A real number s that is irrational is called algebraic if it is the root of a polynomial equation with integer coefficents. For example, s = v2 is alge- braic since it is a root of r2-2 0. Numbers that are irrational but not algebraic are called transcendental. (a) Show that the set of all algebraic numbers is countable. Hint: show that there are countably many polynomials with integer coeffi- cients. You may assume that a polynomial of degree n has n roots. You may also assume class theorems, and that if A is a countable set, then for n > 1, the set of n-tuples of elements of A, A" = {(a1, a2, .., a„) : each a, E A} is also countable. (b) Show that the set of all transcendental numbers is uncountable.

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These…

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are…

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) ge…

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.