Understanding Complex Numbers

• Complex numbers extend real numbers by including an imaginary unit ($i$), where $i^2 = -1$.
• A complex number is expressed as $z = a + bi$, where $a$ is the real part (Re(z)), and $b$ is the imaginary part (Im(z)).

Operations

• To add complex numbers: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
• To subtract complex numbers: $(a + bi) - (c + di) = (a - c) + (b - d)i$.
• To multiply complex numbers: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
• To divide complex numbers: $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$.

Complex Conjugate

• The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$.

Modulus

• The modulus of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$.

Geometric Representation

• Complex numbers are graphically represented on a complex plane (Argand diagram).
• The horizontal axis signifies the real part, while the vertical axis represents the imaginary part.

Polar Form

• A complex number $z = a + bi$ can be expressed in polar form as $z = r(\cos\theta + i\sin\theta)$.
• In polar form, $r = |z| = \sqrt{a^2 + b^2}$ is the modulus, and $\theta = \arctan(\frac{b}{a})$ is the argument.
• Using Euler's formula, the polar form is $z = re^{i\theta}$.

Euler's Formula

• Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$.
• This formula links complex exponentials and trigonometric functions.

Applications

• Complex numbers are utilized in electrical engineering for analyzing AC circuits.
• They are also used in quantum mechanics to describing wave functions.
• They help out solving flow problems in fluid dynamics.
• Complex numbers are utilized in signal processing to analyze signals in the frequency domain.
• They have applications in complex analysis and number theory in mathematics.