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Logical Structure, Definitions, Axioms and Proofs

This level of abstraction occurred in ancient Greece in the 6th century B.C.

Thales contributed geometric proofs. The Pythagoras school gave us the Pythagorean thereom, which relates to trigonometry. This theorem states that for every right-angled triangle, with sides A and B and hypotenuse C, there is a relationship between the two sides and the hypotenuse, such that: A2+B2=C2.

Proof:

pythagorasproof


Let the red triangle be the right-angled triangle. The yellow triangles each have area AB/2, and the blue square has area (B-A)2. Thus C2=2AB+(B-A)2=A2+B2.

Early Signs of Mathematics

Primitive counting is developed. Base 10 is the most common, probably because of number of fingers on both hands.

Geometric concepts are exhibited in prehistoric pottery, textiles and cave paintings.


Organized Math Concepts

Earliest records in Babylonia and Egypt around 3000 B.C. It mainly consists of arithmetic with an emphasis on measurement and calculation.


Irrational Numbers Are Discovered

Unknown mathematician proves that there cannot be a whole number or unit measurement for the sides and the diagonal at the same time.

Eudoxus expands on this idea to develop approximation techniques for areas and volumes.


Euclid's Elements

Euclid's Elements are published by this famous scribe at the Alexandria library in Egypt. They contain the foundations for modern geometry.


Archimedes

Expands on Eudoxus' work of volume approximations by extending them to conics.

Studies center of gravity for objects and levels in water.


Conics Book Published

Appollonious writes a book on conic sections. Conic sections are created by cross sectioning an hourglass shape with a plane, like a parabola, hyperbola and circle.


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