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Complex Number Operations:
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Complex numbers are numbers that consist of a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit with the property that i² = -1.
The calculator performs four basic operations on complex numbers:
Addition: \[ (a + bi) + (c + di) = (a + c) + (b + d)i \]
Subtraction: \[ (a + bi) - (c + di) = (a - c) + (b - d)i \]
Multiplication: \[ (a + bi) \times (c + di) = (ac - bd) + (ad + bc)i \]
Division: \[ \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]
Details: Complex numbers are fundamental in electrical engineering, quantum physics, signal processing, and many areas of mathematics. They provide a way to represent two-dimensional quantities and rotations.
Tips: Enter the real and imaginary parts of both complex numbers, select the operation, and click Calculate. For division, ensure the denominator (second complex number) is not zero.
Q1: What is the imaginary unit 'i'?
A: It's defined as the square root of -1 (i² = -1), which allows solutions to equations that have no real solutions.
Q2: Can complex numbers be compared?
A: Unlike real numbers, complex numbers don't have a natural ordering, so we can't say one is "greater" than another.
Q3: What is the complex conjugate?
A: For a complex number a + bi, its conjugate is a - bi. Multiplying a complex number by its conjugate gives a real number (a² + b²).
Q4: How are complex numbers represented graphically?
A: On the complex plane, with the real part on the x-axis and imaginary part on the y-axis (Argand diagram).
Q5: What's the difference between polar and rectangular form?
A: Rectangular form is a + bi, while polar form uses magnitude (r) and angle (θ): r(cosθ + i sinθ) or re^(iθ).