Monthly Archives: June 2014

Math from scratch

Math from scratch!

Math differs from all other pure sciences in the sense that it can be developed from *scratch*. In math jargon, it is called “to develop from first principles”. You might see such questions in Calculus and also Physics.

This is called Axiomatic-Deductive Logic and was first seen in the works of Euclid’s Elements (Plane Geometry) about 2500 years ago. The ability to think from “first principles” can be developed in high-school with proper understanding and practise of Euclid’s Geometry.

For example, as a child Albert Einstein was captivated to see a proof from *scratch* in Euclid’s Geometry that “the three medians of a triangle are concurrent”. (This, of course, does not need anyone to *verify* by drawing thousands of triangles and their mediansJ) That hooked the child Albert Einstein to Math, and later on to Physics.

An axiom is a statement which means “self-evident truth”. We accept axioms at face-value. So, there are axioms, definitions, propositions, lemmas, theorems and corollaries.

Euclid’s geometry rests on the following fundamental axioms:

1)      There can be one and only one straight line joining two given points.

2)      (a) If O is a point in a straight line AB, then a line OC, which turns about O from the position OA to the  position OB must pass through one position, and only one,, in which it is perpendicular to AB.

(b) All right angles are equal.

3) (a) If a point O moves from A to B along the straight line AB, it must pass through one         position in which it divides AB into two equal parts.

3)(b) If a line  OP, revolving about O, turns from OA to OB, it must pass through one position in which it divides the angle AOB into two equal parts.

4) Magnitudes which can be made to coincide with one another are equal.

5) Playfair’s Axiom: Through a given point, there can be only one straight line parallel to a given straight line.

More later…

-Nalin