1.4 – Complex Numbers

Real numbers have a small issue; no symmetry of their roots – To remedy this, we introduce an “imaginary” unit, so it does work The number i is defined such that

Simplifying Negative Roots For a positive number a, Follow all other rules to simplify the remaining radical Example. Simplify:

Complex Numbers A complex number, a+bi, has the following: – Real part, a – Imaginary part, bi – Only equal if both parts are equal (real/imaginary) – i With imaginary numbers, only combine the like terms (real with real, imaginary with imaginary) Multiplication, follow same rules as polynomials (FOIL, like terms, etc.)

Example. Simplify:

Quotients Similar to radical expressions, denominators of fractions cannot contain imaginary numbers or a complex number Use the complex conjugate = for given complex number a+bi, the complex conjugate is a-bi

Example. Simplify the quotient: Note the denominator contains the complex number, 4-3i What is the complex conjugate? 4+3i

Example. Simplify the following:

Roots and Complex Numbers When dealing with negative roots, we can simplify using the rules introduced Now, we can simplify radicals in a second way Example. Simplify: How can we write ?

Powers of i The imaginary number, i, has a particular pattern i 2 = -1 i 3 = i 2 x i = -1 x i = -i i 4 = i 2 x i 2 = -1 x -1 = 1 i = i Pull out powers that are multiples of 4; those will become 1

Example. Simplify: i 15 = i 12 x i 3 = 4i 25

Assignment Page 61 #1-41 odd