Multiplication of complex numbers When multiplying complex numbers, we use the general algebraic method for multiplication. Example 1: Consider the complex numbers and where = 3 + 2i and = 5–4i. z w, z w Then = (3 + 2i)(5– 4i) zw = 3(5 4i) + 2i(5 4i)………[Use distributive property] – – = 15 12i + 8i2 each bracket] 10i – – ……[Expand = 15 2i 8(–1) Note that = i2 – – ……..[Simplify. –1] = 15 2i + 8………….[Simplify. (–8)(–1) = + 8]] – = 23 –2i …………..[Simplify] Simplify ( 5 and give your answer in the form + where and Example 2: i)2 a bi, a b – are real numbers. Solution: (5 = (5 i)2 i)(5 i) – – – = 5 ( 5 the distributive property] i) i(5 i)…….[Use – – – = 25 5i 5i + i2 – – ……..[Expand] = 25 10i + (–1) like terms. Note the value of = i2 – ……..[Add –1] = 25 1- 10i – = 24 - 10i …………..[Simplify] Challenge to the reader. Determine the product of: ( 3 + 4i)(2 5i)(1 2i) – – Explain your method to your colleague. Activity 5 Simplify each of the following expressions. Write your answer in the form a + bi, where and are real numbers. a b a) (3 + + 3i) b) (4 2i)(5 + 3i) i)(2 – c) (8 + 2i) d) (–3i + 9)(5 i)(9 i) – – e) 3i)(i + 4) f) (3 2i)(7–5i) i(2 – – Simplify each of the following expressions, giving your answer in the form a + bi, where and are real numbers. a b
Multiplication of complex numbers When multiplying complex numbers, we use the general algebraic method for multiplication. Example 1: Consider the complex numbers z and w, where z = 3 + 2i and w = 5–4i. Then zw = (3 + 2i)(5– 4i) = 3(5 – 4i) + 2i(5 – 4i)………[Use distributive property] = 15 – 12i + 10i – 8i2 ……[Expand each bracket] = 15 – 2i – 8(–1) ……..[Simplify. Note that i2 = –1] = 15 – 2i + 8………….[Simplify. (–8)(–1) = + 8]] = 23 –2i …………..[Simplify] Example 2: Simplify ( 5 – i)2 and give your answer in the form a + bi, where a and b are real numbers. Solution: (5 – i)2 = (5 – i)(5 – i) = 5 ( 5 – i) – i(5 – i)…….[Use the distributive property] = 25 – 5i – 5i + i2 ……..[Expand] = 25 – 10i + (–1) ……..[Add like terms. Note the value of i2 = –1] = 25 – 1- 10i = 24 - 10i …………..[Simplify] Challenge to the reader. Determine the product of: ( 3 + 4i)(2 – 5i)(1 – 2i) Explain your method to your colleague. Activity 5 Simplify each of the following expressions. Write your answer in the form a + bi, where a and b are real numbers. a) (3 + i)(2 + 3i) b) (4 – 2i)(5 + 3i) c) (8 – i)(9 + 2i) d) (–3i + 9)(5 – i) e) i(2 – 3i)(i + 4) f) (3 – 2i)(7–5i) Simplify each of the following expressions, giving your answer in the form a + bi, where a and b are real numbers.
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