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.
Calculator
Adjust values & calculatez1 = a1 + b1i
z2 = a2 + b2i
Formula
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.
Last reviewed: December 2025
Worked Examples
Example 1: Multiplying (3 + 4i)(1 - 2i)
Example 2: Dividing (5 + 3i) by (2 + i)
Background & Theory
The Complex Number Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Complex Number Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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)\n= 3(1) + 3(-2i) + 4i(1) + 4i(-2i)\n= 3 - 6i + 4i - 8i^2\n= 3 - 2i - 8(-1)\n= 3 - 2i + 8\n= 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:\n(5 + 3i)(2 - i) / ((2 + i)(2 - i))\nNumerator: 10 - 5i + 6i - 3i^2 = 10 + i + 3 = 13 + i\nDenominator: 4 + 1 = 5\nResult: (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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy