Draw complex arithmetic and midpoints

Try

Plot z=1+2i and w=−2+i. The horizontal axis records real part and the vertical axis imaginary part, with equal units. Their sum is −1+3i. To draw it, place an arrow representing w at the tip of the arrow representing z. The final tip is (−1,3), and the arrow from the origin to that tip represents z+w.

For subtraction, ask a different question: what displacement takes the tip of w to the tip of z? It is (3,1), representing z−w=3+i. The arrow order matters.

Predict

Predict the locations of z̄ and iz for this z. Conjugation reverses the vertical coordinate, giving (1,−2). Multiplication by i gives i−2, or (−2,1). One operation is a reflection and the other a quarter-turn; they are not interchangeable.

Predict the midpoint of z and w. An average of both coordinates gives (−1/2,3/2). Equal distances to the two endpoints are necessary but not sufficient for a midpoint: many points on the perpendicular bisector have equal distances. You must also establish that the point lies on the joining segment.

Experiment

Inspect the construction below, then reproduce it with labeled coordinates or an exact list of points and directed segments. In either format specify the real and imaginary axes, the shared scale, and the direction of each arrow.

Now construct z+w separately. Do not reuse the subtraction arrow and merely change its label. Check the endpoint by component arithmetic.

Understand

For z=a+bi and w=c+di, the displacement from z to w has components (c−a,d−b). The distance is √((c−a)²+(d−b)²)=|w−z|. Traveling half that displacement from z gives ((a+c)/2,(b+d)/2), so the average (z+w)/2 lies on the segment and has equal half-lengths. This proves the midpoint rule generally.

Conjugation sends (a,b) to (a,−b), reflection across the real axis. Multiplication by i sends it to (−b,a), a 90° counterclockwise rotation. Both preserve a²+b². Multiplication by a positive real r scales both coordinates and the modulus by r. Together these calculations give geometric interpretations of multiplication without requiring advanced complex analysis.

A polar description uses nonnegative distance r and a direction angle θ: the coordinates are (r cosθ,r sinθ). For the nonzero point √3+i, r=2 and θ=30° give exactly those coordinates. The zero number has r=0 but no distinguished direction. Directions differing by a full turn describe the same nonzero point.

Apply

Create a fresh subtraction construction in practice and record both methods: the tip-to-tip arrow from the subtracted number to the first number, and addition of its negative at the first tip. State the final displacement and compare it with algebra. A drawing rubric checks axis labels, scale, point locations and arrow direction; an attractive picture alone is not sufficient.

For a coordinate-only response, list each endpoint and the operation that creates it. This is an accessible geometric construction, not an exemption from showing the geometry.

Continue

These constructions address N-CN.B.5–6, with a rectangular/polar check supporting B.4. For further polar powers, use the existing complex polar form lesson.

Next, use conjugate pairs to factor quadratic polynomials. Geometric locations and polynomial roots are related representations, but proving an exact factorization requires multiplying the factors, not judging the drawing.