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.

Note that these are the only basic assumptions to be used in geometric constructions also with ruler and compass.

PS: article reblogged and slightly modified:

Reference: To start geometry from scratch, you can start working from the first page of a classic text ” A School Geometry” by Hall and Stevens, Metric Edition; For example, Amazon India link is:

https://www.amazon.in/School-Geometry-H-S-Hall/dp/9385923331/ref=sr_1_4?crid=6QK90ZAHHFAU&keywords=a+school+geometry+hall+and+stevens&qid=1561783333&s=books&sprefix=A+School%2Caps%2C253&sr=1-4

I would like to add a few more details as there are some students/readers who want to pursue this further. {By the way, I have used the above reference only. One more thing,…Dover publications still prints/publishes/sells the original volumes of Euclids books}:

$\textbf{Hypothetical Constructions}$

From the above axioms, it follows that we may suppose:

i) A straight line can be drawn perpendicular to a given straight line from any point in it.

ii) A finite straight line (that is, a segment) can be bisected.

iii) Any angle can be bisected by a line (we call such a line its angle bisector).

$\textbf{Superposition and Equality}$

AXIOM: Magnitudes which can be made to coincide with one another are equal.

This axiom implies that any line, angle, or figure may be taken up from its position, and without change in size or form, laid down upon a second line, angle, or figure, for the purpose of comparison, and ti states that two such magnitudes are equal when one can be exactly placed over the other without overlapping.

This process is called superposition, and the first magnitude is said to be applied to the other. (Note: this is the essence of “congruency” relation in geometry).

$\textbf{Postulates}$

In order to draw geometric figures, certain instruments are required. These are — a straight ruler, and a pair of compasses. The following postulates (or requests) claim the use of these instruments, and assume that with their help the processes mentioned below may be duly performed:

Let it be granted:

1. That a straight line may be drawn from any one point to any other point.
2.  That a finite (or terminated) straight line may be produced (that is, prolonged) to any length in that straight line.
3. That a circle may be drawn with any point as centre and with a radius of any length.
4. some notes: Postulate 3 above implies that we may adjust the compasses to the length of any straight line PQ, and with a radius of this length draw a circle with any point O as centre. That is to say, the compasses may be used to transfer distances from one part of a diagram to another.
5. Hence, from AB, (a given terminated line), the greater of two straight lines, we may cut off a part equal to PQ the less. Because, if with centre A, and radius equal to PQ, we draw an arc of a circle cutting AB at X, it is obvious that AX is equal to PQ.

More later,

Regards,

Nalin Pithwa.