Circles and Parametrization The blog post about circles, trigonometry, and groups defined a circle to be the set of points in a plane lying at fixed radius \(r\) from a point, the center. In rectangular coordinates, if \(c = (x_{0}, y_{0})\) is the center, then a point \((x, y)\) lies on the circle of radius…
Tag: calculus
Polar Coordinates and Polar Graphs
Polar Coordinates Mathematicians translate between geometry and algebra using “coordinates”, assignments of ordered lists of numbers to geometric locations. To define rectangular coordinates in the plane, see our blog post on functions, coordinates, and graphs, we fix perpendicular “axes”: a horizontal and a vertical number line meeting at a point called the “origin”, let \(x\) denote…
Complex Numbers
Formally, a complex number is an expression \(a + bi\), with \(a\) and \(b\) real numbers and \(i\) an entity satisfying \(i^{2} = -1\). The square of an arbitrary real number is non-negative, so whatever \(i\) may be, it is not a real number. Complex Addition and Multiplication Suppose \(z = 3 + 4i\) and \(w =…
Functions, Coordinates, and Graphs
Functions In mathematics, a function represents a causal relationship between “input” and “output.” For example, the area \(A\) of a square of side length \(s\) is given by the formula \(A = s^{2}\), read “\(A\) equals \(s\) squared.” If \(s\) is known, the area is determined. The name of the algebraic operation, squaring, matches its…