Geometric Representation

Complex Analysis · Axiom Academy

LESSON Geometric Representation Every complex number is a point — and an arrow — in the plane. Adding them, scaling them, and measuring them all become geometry. Take z = 3 + 2i . Go 3 to the right (the real part) and 2 up (the imaginary part), and you land on the point (3, 2) . That same number is also the arrow from the origin to that point — a position vector . The arrow's length is the modulus |z| . the point: real part across, imaginary part up the length of the arrow is the modulus 2. Adding Is Just Stacking Arrows The vector view makes addition obvious. Draw z_1 = 3 + 2i as an arrow, then move z_2 = 1 + 3i so its tail starts at the tip of z_1 . Where the second arrow ends is the sum — and the four arrows close into a parallelogram . The components simply add on their own: 3 + 1 across, 2 + 3 up. 3 + 1 = 4 — the horizontal reach of the two arrows just stacks up. 2 + 3 = 5 — the vertical climb of the two arrows stacks the same way. Slide z_2 to the tip of z_1 ; its new tip is the sum z_1 + z_2 . z_1 , z_2 , and the sum frame a parallelogram — order doesn't matter. Because subtraction is adding the opposite arrow, |z_1 - z_2| is the length of the arrow between the two points — the distance between two complex numbers, read straight off the plane. 3. A Real Multiplier Just Stretches It

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