MATHEMATICAL SYSTEM

MATHEMATICS ONLINE CLASS

Ma'am Lorena S. Maglonzo

MATERIALS NEEDED

• Ballpen

• Paper

• Module 1

REVIEW

Undefined Terms

Multiple Select

Which of the following are undefined terms? Choose 3.

• polygons

• shapes

• lines

• planes

• points

Defined Terms

• Collinear Points

• Coplanar Points

• Intersecting Lines

• Perpendicular Lines

• Intersecting Planes

• Skew Lines

• Parallel Lines and Parallel Planes

MATHEMATICAL SYSTEMS

Mathematical System is a structure formed from one or more sets of undefined objects, various concepts that may or may not be defined, and a set of axioms relating these objects and concepts. (McGraw-Hill Dictionary)

UNDEFINED TERMS

These are mathematical terms which cannot be precisely defined.

They can only be explained using examples and descriptions. In geometry, these are lines, points, and planes.

DEFINITION

Using the undefined terms, other terms can be generated and define precisely. For example, angles, line segments, circles, etc. A good definition must possess four important characteristics:

1) The term being defined is named.

2) The term being described is placed under a larger category.

3) The distinguishing quality is included.

4) It is reversible.

AXIOMS and POSTULATES

These refer to statements or propositions which are regarded as being established, accepted, or self-evidently true.

THEOREMS

These are statements that we can arrive at by successive application of the rule of implication to the axioms and postulates. These are statements that can be proven to be true.

COROLLARY

A proposition that follows from (and is often appended to) one already proven.

LEMMA

This is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".

EUCLID

“Father of Geometry”. He had written the most influential book in geometry entitled the Elements. This presented a set of formal logical arguments based on a few basic terms and axioms, provided a systematic method of rational exploration that guided mathematicians, philosophers, and scientists well into the 19th century.

EUCLID'S DEFINITION

• • A point is that which has no part.

• A line is breathless length.

• The ends of a line segment are points.

• A straight line extends indefinitely in both the direction.

• A surface is that which has length and breadth only.

• A plane surface is a surface which lies evenly with the straight lines on itself.

EUCLID'S AXIOM

• Things which are equal to the same thing are equal to one another.

• If equals are added to equals, the wholes are equal.

• If equals are subtracted from equals, then the remainders are equal.

• Things which coincide with one another are equal.

• The whole is greater than the part.

• Things which are double of the same things are equal.

• Things which are halves of the same things are equal.

EUCLID'S POSTULATES

Postulate 1: A straight line may be drawn from any point to any other point.

Postulate 2: A terminated line can be produced indefinitely.

Postulate 3: A circle can be drawn with any center and any radius.

Postulate 4: All right angles are equal to one another.

Postulate 5: If a straight line falling into straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

ASSESSMENT

PAGES 12-14