Products and Quotients of Complex Numbers

Products and Quotients of Complex Numbers

Multiplying and Dividing 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 $i^2 = -1$. Operations such as multiplication and division can be performed using algebraic rules and properties of i.

Multiplication of Complex Numbers

• Definition: To multiply two complex numbers, use the distributive property (FOIL method) and simplify using $i^2 = -1$.

• Formula: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Example: $(2 + 3i)(4 - 5i)$ $= 2 \times 4 + 2 \times (-5i) + 3i \times 4 + 3i \times (-5i)$ $= 8 - 10i + 12i - 15i^2$ $= 8 + 2i + 15$ (since $i^2 = -1$) $= 23 + 2i$

Division of Complex Numbers

• Definition: To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Formula: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Conjugate: The conjugate of $c + di$ is $c - di$.

• Example: $\frac{2 + 3i}{4 - 5i}$ Multiply numerator and denominator by $4 + 5i$: $= \frac{(2 + 3i)(4 + 5i)}{(4 - 5i)(4 + 5i)}$ Numerator: $(2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i$ Denominator: $(4 - 5i)(4 + 5i) = 16 + 20i - 20i - 25i^2 = 16 + 25 = 41$ $= \frac{-7 + 22i}{41}$

Practice Problems

Type	Problem	Solution
Product	$(3 + 2i)(5 - 6i)$	$3 \times 5 + 3 \times (-6i) + 2i \times 5 + 2i \times (-6i) = 15 - 18i + 10i - 12i^2 = 15 - 8i + 12 = 27 - 8i$
Quotient	$\frac{3 + 2i}{5 - 6i}$	Multiply numerator and denominator by $5 + 6i$: Numerator: $(3 + 2i)(5 + 6i) = 15 + 18i + 10i + 12i^2 = 15 + 28i - 12 = 3 + 28i$ Denominator: $(5 - 6i)(5 + 6i) = 25 + 30i - 30i - 36i^2 = 25 + 36 = 61$ $= \frac{3 + 28i}{61}$

Additional info: Mastery of these operations is essential for solving equations involving complex numbers and for applications in trigonometry, engineering, and physics.