1. What is Imaginary Number Multiplication?

Imaginary number multiplication follows specific rules in complex arithmetic. When multiplying two purely imaginary numbers, the result is a real number equal to the negative product of their coefficients.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( a \) — Coefficient of first imaginary number
• \( b \) — Coefficient of second imaginary number
• \( i \) — Imaginary unit (√-1)

Explanation: The product of two imaginary numbers results in a real number because \( i \times i = i^2 = -1 \).

3. Importance of Imaginary Numbers

Details: Imaginary numbers are fundamental in complex analysis, electrical engineering, quantum mechanics, and many areas of advanced mathematics and physics.

4. Using the Calculator

Tips: Enter the coefficients (a and b) of your imaginary numbers. The calculator will compute the product which will be a real number.

5. Frequently Asked Questions (FAQ)

Q1: Why does multiplying two imaginary numbers give a real number?
A: Because \( i^2 = -1 \), so (a i) × (b i) = a b × i² = a b × (-1) = -a b.

Q2: What if I multiply a real and imaginary number?
A: The product of a real number c and imaginary number b i is c b i (purely imaginary).

Q3: Where are imaginary numbers used in real life?
A: Electrical engineering (AC circuit analysis), signal processing, quantum mechanics, and control theory.

Q4: What about multiplying complex numbers?
A: Complex number multiplication follows the distributive property and includes both real and imaginary parts.

Q5: Can the result ever be positive?
A: Yes, if either a or b is negative, their product will be positive, and the final result will be negative of that positive product.