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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, and $i$ is the imaginary unit, defined as $i = \sqrt{-1}$. In other words, $i^2 = -1$. * $a$ is called the **real part** of $z$, denoted as $\operatorname{Re}(z)$. *...

# Complex Numbers ## Definition A **complex number** is a number of the form $z = a + bi$ where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i = \sqrt{-1}$. In other words, $i^2 = -1$. * $a$ is called the **real part** of $z$, denoted as $\operatorname{Re}(z)$. * $b$ is called the **imaginary part** of $z$, denoted as $\operatorname{Im}(z)$. ## Examples * $3 + 2i$: Real part is 3, imaginary part is 2. * $-5 - i$: Real part is -5, imaginary part is -1. * $4i$: Real part is 0, imaginary part is 4. * $6$: Real part is 6, imaginary part is 0. ## Complex Plane A complex number $z = a + bi$ can be represented as a point $(a, b)$ in the **complex plane**. * The horizontal axis is the **real axis**. * The vertical axis is the **imaginary axis**. ### Example The complex number $3 + 4i$ is represented by the point $(3, 4)$ in the complex plane. ## 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 $(5 + 4i) - (2 + i) = (5 - 2) + (4 - 1)i = 3 + 3i$ ### Multiplication To multiply two complex numbers, use the distributive property and the fact that $i^2 = -1$: $(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$ #### Example $(1 + 2i)(3 - i) = 3 - i + 6i - 2i^2 = (3 + 2) + (-1 + 6)i = 5 + 5i$ ### 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 $\frac{1 + 2i}{3 + i} = \frac{(1 + 2i)(3 - i)}{(3 + i)(3 - i)} = \frac{3 - i + 6i - 2i^2}{9 - i^2} = \frac{(3 + 2) + (-1 + 6)i}{9 + 1} = \frac{5 + 5i}{10} = \frac{1}{2} + \frac{1}{2}i$ ## Complex Conjugate The **complex conjugate** of a complex number $z = a + bi$ is denoted as $\bar{z}$ and is defined as: $\bar{z} = a - bi$ ### Example If $z = 2 + 3i$, then $\bar{z} = 2 - 3i$. ### Properties of Complex Conjugate * $\overline{z + w} = \bar{z} + \bar{w}$ * $\overline{z \cdot w} = \bar{z} \cdot \bar{w}$ * $\overline{\left(\frac{z}{w}\right)} = \frac{\bar{z}}{\bar{w}}$ * $z \cdot \bar{z} = a^2 + b^2$ * $\operatorname{Re}(z) = \frac{z + \bar{z}}{2}$ * $\operatorname{Im}(z) = \frac{z - \bar{z}}{2i}$ ## Modulus of a Complex Number The **modulus** (or absolute value) of a complex number $z = a + bi$ is denoted as $|z|$ and is defined as: $|z| = \sqrt{a^2 + b^2}$ ### Example If $z = 3 + 4i$, then $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. ### Properties of Modulus * $|z| \geq 0$ for all complex numbers $z$. * $|z| = 0$ if and only if $z = 0$. * $|z \cdot w| = |z| \cdot |w|$ * $\left|\frac{z}{w}\right| = \frac{|z|}{|w|}$ * $|z + w| \leq |z| + |w|$ (Triangle Inequality) These concepts and operations form the foundation for working with complex numbers in various mathematical and engineering applications.