Figure 8-1: Graph of |z| < r
While complex numbers cannot be ordered in the same manner as real numbers; their absolute values can be ordered. The absolute value, or modulus, of a complex number can be interpreted as the distance to the origin in the complex plane. The absolute value of the complex number,
is defined as:z = x + yi,
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|z| = |x + yi| = √ x2 + y2
The distance between z and a point p in the complex plane is |z - p|.
If, p = a + bi, then:
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|z - p| = √(x - a)2 + (y - b)2
Example, solve: |z + 2 - i| = 2
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|z + 2 - i| = √(x + 2)2 + (y - 1)2 = 2
(x + 2)2 + (y - 1)2 = 4
This would be a circle in the complex plane with radius 2 centered at point:
-2 + i
|z·w| = |z||w| |z/w| = |z|/|w| |z + w| ≤ |z| + |w| triangle inequality
Figure 8-1: Graph of |z| < r
Similarly, the inequality, |z| > r, would be satisfied by all points that are greater than r units from the origin. The inequality, |z - p| < r, would be satisfied by all points that are less than r units from the point p.
Example, solve: |z - 1| < 1
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|z - 1| = √(x - 1)2 + y2 < 1
The inequality is true for all points inside the circle with radius 1 and center at point:
1 + 0i