Complex Numbers: Multiplication and Applications

Complex Numbers

Multiplying 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 i2 = -1. Multiplying complex numbers involves using the distributive property (also known as FOIL for binomials) and simplifying using the property of i.

• Key Point 1: To multiply two complex numbers, expand the product as you would with binomials, then simplify using i^2 = -1.

• Key Point 2: The product of two complex numbers (a + bi)(c + di) is given by:

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

• Example: Multiply (7 - 6i) and (3 - 4i).

$(7 - 6i)(3 - 4i) = 7 \times 3 + 7 \times (-4i) - 6i \times 3 - 6i \times (-4i)$ $= 21 - 28i - 18i + 24i^2$ $= 21 - 46i + 24(-1)$ $= 21 - 46i - 24$ $= -3 - 46i$

• Application: Multiplying complex numbers is essential in solving equations involving imaginary numbers and in fields such as engineering and physics.