Multiplying and dividing complex numbers

Complex numbers are of the form z = a + bi, where a and b are real numbers and i is the imaginary unit with the property that i^2 = -1.
To multiply complex numbers, expand the parentheses as you would with binomial expressions in algebra, then simplify by substituting i^2 with -1.
For example, to multiply (2 + 3i)(4 - 5i), you would distribute to obtain 8 - 10i + 12i - 15i^2, which simplifies to 8 + 2i + 15 (because i^2 is -1), and further simplifies to 23 + 2i.
To divide complex numbers, you would normally need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi.
To divide (1 + i) by (2 - i), for example, you would multiply numerator and denominator by (2 + i), resulting in ((1 + i)(2 + i))/((2 - i)(2 + i)). Simplifying this gives (2 + i + 2i + i^2)/(4 + i^2), and further simplifying with i^2 = -1 gives (2 + 3i - 1)/(4 - 1) which simplifies to (1 + 3i)/3, or 1/3 + i.
So, (1 + i)/(2 - i) equals 1/3 + i.
Notice that when you multiply a complex number by its conjugate, you get a real number. This is because the imaginary parts cancel out.
Mastery of these techniques is necessary for working with complex numbers in various contexts including solving equations, graphing functions, and understanding the behaviour of electrical circuits, among other applications in pure and applied mathematics.
Solve plenty of practise problems to gain confidence in performing these operations with complex numbers. Over time, it will become easier and feel more natural. Always remember to replace i^2 with -1 where it appears in your calculations.