Math & Engineering
Complex Number & AC Phasor Math: Magnitude, Phase, Multiply & Divide
Model complex impedances and AC phasors a + bi in numeric Real and Imaginary columns to compute modulus (IMABS), phase angle in degrees (IMARGUMENT), complex multiplication (IMPRODUCT), and rationalized division (IMDIV).
Why numeric (Re, Im) columns beat text-encoded "3+4i" strings
Excel’s legacy IM* functions store complex numbers as text strings ("3+4i"), which cannot be formatted with numeric decimal places, plotted directly on scatter diagrams, or summed with standard SUM/AVERAGE.
Keeping Real (a) and Imaginary (b) components in numeric columns makes every operation fast, exact, and chart-ready: modulus |z| = SQRT(a^2 + b^2), phase angle theta = DEGREES(ATAN2(a, b)), product (a + bi)(c + di) = (ac - bd) + (ad + bc)i, and quotient (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i] / (c^2 + d^2) via conjugate rationalization.
| Operation on z1 = 3 + 4i, z2 = 1 - 2i | Real Component | Imaginary Component | Rectangular String (COMPLEX) |
|---|---|---|---|
| Modulus |z1| & Phase deg | |z1| = 5.0000 | Phase = 53.1301° | Polar: 5 ∠ 53.13° |
| Sum z1 + z2 (IMSUM) | 3 + 1 = 4 | 4 + (-2) = 2 | "4+2i" |
| Product z1 * z2 (IMPRODUCT) | 3*1 - 4*(-2) = 11 | 3*(-2) + 4*1 = -2 | "11-2i" |
| Quotient z1 / z2 (IMDIV) | (3 - 8) / 5 = -1 | (4 - (-6)) / 5 = 2 | "-1+2i" |
Modulus |z| (IMABS) and Phase Angle in Degrees (IMARGUMENT)
Computes exact Euclidean magnitude SQRT(Re^2 + Im^2) of z_1 = B2 + C2*i.
=ROUND(SQRT(B2 ^ 2 + C2 ^ 2), 4)Complex Multiplication (IMPRODUCT) & Conjugate-Rationalized Division (IMDIV)
Divides z_1 = B2 + C2*i by z_2 = B3 + C3*i and formats the exact rectangular result a+bi while guarding division by 0 + 0i.
=LET(denom, B3 ^ 2 + C3 ^ 2, IF(denom = 0, "Divide by Zero Phasor", LET(re, ROUND((B2 * B3 + C2 * C3) / denom, 4), im, ROUND((C2 * B3 - B2 * C3) / denom, 4), re & IF(im >= 0, "+", "") & im & "i")))Try the example
Use the original values or change them to test your understanding.