Definitions in Mathematics

Introduction

Definitions per se are not something specific to the field of mathematics, but are rather prevalent in almost all subjects (for example English Grammar). An obvious need for definitions is to express a new term using already existing vocabulary. In mathematics however they have much more importance than in any other subject. Definitions in mathematics are rather precise and clearly express what a thing is or is not. From now on we will restrict ourselves to the form of definition used in mathematics.

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Conics and the Cone: Part 3

After having dealt with the case of circular sections of an oblique cone in previous post, let's now focus on the conics. We will first treat the case of a parabola as this is the simplest case after the case of a circular section.

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Conics and the Cone: Part 2

In the previous post we established that sections of a right circular cone are the familiar curves (ellipse, parabola, hyperbola) having the focus directrix property. Now we will have a look at the more general case when the cone is not right but oblique. Our approach will be identical to the one followed by the great Greek geometer Apollonius. However we will not be developing a systematic theory of conics as described by Apollonius, but rather focus on the interesting results which will help us to connect them with the modern definition of conics. In doing so we will observe that the main tool used by Apollonius is the similarity of triangles.

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Conics and the Cone: Part 1

Introduction

While studying co-ordinate geometry (aka analytic geometry) in intermediate classes we normally arrive at the study of conic sections or in short "conics". Three new curves namely "ellipse", "parabola", and "hyperbola" come into picture and their theory is quite unlike those of the elementary geometrical objects (line, triangle, circle etc) studied in secondary classes. In case of the elementary geometrical objects like points, lines, triangles, circles we have two approaches: 1) using the axioms of Euclid and then deducing the properties of these objects logically from Euclid's axioms and, 2) using the language of coordinate geometry which transforms the subject of geometry into algebra. Unfortunately the beautiful approach of using Euclid's axioms is discarded in higher secondary classes in favor of the approach using coordinate geometry which makes the subject dull with huge amount of laborious algebraical manipulations. In fact most students think that the only way to study new curves like ellipses, parabola and hyperbola is through coordinate geometry.

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