Rational coefficients are numbers that can be expressed as the quotient of two integers, where the denominator is not zero, and are used in algebraic expressions and equations to allow for precise manipulation and solution. They are essential in polynomial equations, enabling operations such as addition, subtraction, multiplication, and division to be performed while maintaining the rationality of the coefficients.
Constructible numbers are those that can be obtained using a finite number of operations involving addition, subtraction, multiplication, division, and square root extractions, starting from a given set of numbers, typically the integers. They are intimately connected to classical problems of geometry, such as trisecting an angle or doubling a cube, which can be reduced to questions about the constructibility of certain numbers.
Compass and straightedge constructions are classical methods in geometry used to create various geometric figures using only an unmarked straightedge and a compass. These constructions are governed by strict rules that allow for the drawing of lines, circles, and points of intersection, leading to solutions for problems such as bisecting angles, constructing perpendiculars, and duplicating segments.