Complex Number Calculator
Free Complex number Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. Includes formulas and worked examples.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Complex Number Calculator
Calculator
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Formula: z1 op z2 = (a1 + b1i) op (a2 + b2i)
Worked example — (3 + 4i)(1 - 2i) = 11 - 2i
Formula
z1 op z2 = (a1 + b1i) op (a2 + b2i)
Complex arithmetic: Addition adds real and imaginary parts separately. Multiplication uses (a1+b1i)(a2+b2i) = (a1a2-b1b2) + (a1b2+b1a2)i. Division multiplies by the conjugate of the denominator. The modulus |z| = sqrt(a^2+b^2) and argument arg(z) = atan2(b,a) give the polar representation.
Worked Examples
Example 1: Multiplying (3 + 4i)(1 - 2i)
Problem:Find the product of z1 = 3 + 4i and z2 = 1 - 2i.
Solution:(3 + 4i)(1 - 2i) = 3(1) + 3(-2i) + 4i(1) + 4i(-2i) = 3 - 6i + 4i - 8i^2 = 3 - 2i - 8(-1) = 3 - 2i + 8 = 11 - 2i
Result:(3 + 4i)(1 - 2i) = 11 - 2i
Example 2: Dividing (5 + 3i) by (2 + i)
Problem:Compute (5 + 3i) / (2 + i).
Solution:Multiply by conjugate of denominator: (5 + 3i)(2 - i) / ((2 + i)(2 - i)) Numerator: 10 - 5i + 6i - 3i^2 = 10 + i + 3 = 13 + i Denominator: 4 + 1 = 5 Result: (13 + i)/5 = 2.6 + 0.2i
Result:(5 + 3i)/(2 + i) = 2.6 + 0.2i
Frequently Asked Questions
What is a complex number and what are its parts?
A complex number is a number of the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i^2 = -1. The real part is denoted Re(z) = a and the imaginary part is denoted Im(z) = b (note that the imaginary part itself is a real number, not including the i factor). Complex numbers extend the real number system to include solutions to equations like x^2 + 1 = 0 that have no real solutions. Every real number is a complex number with b = 0, and every purely imaginary number has a = 0. The set of all complex numbers is denoted by the symbol C.
How do you add and subtract complex numbers?
Adding complex numbers is done component-wise: (a + bi) + (c + di) = (a + c) + (b + d)i. Simply add the real parts together and the imaginary parts together. Subtraction works the same way: (a + bi) - (c + di) = (a - c) + (b - d)i. For example, (3 + 4i) + (1 - 2i) = 4 + 2i. Geometrically, complex addition corresponds to vector addition in the complex plane, using the parallelogram rule. This means the sum of two complex numbers is the diagonal of the parallelogram formed by the two numbers treated as position vectors from the origin. Subtraction similarly corresponds to vector subtraction.
How do you multiply two complex numbers?
To multiply complex numbers, use the distributive property (FOIL method) and the fact that i^2 = -1: (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i. For example, (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i. In polar form, multiplication is much simpler: multiply the moduli and add the arguments. If z1 = r1*e^(i*t1) and z2 = r2*e^(i*t2), then z1*z2 = r1*r2*e^(i*(t1+t2)). This geometric interpretation means multiplication scales by one modulus and rotates by one argument, which is why complex multiplication is connected to rotation.
How do you divide complex numbers?
To divide complex numbers, multiply both numerator and denominator by the conjugate of the denominator to rationalize it: (a + bi)/(c + di) = (a + bi)(c - di)/((c + di)(c - di)) = [(ac + bd) + (bc - ad)i]/(c^2 + d^2). The denominator becomes the real number c^2 + d^2. For example, (3 + 2i)/(1 + i) = (3 + 2i)(1 - i)/((1)(1 + 1)) = (3 - 3i + 2i - 2i^2)/2 = (5 - i)/2 = 2.5 - 0.5i. In polar form, division is straightforward: divide the moduli and subtract the arguments. This technique is essential for simplifying complex fractions and is used extensively in circuit analysis and signal processing.
What is the modulus and argument of a complex number?
The modulus (or absolute value) of z = a + bi is |z| = sqrt(a^2 + b^2), representing the distance from z to the origin in the complex plane. The argument (or angle) is arg(z) = atan2(b, a), the angle from the positive real axis to the line from the origin to z, measured counterclockwise. Together they give the polar form z = |z| * e^(i*arg(z)). The principal argument is usually taken in the range (-pi, pi] or [0, 2*pi). The modulus is always non-negative, and it equals zero only when z = 0. These polar coordinates provide geometric insight and simplify multiplication, division, and exponentiation of complex numbers.
What is the complex plane and how are complex numbers plotted?
The complex plane (also called the Argand plane or Gauss plane) is a two-dimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number. The number z = a + bi is plotted as the point (a, b). This geometric representation transforms complex arithmetic into geometric operations: addition becomes vector addition, multiplication by e^(i*theta) becomes rotation by angle theta, and multiplication by a real number r becomes scaling by r. The complex plane unifies algebra and geometry, allowing visual intuition about complex operations. Circles, lines, and curves in the complex plane correspond to important algebraic relationships.
What is Euler's formula and how does it connect exponentials to complex numbers?
Euler's formula states that e^(i*theta) = cos(theta) + i*sin(theta), connecting the exponential function to trigonometric functions through complex numbers. Setting theta = pi gives the famous identity e^(i*pi) + 1 = 0, linking five fundamental constants. This formula means any complex number can be written as z = r*e^(i*theta) where r = |z| and theta = arg(z). Euler's formula is not just a notational convenience but reflects deep mathematical structure: the complex exponential function maps the imaginary axis to the unit circle. It simplifies many calculations in engineering, physics, and signal processing by converting trigonometric manipulations into simpler algebraic operations with exponentials.
How are complex numbers used in electrical engineering?
In electrical engineering, complex numbers represent AC voltages, currents, and impedances. Voltage and current phasors are complex numbers where the modulus represents amplitude and the argument represents phase angle. Impedance Z = R + jX (engineers use j instead of i) combines resistance R (real part) and reactance X (imaginary part). A capacitor has impedance 1/(j*omega*C) and an inductor has impedance j*omega*L. Ohm's law in AC circuits becomes V = I*Z using complex arithmetic. Complex power S = P + jQ separates real power P (watts) from reactive power Q (vars). This framework dramatically simplifies AC circuit analysis by converting differential equations into algebraic equations.
What are the fundamental algebraic properties of complex numbers?
Complex numbers form a field, meaning they satisfy all the standard arithmetic properties. Addition and multiplication are both commutative (z1 + z2 = z2 + z1 and z1*z2 = z2*z1) and associative. Multiplication distributes over addition. The additive identity is 0 (= 0 + 0i) and the multiplicative identity is 1 (= 1 + 0i). Every complex number has an additive inverse (-z) and every nonzero complex number has a multiplicative inverse (1/z = conj(z)/|z|^2). However, unlike real numbers, complex numbers cannot be ordered (there is no consistent way to say one complex number is greater than another). The Fundamental Theorem of Algebra guarantees that every non-constant polynomial has at least one complex root.
How do you compute powers and roots of complex numbers?
Powers of complex numbers are easiest in polar form: if z = r*e^(i*theta), then z^n = r^n * e^(i*n*theta). This is De Moivre's theorem, which shows that exponentiation multiplies the argument by n and raises the modulus to the nth power. For roots, the nth root of z = r*e^(i*theta) gives n distinct values: r^(1/n) * e^(i*(theta + 2*pi*k)/n) for k = 0, 1, ..., n-1. These n roots are equally spaced around a circle of radius r^(1/n) in the complex plane. For example, the cube roots of 8 are 2, 2*e^(i*2*pi/3), and 2*e^(i*4*pi/3), which are evenly distributed on a circle of radius 2 at 120-degree intervals.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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