Sums, Products, and Quotients of Complex Numbers

Sums, Products, and Quotients of Complex Numbers

Introduction to Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by the property that $i^2 = -1$. Operations with complex numbers include addition, subtraction, multiplication, and division, which are essential for solving equations and modeling phenomena in trigonometry and other mathematical fields.

Addition and Subtraction of Complex Numbers

• Definition: To add or subtract complex numbers, combine their real parts and their imaginary parts separately.

• Formula: For $z_1 = a + bi$ and $z_2 = c + di$:

$z_1 + z_2 = (a + c) + (b + d)i$ $z_1 - z_2 = (a - c) + (b - d)i$

• Example: $(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i$

Multiplication of Complex Numbers

• Definition: Multiply complex numbers using the distributive property (FOIL method), remembering that $i^2 = -1$.

• Formula: For $z_1 = a + bi$ and $z_2 = c + di$:

$z_1 z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

• Example: $(2 + 3i)(1 - 2i) = 2 imes 1 + 2 imes (-2i) + 3i imes 1 + 3i imes (-2i)$ $= 2 - 4i + 3i - 6i^2$ $= 2 - 4i + 3i + 6$ (since $i^2 = -1$) $= (2 + 6) + (-4i + 3i)$ $= 8 - i$

Division (Quotients) of Complex Numbers

• Definition: To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

• Formula: For $z_1 = a + bi$ and $z_2 = c + di$ (with $z_2 \neq 0$):

$\frac{z_1}{z_2} = \frac{a + bi}{c + di} \times \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• The denominator simplifies to $c^2 + d^2$.

• Example: $\frac{2 + 3i}{1 - 2i}$ Multiply numerator and denominator by the conjugate $1 + 2i$: $\frac{2 + 3i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(2 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)}$ Numerator: $2 \times 1 + 2 \times 2i + 3i \times 1 + 3i \times 2i = 2 + 4i + 3i + 6i^2 = 2 + 7i - 6 = -4 + 7i$ Denominator: $1^2 - (2i)^2 = 1 - 4i^2 = 1 + 4 = 5$ Final answer: $\frac{-4 + 7i}{5} = -\frac{4}{5} + \frac{7}{5}i$

Table: Operations with Complex Numbers

Operation	Formula	Example
Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	$(2 + 3i) + (1 - 2i) = 3 + i$
Subtraction	$(a + bi) - (c + di) = (a - c) + (b - d)i$	$(2 + 3i) - (1 - 2i) = 1 + 5i$
Multiplication	$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$	$(2 + 3i)(1 - 2i) = 8 - i$
Division	$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$	$\frac{2 + 3i}{1 - 2i} = -\frac{4}{5} + \frac{7}{5}i$

Applications

• Complex numbers are used in trigonometry to represent points and vectors in the complex plane, solve quadratic equations with negative discriminants, and model oscillatory phenomena such as waves and alternating current circuits.