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# Complex Numbers ## Definition A **complex number** is a number of the form $a + bi$, where: * $a$ and $b$ are real numbers. * $i$ is the imaginary unit, defined as $i^2 = -1$. ### Terms Used * $a$ is the **real part** of the complex number. * $b$ is the **imaginary part** of the compl...

# Complex Numbers ## Definition A **complex number** is a number of the form $a + bi$, where: * $a$ and $b$ are real numbers. * $i$ is the imaginary unit, defined as $i^2 = -1$. ### Terms Used * $a$ is the **real part** of the complex number. * $b$ is the **imaginary part** of the complex number. ### Examples * $3 + 2i$ * $-1 - i$ * $4i$ (pure imaginary number) * $5$ (real number, since $5 = 5 + 0i$) ## Operations with Complex Numbers ### Addition To add two complex numbers, add their real parts and their imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$ **Example:** $(2 + 3i) + (1 - i) = (2 + 1) + (3 - 1)i = 3 + 2i$ ### Subtraction To subtract two complex numbers, subtract their real parts and their imaginary parts separately: $(a + bi) - (c + di) = (a - c) + (b - d)i$ **Example:** $(5 - 2i) - (3 + i) = (5 - 3) + (-2 - 1)i = 2 - 3i$ ### Multiplication To multiply two complex numbers, use the distributive property and the fact that $i^2 = -1$: $(a + bi)(c + di) = a(c + di) + bi(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$ **Example:** $(1 + 2i)(3 - i) = 1(3 - i) + 2i(3 - i) = 3 - i + 6i - 2i^2 = 3 - i + 6i + 2 = 5 + 5i$ ### Division To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ **Example:** $\frac{2 + i}{1 - i} = \frac{(2 + i)(1 + i)}{(1 - i)(1 + i)} = \frac{2 + 2i + i + i^2}{1 - i^2} = \frac{2 + 3i - 1}{1 + 1} = \frac{1 + 3i}{2} = \frac{1}{2} + \frac{3}{2}i$ ## Complex Conjugate The **complex conjugate** of a complex number $a + bi$ is $a - bi$, denoted as $\overline{a + bi}$. ### Properties * $\overline{z + w} = \overline{z} + \overline{w}$ * $\overline{z \cdot w} = \overline{z} \cdot \overline{w}$ * $z \cdot \overline{z} = a^2 + b^2$, where $z = a + bi$ ## Modulus (Absolute Value) The **modulus** (or absolute value) of a complex number $z = a + bi$ is the distance from the origin to the point $(a, b)$ in the complex plane: $|z| = \sqrt{a^2 + b^2}$ ### Properties * $|z| \geq 0$ for all complex numbers $z$. * $|z| = 0$ if and only if $z = 0$. * $|z \cdot w| = |z| \cdot |w|$ * $|z + w| \leq |z| + |w|$ (Triangle Inequality) ## Polar Form of Complex Numbers A complex number $z = a + bi$ can be represented in polar form as: $z = r(\cos{\theta} + i\sin{\theta})$ Where: * $r = |z| = \sqrt{a^2 + b^2}$ is the modulus of $z$. * $\theta = \arctan{(\frac{b}{a})}$ is the argument of $z$. ### Euler's Formula Euler's formula connects complex exponentials to trigonometric functions: $e^{i\theta} = \cos{\theta} + i\sin{\theta}$ Using Euler's formula, the polar form can be written as: $z = re^{i\theta}$ ### Operations in Polar Form * **Multiplication:** $z_1 \cdot z_2 = r_1r_2e^{i(\theta_1 + \theta_2)}$ * **Division:** $\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$ * **Power:** $z^n = r^ne^{in\theta}$ (De Moivre's Theorem) ## Complex Plane The **complex plane** (or Argand diagram) is a graphical representation of complex numbers: * The horizontal axis represents the real part. * The vertical axis represents the imaginary part. A complex number $a + bi$ is plotted as the point $(a, b)$ in the complex plane. ## Applications Complex numbers have applications in various fields, including: * Electrical engineering * Quantum mechanics * Fluid dynamics * Signal processing * Control theory