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Complex Number Operations:
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Complex numbers are numbers that consist of a real part (a) and an imaginary part (bi), where i is the imaginary unit with the property that i² = -1. They are written in the form a + bi and are fundamental in many areas of mathematics, physics, and engineering.
The calculator performs operations on complex numbers using standard mathematical formulas:
The calculator supports addition, subtraction, multiplication, division, magnitude calculation, and complex conjugation.
Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
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)/(c²+d²)] + [(bc-ad)/(c²+d²)]i
Magnitude: |a+bi| = √(a² + b²)
Conjugate: (a+bi)* = a - bi
Instructions: Enter two complex numbers in a+bi form, select the operation, and click Calculate. For magnitude and conjugate operations, only the first number is used.
Q1: What is the imaginary unit i?
A: The imaginary unit i is defined as the square root of -1 (i² = -1). It's the fundamental building block of complex numbers.
Q2: What are complex numbers used for?
A: Complex numbers are used in electrical engineering, quantum physics, signal processing, fluid dynamics, and many other fields where oscillatory phenomena occur.
Q3: How is division of complex numbers performed?
A: Division is performed by multiplying numerator and denominator by the complex conjugate of the denominator to eliminate i from the denominator.
Q4: What does the magnitude of a complex number represent?
A: The magnitude (or modulus) represents the distance from the origin to the point (a,b) in the complex plane.
Q5: What's the significance of the complex conjugate?
A: The conjugate reflects the number across the real axis and is useful for division and finding magnitudes (|z|² = z × z*).