Tangential to CSC 253/453 on software design but fun to explain when there is enough time in a lecture are these two math problems which are general composite properties that can be proved easily using simple building blocks.

Theorem: If a complex value is a root of a polynomial of real-valued coefficients, so is its conjugate.

To prove this for ALL such polynomials, we need just two properties of complex number arithmetic: (1) the conjugate of the sum is the sum of the conjugates, and (2) the conjugate of the product is the product of the conjugates. These are binary operations and can be shown easily. Then for any real-valued polynomial f(x), we have f(~x) = ~f(x) = ~0 = 0, and the theorem is proved.

Theorem: In a triangle, a median is a line from an end point to the center of the opposite side. For any triangle, its three medians meet at one point which is 1/3 the way from the end point to the edge.

To prove this for ALL triangles, we use a property of a single line segment. Let P be a mid-point on the line segment P1P2. Let the (complex-plane) coordinates of P1, P2, P be x, y, z, and the ratio r = P1P / PP2, then we have z = (x + ry) / (1 + r). If we use this equation to compute the coordinate of the 1/3-way point of the three medians, we’ll see that they are identical: 1/3(x1+x2+x3), where xs are the coordinates of the three end-points of the triangle.

Source: An Imaginary Tale — The Story of i, by Paul J. Nahin, Princeton U Press, 1998.

Photo credit: AI generated by Kaave Hosseini for CSC 484 for “dimension reduction”