The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Mathematical Background Outline

  1. Polynomials
    1. Terminology: coefficients, terms, factors, degree
    2. Relating roots and coefficients of quadratic polynomials
    3. Average of roots and average of roots of derivative are the same
  2. Complex Numbers
    1. Basic definitions and properties
    2. Polar form and geometric interpretations: writing any complex number as x + iy = reiq; multiplying and adding complex numbers geometrically
    3. Geometric condition for zw to be a negative real, where z and w are complex numbers
  3. Linear and geometric transformations of the plane
    1. Translation, rotation, and scaling figures
    2. Representing transformations with matrices
    3. Shears and one directional scaling
    4. Properties of linear transformations
  4. Ellipses
    1. Overview concepts of ellipses and conic sections
    2. Definition of ellipse as a deformed circle
    3. Standard equation for an ellipse: 
    4. Equivalent definition: set of points the sum of whose distances from two given points is fixed
    5. Linear transformations take ellipses to ellipses
    6. First uniqueness property: Given two points E and F and a point P not on the segment between E and F, there is a unique ellipse through P having foci at E and F
    7. Second uniqueness property: Given two points E and F and a line L not intersecting the segment between E and F, there is a unique ellipse tangent to L having foci at E and F
    8. Optical Property: At any point P of an ellipse, the lines to P from the foci make equal acute angles with the tangent to the ellipse at P.
    9. Extended Optical Property: At any point P outside of an ellipse, the lines to P from the foci make equal acute angles with the two tangent lines to the ellipse from the point P.
  5. Every triangle has a unique inscribed ellipse, tangent at midpoints