Johann Carl Friedrich Gauss

1. Johann Carl Friedrich Gauss 1777 – 1855 Germany

2. Johann Carl Friedrich Gauss • Geodesy • Geophysics • Electrostatics • Astronomy • Optics • Number Theory • Algebra • Statistics • Analysis • Differential Geometry 1777 – 1855 Germany

3. Disquisitiones Arithmeticae -1801

4. Disquisitiones Arithmeticae -1801 • Division of a circle of polygon was dependant on Higher Arithmetic. • 17-Gon Construction with compass and ruler

5. Disquisitiones Arithmeticae -1801 • Division of a circle of polygon was dependant on Higher Arithmetic. • 17-Gon Construction with compass and ruler • Systematic Introduction to Modular Arithmetic • Similar to how Euclid broke down the Elements

6. Disquisitiones Arithmeticae -1801 • Fundamental Theorem of Algebra • Every Non-Constant Single Variable Polynomial with Complex Coefficients had as least one Complex Root

7. Disquisitiones Arithmeticae -1801 • Fundamental Theorem of Algebra • Every Non-Constant Single Variable Polynomial with Complex Coefficients had as least one Complex Root • Quadratic Reciprocity • If p = 1 (mod4) or q = 1 (mod4) then p/q = q/p • If p,q = 3 (mod4) then p/q = -p/q

8. 17-Gon

9. 17-Gon PREREQUISITES: • Euclidean Geometric Constructions • Doubling sides technique.

10. 17-Gon PREREQUISITES: • Euclidean Geometric Constructions • Doubling sides technique. • Imaginary Plane – Caspar Wessel • z = x + iy • z = r (cos θ + i sin θ)

11. 17-Gon PREREQUISITES: • Euclidean Geometric Constructions • Doubling sides technique. • Imaginary Plane – Caspar Wessel • z = x + iy • z = r (cos θ + i sin θ) • Vertices of the n-gon is a root of unity. • An nth root of unity is any complex number s.t. z^n = 1 • (Primitive root if no other roots z^k = 1 exist where k < n)