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# Complex Numbers ## Definition A **complex number** is a number of the form: $z = a + bi$ where: * $a$ and $b$ are real numbers * $i$ is the imaginary unit, defined as $i = \sqrt{-1}$ ## Components * **Real Part:** The real part of *z* is denoted as Re(z) = a. * **Imaginary Part:** T...

# Complex Numbers ## Definition A **complex number** is a number of the form: $z = a + bi$ where: * $a$ and $b$ are real numbers * $i$ is the imaginary unit, defined as $i = \sqrt{-1}$ ## Components * **Real Part:** The real part of *z* is denoted as Re(z) = a. * **Imaginary Part:** The imaginary part of *z* is denoted as Im(z) = b. ## Complex Plane Complex numbers can be represented graphically on the complex plane: * **Horizontal Axis:** Represents the real part (a). * **Vertical Axis:** Represents the imaginary part (b). * **Point:** The complex number $z = a + bi$ is represented by the point (a, b). ## Forms of Complex Numbers ### Rectangular Form The standard form of a complex number: $z = a + bi$ where a and b are real numbers. ### Polar Form A complex number can also be represented in polar form: $z = r(cos\theta + isin\theta)$ Where: * $r = \sqrt{a^2 + b^2}$ is the magnitude (or modulus) of z. * $\theta = arctan(\frac{b}{a})$ is the argument (or angle) of z. ### Exponential Form (Euler's Form) Using Euler's formula, the polar form can be written in exponential form: $z = re^{i\theta}$ where: * $r$ is the magnitude of z. * $\theta$ is the argument of z. * $e$ is the base of the natural logarithm. ## Operations with Complex Numbers ### Addition To add two complex numbers, add their real and imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$ ### Subtraction To subtract two complex numbers, subtract their real and imaginary parts separately: $(a + bi) - (c + di) = (a - c) + (b - d)i$ ### Multiplication To multiply two complex numbers, use the distributive property: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$ ### Division To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ ### Complex Conjugate The complex conjugate of a complex number $z = a + bi$ is: $\bar{z} = a - bi$ ### Magnitude (Modulus) The magnitude (or modulus) of a complex number $z = a + bi$ is: $|z| = \sqrt{a^2 + b^2}$ ### Argument The argument (or angle) of a complex number $z = a + bi$ is the angle $\theta$ such that: $\theta = arg(z) = arctan(\frac{b}{a})$ ## Euler's Formula Euler's formula relates the exponential function to trigonometric functions: $e^{i\theta} = cos\theta + isin\theta$ This formula is fundamental in complex analysis and provides a way to convert between exponential and trigonometric forms.