Absolute Value AWS

Let 𝑧 = 𝑎 + 𝑏𝑖, then the absolute value of z is |𝑧| = √𝑎2 + 𝑏2 Theorem (Absolute Value) Let 𝑧 = 𝑎 + 𝑏𝑖 and 𝑤 = 𝑐 + 𝑑𝑖 be in ℂ, then    

|𝑧| ≥ 0 |𝑧| = 0 if and only if 𝑧 = 0 𝑧𝑧̅ = |𝑧|2 |𝑧𝑤| = |𝑧||𝑤| 𝑧

|𝑧|

 |𝑤| = |𝑤| if 𝑤 ≠ 0 1

1

 𝑧 −1 = 𝑧 = |𝑧|2 𝑧̅ (inverse of 𝑧) Example Find the absolute value of the following: a.

𝑧 = 2 − 5𝑖 |𝑧| = √(2)2 + (−5)2 = √4 + 25 = √29

b.

𝑧 = −3 |𝑧| = √(−3)2 + (0)2 = √9 = 3 (which is what we expect)

c.

𝑧 = 2𝑖 |𝑧| = √(0)2 + (2)2 = √4 = 2

Practice Problems 𝑧̅ −𝑤

1. If 𝑧 = 2 − 𝑖 and 𝑤 = 3 + 2𝑖, find |

𝑧

|.