Egypt : 3000B.C.
– Positional number system, base 10
– Addition, multiplication, division. Fractions.
– Complicated formalism; limited algebra.
– Only perfect squares (no irrational numbers).
Babylon : 1700‐300B.C.
– Positional number system (base 60; sexagesimal)
– Addition, multiplication, division. Fractions.
– Solved systems of equations with many unknowns
– No negative numbers. No geometry.
– Squares, cubes, square roots, cube roots
– Solve quadratic equations (but no quadratic formula)
– Uses: Building, planning, selling, astronomy (later)
Greece : 600B.C. – 600A.D. (Papyrus created)
– Pythagoras : mathematics as abstract concepts,
properties of numbers, irrationality of √2,
Pythagorean Theorem a²+b²=c², geometric areas
– Zeno paradoxes : infinite sum of numbers is finite
– Constructions with ruler and compass; ‘Squaring
the circle’, ‘Doubling the cube’, ‘Trisecting the
angle’
– Plato : plane and solid geometry
Greece : 600B.C. – 600A.D.
– Aristotle : mathematics and the physical world (astronomy, geography,
mechanics), mathematical formalism (definitions, axioms, proofs via
construction)
– Euclid : Elements–13 books. Geometry, algebra, theory of numbers
(prime and composite numbers, irrationals), method of exhaustion
(calculus), Euclid’s Algorithm for finding greatest common divisor, proof
that there are infinitely many prime numbers, Fundamental Theorem of
Arithmetic(all integers can be written as a product of prime numbers)
– Apollonius : conic sections
– Archimedes : surface area and volume, centre of gravity, hydrostatics
– Hipparchus and Ptolemy : Trigonometry (circle has 360°, sin, cos, tan;
sin² + cos² =1), the Almagest (astronomy; spherical trigonometry).
– Diophantus : introduction of symbolism in algebra, solves polynomial
equations
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