IMG_2038.HEIC

# Complex Numbers ### Definition - A Complex Number has two parts: a **real part** and an **imaginary part**. - Complex numbers can be added, subtracted, multiplied, and divided. ### Form - Complex numbers are written in the form $a + bi$, where: - $a$ is the real part. - $b$ is the imagina...

# Complex Numbers ### Definition - A Complex Number has two parts: a **real part** and an **imaginary part**. - Complex numbers can be added, subtracted, multiplied, and divided. ### Form - Complex numbers are written in the form $a + bi$, where: - $a$ is the real part. - $b$ is the imaginary part. - $i$ is the imaginary unit, defined as $i = \sqrt{-1}$ ### Example - $3 + 4i$ - Real Part $= 3$ - Imaginary Part $= 4$ ### Operations with Complex Numbers #### Addition - To add complex numbers, add the real parts and the imaginary parts separately $(a + bi) + (c + di) = (a + c) + (b + d)i$ - **Example** $(2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i$ #### Subtraction - To subtract complex numbers, subtract the real parts and the imaginary parts separately. $(a + bi) - (c + di) = (a - c) + (b - d)i$ - **Example** $(5 + 6i) - (1 + 2i) = (5 - 1) + (6 - 2)i = 4 + 4i$ #### Multiplication - To multiply complex numbers, apply the distributive property and remember that $i^2 = -1$ $(a + bi)(c + di) = ac + adi + bci + bdi^2$ $= ac + (ad + bc)i - bd$ $= (ac - bd) + (ad + bc)i$ - **Example** $(2 + 3i)(4 + 5i) = 2 \cdot 4 + 2 \cdot 5i + 3i \cdot 4 + 3i \cdot 5i$ $= 8 + 10i + 12i + 15i^2$ $= 8 + 22i - 15$ $= -7 + 22i$ #### Division - To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$ $\frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ - **Example** $\frac{2 + 3i}{4 + 5i} = \frac{(2 + 3i)(4 - 5i)}{(4 + 5i)(4 - 5i)}$ $= \frac{8 - 10i + 12i - 15i^2}{16 - 20i + 20i - 25i^2}$ $= \frac{8 + 15 + (-10 + 12)i}{16 + 25}$ $= \frac{23 + 2i}{41}$ $= \frac{23}{41} + \frac{2}{41}i$ ### Complex Conjugate - The complex conjugate of a complex number $a + bi$ is $a - bi$ - Multiplying a complex number by its conjugate results in a real numbers: $(a + bi)(a - bi) = a^2 + b^2$ #### Example - The complex conjugate of $3 + 4i$ is $3 - 4i$ $(3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9 + 16 = 25$ ### Absolute Value (or Modulus) - The absolute value (or modulus) of a complex number $a + bi$ is the distance from the origin to the point $(a, b)$ in the complex plane $|a + bi| = \sqrt{a^2 + b^2}$ #### Example $|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$