Complex Numbers

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.

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.

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.

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).

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.

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.