Master complex numbers with precision. Our calculator with imaginary numbers allows you to perform basic arithmetic operations on numbers in the form $a + bi$, providing step-by-step solutions for academic and professional use.
Calculator with Imaginary Numbers
Calculator with Imaginary Numbers Formula
Subtraction: (a + bi) – (c + di) = (a – c) + (b – d)i
Multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i
Division: (a + bi) / (c + di) = [(ac + bd) + (bc – ad)i] / (c² + d²)
Variables:
- a & c: The Real components of the two complex numbers.
- b & d: The Imaginary components (coefficients of i).
- i: The imaginary unit, defined as √-1.
What is Calculator with Imaginary Numbers?
An imaginary number calculator is a specialized tool designed to handle mathematical operations involving complex numbers. A complex number consists of a real part and an imaginary part, usually expressed in the standard form $a + bi$.
Unlike standard arithmetic, multiplying imaginary numbers requires the knowledge that $i^2 = -1$. This calculator automates the “FOIL” method for multiplication and the use of the complex conjugate for division, ensuring highly accurate results for engineering, physics, and advanced mathematics.
How to Calculate Calculator with Imaginary Numbers (Example)
Let’s multiply $(2 + 3i)$ by $(4 – 1i)$:
- Identify variables: $a=2, b=3, c=4, d=-1$.
- Apply Multiplication Formula: $(ac – bd) + (ad + bc)i$.
- Calculate Real Part: $(2 \times 4) – (3 \times -1) = 8 – (-3) = 11$.
- Calculate Imaginary Part: $(2 \times -1) + (3 \times 4) = -2 + 12 = 10$.
- Final Result: $11 + 10i$.
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Frequently Asked Questions (FAQ)
Is $i$ always equal to √-1? Yes, in the context of complex numbers, the imaginary unit $i$ is defined such that $i^2 = -1$.
Can this calculator handle negative imaginary parts? Absolutely. Simply enter a negative value in the imaginary input box (e.g., -5 for $3 – 5i$).
What happens if the denominator is zero in division? If both the real and imaginary parts of the divisor are zero, the operation is undefined (division by zero error).
Why are complex numbers used in the real world? They are essential in electrical engineering (AC circuit analysis), quantum mechanics, and fluid dynamics.