Perform complex number addition, subtraction, multiplication, division, and exponentiation. Converts rectangular \(a + bi\) to polar and exponential forms with live Argand plane vector plotting.
4.4721 (cos 26.57° + i sin 26.57°)
4.4721 · e^(i 0.4636 rad)
Input the real and imaginary parts of \(z_1 = a + bi\) (and \(z_2 = c + di\) or power \(n\)).
Toggle between addition, subtraction, multiplication, division, or integer powers.
Inspect the rectangular result, modulus \(r\), argument \(\theta\), polar form, and vector on the Argand plane.
In electrical engineering, AC circuits represent voltage, current, and impedance as complex phasors \(Z = R + jX\). Multiplication computes voltage \(V = I \cdot Z\).
Eliminates manual division errors by automating the multiplication of numerator and denominator by the complex conjugate \((c - di)\).
Computes the exact quadrant-aware angle using \(\operatorname{atan2}(b, a)\), avoiding the classic 180-degree quadrant ambiguity.
Expanding \((1 + i)^8\) by hand requires repeated binomial multiplication; De Moivre's theorem computes \(r^n e^{i n \theta}\) instantaneously.
Seamlessly add, subtract, multiply, divide, and exponentiate complex numbers in real time.
Auto-scaling vector diagram showing the real and imaginary axes, operands, and resulting vector.
Displays magnitude, angle in degrees and radians, polar notation \(r\operatorname{cis}\theta\), and Euler exponential \(r e^{i\theta}\).
Despite the unfortunate historical name "imaginary", complex numbers are as real and physical as negative numbers. They provide the complete two-dimensional language of modern electronics, aerodynamics, and quantum mechanics.
On a standard 1D number line, multiplying by \(-1\) flips your direction by \(180^\circ\) (facing East to facing West). What operation done twice in a row flips you by \(180^\circ\)? Two successive 90° counter-clockwise turns! That is exactly what \(i\) is: a \(90^\circ\) rotation into the second dimension. Multiplying by \(i\) once turns you North. Multiplying by \(i\) a second time turns you West: \(i \times i = i^2 = -1\). It's geometry, not magic!
Treat complex numbers like 2D vectors: \(a\) steps East/West along the real axis, \(b\) steps North/South along the imaginary axis. Adding complex numbers is just adding like terms: \((a + c) + (b + d)i\).
In polar form \(r e^{i\theta}\), complex multiplication reveals its true elegance: multiply the magnitudes, add the angles! \((r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = (r_1 r_2)e^{i(\theta_1 + \theta_2)}\). Powers and roots become simple arithmetic.
Electrical engineers represent alternating voltage and current as rotating phasors, unifying resistance and reactance into complex impedance \(Z = R + jX\). In quantum physics, particle states are complex wavefunctions \(\psi(x, t)\).
When expanding \((3 + 2i)(4 - 5i)\), the last product is \(-10i^2\). Because \(i^2 = -1\), this becomes \(-10(-1) = +10\). Forgetting to flip the negative sign is the #1 mistake students make.
You cannot divide directly by \(c + di\). You must "rationalize" the denominator by multiplying both top and bottom by the complex conjugate \(c - di\), creating a purely real number \(c^2 + d^2\) in the denominator.
Blindly computing \(\theta = \arctan(b/a)\) fails when \(a < 0\). For \(-1 + i\), \(\arctan(1/-1) = -45^\circ\), which points in Quadrant IV! You must add \(180^\circ\) (\(\pi\) radians) to land in the true Quadrant II (\(135^\circ\)).
| Form | Standard Notation | Best Used For | Geometric Intuition |
|---|---|---|---|
| Rectangular (Cartesian) | \(z = a + bi\) | Addition and Subtraction | Grid displacement \((x, y)\) along horizontal & vertical axes |
| Polar (Trigonometric) | \(z = r(\cos\theta + i\sin\theta)\) | Multiplication, Division, De Moivre roots | Radius distance \(r\) and counter-clockwise rotation angle \(\theta\) |
| Exponential (Euler) | \(z = r e^{i\theta}\) | Calculus, Signal Processing, AC circuit analysis | Continuous rotation phasor on the complex unit circle |
Step 1: FOIL expansion: \(2(4) + 2(-5i) + 3i(4) + 3i(-5i)\).
Step 2: Simplify terms: \(8 - 10i + 12i - 15i^2\).
Step 3: Since \(i^2 = -1\), \(-15i^2 = +15\).
Step 4: Combine like terms: \((8 + 15) + (-10i + 12i) = 23 + 2i\).
Step 1: Conjugate of denominator \(1 - i\) is \(1 + i\).
Step 2: Multiply numerator & denominator by \((1 + i)\): \(\frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}\).
Step 3: Numerator: \(1 + 2i + i^2 = 1 + 2i - 1 = 2i\).
Step 4: Denominator: \(1^2 - i^2 = 1 - (-1) = 2\).
Result: \(\frac{2i}{2} = i\) (or \(0 + 1i\)).
