Multiply Complex Numbers Calculator

Multiply two complex numbers using the distributive rule and i² = −1.

How the Math Works

Multiplying complex numbers follows the distributive property, extending the FOIL method for binomials. When calculating (a + bi)(c + di), we expand the product into four terms: ac, adi, bci, and bdi². Since i² equals -1, the last term becomes -bd. Combining real parts (ac - bd) and imaginary parts (ad + bc)i yields the final result. This process systematically accounts for both magnitude and phase interactions between complex numbers.

Practical Applications

This calculation is essential in electrical engineering for analyzing alternating current (AC) circuits, where impedance combines resistance and reactance. It also appears in signal processing for Fourier transforms, quantum mechanics for wave function calculations, and computer graphics for 2D transformations. Engineers use it to solve systems involving oscillations, vibrations, or electromagnetic waves by representing periodic behavior as complex exponentials for easier manipulation.

Day-to-Day Use

While complex numbers may seem abstract, they underpin technologies you encounter daily. The smartphone in your pocket uses complex calculations in its antenna design and signal processing to maintain clear connections. Similarly, GPS systems, medical imaging devices, and even music streaming services rely on mathematical models involving complex numbers to function accurately. Understanding these calculations helps optimize the digital tools that simplify modern life.

Worked example

(2 + 3i)(1 − 4i) = 14 − 5i.

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