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Definition:
- A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property i^2 = -1.
- The real part of the complex number is a, and the imaginary part is bi.
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Operations:
- Addition and Subtraction: Complex numbers are added or subtracted by adding or subtracting their real and imaginary parts separately.
- Multiplication: To multiply complex numbers (a + bi) and (c + di), distribute and combine like terms, then simplify using i^2 = -1.
- Division: To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator, then simplify.
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Complex Conjugate:
- The complex conjugate of a complex number a + bi is a - bi.
- When multiplied together, a complex number and its conjugate yield a real number: (a + bi)(a - bi) = a^2 + b^2.
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Polar Form:
- Complex numbers can also be represented in polar form as r(cos θ + i sin θ), where r is the magnitude (or modulus) of the complex number and θ is the argument (or angle) measured counterclockwise from the positive real axis.
- The magnitude of a complex number a + bi is √(a^2 + b^2), and the argument is arctan(b/a).
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Euler's Formula:
- Euler's formula relates complex numbers, trigonometric functions, and the exponential function: e^(iθ) = cos θ + i sin θ.
- This formula allows complex numbers to be expressed in terms of exponentials, facilitating operations like exponentiation and finding roots.
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Applications:
- Complex numbers have applications in various fields, including electrical engineering, signal processing, quantum mechanics, and fluid dynamics.
- They are used to represent alternating current (AC) circuits, analyze oscillatory motion, solve differential equations, and more.
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