Expert Verified by: David Chen, CFA
Professional financial analyst and mathematics consultant with 15+ years of experience in engineering calculations.
This Complex Number Calculator is designed to perform basic arithmetic operations on complex numbers ($a + bi$). Whether you are working on electrical engineering problems, quantum physics, or advanced algebra, our tool provides precise results and step-by-step breakdowns for addition, subtraction, multiplication, and division.
Complex Number Calculator
Complex Number Calculator Formula:
Addition: $(a+bi) + (c+di) = (a+c) + (b+d)i$
Multiplication: $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$
Division: $\frac{a+bi}{c+di} = \frac{(ac+bd) + (bc-ad)i}{c^2+d^2}$
Variables:
- a (Real Part 1): The real component of the first number.
- b (Imaginary Part 1): The imaginary component of the first number.
- c (Real Part 2): The real component of the second number.
- d (Imaginary Part 2): The imaginary component of the second number.
- i: The imaginary unit, where $i^2 = -1$.
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- Matrix Multiplication Tool
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What is a Complex Number Calculator?
A complex number calculator is a specialized tool used to perform mathematical operations on numbers that consist of both a real part and an imaginary part. In the form $a + bi$, “a” represents the real portion, while “bi” represents the imaginary portion.
These calculators are essential in fields such as electrical engineering for analyzing AC circuits, signal processing, and fluid dynamics. They help simplify the tedious process of multiplying conjugates or finding magnitudes and phases.
How to Calculate Complex Numbers (Example):
- Identify your two complex numbers, e.g., $3 + 2i$ and $1 + 4i$.
- Choose your operation (e.g., Multiplication).
- Apply the FOIL method: $(3 \cdot 1) + (3 \cdot 4i) + (2i \cdot 1) + (2i \cdot 4i)$.
- Simplify: $3 + 12i + 2i – 8$ (since $i^2 = -1$).
- Combine like terms: $(3 – 8) + (12i + 2i) = -5 + 14i$.
Frequently Asked Questions (FAQ):
What is the imaginary unit i? It is defined as the square root of -1.
Can I divide by zero in complex math? No, if the second number is $0 + 0i$, the division is undefined.
What is the conjugate of a complex number? The conjugate of $a + bi$ is $a – bi$.
Does this tool support polar forms? It calculates in rectangular form but displays the polar magnitude and angle for convenience.