The plane that associates a complex number z = a + bi with the point (a, b) is called the complex plane (Gaussian plane).
・Real axis: the axis of a ・Imaginary axis: the axis of b
Absolute value: |z| = √(a²+b²) (the distance from the origin)
Complex conjugate: z̄ = a−bi (reflection across the real axis)
Example ① Find the absolute value and complex conjugate of z = 3 + 4i.
Solution:|z| = √(9+16) = 5, z̄ = 3 − 4i
Solution:|z| = √(9+16) = 5, z̄ = 3 − 4i
Key Points
- z·z̄ = |z|² (which is a real number)
- Operations with conjugates: (z+w)̄ = z̄+w̄, (zw)̄ = z̄·w̄
Practice Problems
- Find the absolute value of z = 1 − √3 i.
Solution and Explanation
- Answer:2