Real Part (c)

Imaginary Part (d)

Results

Result Z_res (Rectangular)

4.0000 +2.0000i

Magnitude |Z|4.4721

Phase Angle θ26.57°

Argand Plane Plotter

Z₁vector | Z₂vector | Result phasor

Step-by-Step Mathematical Solution

Given $Z_1 = 3 + 4i$ and $Z_2 = 1 -2i$:

Polar conversion:

\[Z_1 = 5.000 \angle 53.1^\circ, \quad Z_2 = 2.236 \angle -63.4^\circ\]

Add real parts and imaginary parts separately:

\[Z_1 + Z_2 = (3 + 1) + (4 + -2)i = 4 +2i\]

Formula

Complex Number formulas

Z_1 \cdot Z_2 = (ac-bd) + (ad+bc)i

Polar coordinate math utilizes Euler's formula to simplify multiplication, division, and complex exponentiation.

Step by step

How the calculation works

1Enter coordinates for Z₁ and Z₂. Choose Rectangular input formats.
2Select the math operation (Add, Subtract, Multiply, Divide, or Roots).
3View vector representations plotted on the Argand plane coordinate grid.

Related guides

Learn more about this calculation

Guide