Adapted from Walch Education

 First, find the sum or difference of the real parts of the complex number. Then, to find the sum or difference of the imaginary numbers, add or subtract the coefficients of i. The sum or difference of two complex numbers can be wholly real (having only real parts), wholly imaginary (having only imaginary parts), or complex (having both real and imaginary parts) : Adding and Subtracting Complex Numbers2

 Is (5 + 6i 9 ) – (5 + 3i 15 ) wholly real or wholly imaginary, or does it have both a real and an imaginary part? 4.3.2: Adding and Subtracting Complex Numbers3

 Two expressions, 6i 9 and 3i 15, contain i n.  Divide each power of i by 4 and use the remainder to simplify i n.  9 ÷ 4 = 2 remainder 1, so 9 =  i 9 = i 2 4 i 1 = i  15 ÷ 4 = 3 remainder 3, so 15 =  i 15 = i 3 4 i 3 = –i 4.3.2: Adding and Subtracting Complex Numbers4

 Replace each occurrence of i n in the expressions with the simplified versions, and replace the original expressions in the difference with the simplified expressions.  6i 9 = 6 (i) = 6i  3i 15 = 3 (–i) = –3i  (5 + 6i 9 ) – (5 + 3i 15 ) = (5 + 6i) – [5 + (–3i)] 4.3.2: Adding and Subtracting Complex Numbers5

Distribute the difference through both parts of the complex number. Find the sum or difference of the real parts. 5 – 5 = 0 Find the sum or difference of the imaginary parts. 6i + 3i = 9i 4.3.2: Adding and Subtracting Complex Numbers6

Find the sum of the real and imaginary parts i = 9i Use the form of the sum to determine if it is wholly real or wholly imaginary, or if it has both a real and an imaginary part. 9i has only an imaginary part, 9i, so the difference is wholly imaginary : Adding and Subtracting Complex Numbers7

Is (12 – I 20 ) + (–18 – 4i 18 ) wholly real or wholly imaginary, or does it have both a real and an imaginary part? 4.3.2: Adding and Subtracting Complex Numbers8

~ms. dambreville