Introduction to complex numbers (Overview)

Learning Objectives Define the imaginary unit, i, and understand its fundamental property, i^2 = -1. Express the square root of any negative number in terms of i. Identify the real and imaginary parts of a complex number written in standard form (a + bi). Perform addition and subtraction of complex numbers. Perform multiplication of complex numbers and express the result in standard form. Recognize the cyclical pattern of the powers of i (i, i^2, i^3, i^4). What if I told you there's a valid, mathematical answer to \sqrt{-1}? 🤔 Let's explore a whole new dimension of numbers! This lesson introduces the complex number system, which extends the real numbers to include the 'imaginary unit', i. You will learn what complex numbers are, why they are necessary,...

TermDefinitionExample Imaginary Unit (i)The imaginary unit, denoted by 'i', is defined as the principal square root of negative one. It is the foundation of complex numbers.i = \sqrt{-1} Complex NumberA number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.3 + 2i is a complex number. Standard FormThe standard way to write a complex number, with the real part first and the imaginary part second.The standard form is a + bi. For \sqrt{-9} + 5, the standard form is 5 + 3i. Real PartThe real number 'a' in a complex number a + bi.In the complex number 7 - 4i, the real part is 7. Imaginary PartThe real number 'b' (the coefficient of i) in a complex number a + bi.In the complex...

The Definition of i i = \sqrt{-1} \quad \text{and} \quad i^2 = -1 This is the fundamental identity of the imaginary unit. Whenever you see i^2 in a calculation, you must replace it with -1 to simplify the expression. Addition and Subtraction of Complex Numbers (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i To add or subtract complex numbers, combine the real parts together and combine the imaginary parts together, just like combining like terms in algebra. Multiplication of Complex Numbers (a + bi)(c + di) = (ac - bd) + (ad + bc)i Multiply complex numbers using the distributive property or the FOIL method, as you would with binomials. Remember to replace any instance of i^2 with -1 and then combine like terms.