Multiply Complex Numbers Calculator

Complex Multiplication Formula:

\[ (a + bi)(c + di) = (a c - b d) + (a d + b c)i \]

First complex number (a + bi):

a (real part)

b (imaginary coefficient)

Second complex number (c + di):

c (real part)

d (imaginary coefficient)

Product of Complex Numbers:

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1. What is Complex Number Multiplication?

Complex number multiplication follows specific algebraic rules that account for both real and imaginary components. The product of two complex numbers (a + bi) and (c + di) results in a new complex number with both real and imaginary parts.

2. How Does the Calculator Work?

The calculator uses the complex multiplication formula:

\[ (a + bi)(c + di) = (a c - b d) + (a d + b c)i \]

Where:

\( a \) — Real part of first complex number
\( b \) — Imaginary coefficient of first complex number
\( c \) — Real part of second complex number
\( d \) — Imaginary coefficient of second complex number
\( i \) — Imaginary unit (√-1)

Explanation: The formula accounts for both the real and imaginary components of each complex number, following the distributive property of multiplication while respecting that \( i^2 = -1 \).

3. Importance of Complex Numbers

Details: Complex numbers are fundamental in many areas of mathematics, physics, and engineering, particularly in electrical engineering, quantum mechanics, signal processing, and fluid dynamics.

4. Using the Calculator

Tips: Enter the real and imaginary coefficients for both complex numbers. The calculator will compute the product according to the complex multiplication formula.

5. Frequently Asked Questions (FAQ)

Q1: Why does the formula have a minus sign in the real part?
A: Because \( i^2 = -1 \), when multiplying (a + bi)(c + di), the term bi × di becomes bd × i² = -bd, which affects the real part.

Q2: What are some applications of complex number multiplication?
A: Used in AC circuit analysis, Fourier transforms, quantum mechanics wave functions, and representing rotations in 2D space.

Q3: How is complex multiplication different from real number multiplication?
A: Complex multiplication involves both cross-multiplication of real and imaginary parts and accounts for the special property of the imaginary unit (i² = -1).

Q4: Can complex numbers be multiplied in polar form?
A: Yes, in polar form multiplication is simpler: multiply magnitudes and add angles (arguments).

Q5: What happens when you multiply a complex number by its conjugate?
A: The product is a real number equal to the sum of the squares of the real and imaginary parts (a² + b²).