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The book is organized into three sections. [2] [3] The first section provides introductory material, describing different mathematical situations in which multiple curves might meet, and providing different possible explanations for this phenomenon, including symmetry, geometric transformations, and membership of the curves in a pencil of curves. [4]
The formula above is a geometric series—each successive term is one fourth of the previous term. In modern mathematics, that formula is a special case of the sum formula for a geometric series. Archimedes evaluates the sum using an entirely geometric method, [8] illustrated in the adjacent picture. This picture shows a unit square which has ...
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases ...
The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. [1]
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
For example, one can teach the method of completing the square by using algebra tiles. Cuisenaire rods can be used to teach fractions, and pattern blocks can be used to teach geometry. Using mathematical manipulatives helps students gain a conceptual understanding that might not be seen immediately in written mathematical formulas. [15]
The object of thought is deductive reasoning (simple proofs), which the student learns to combine to form a system of formal proofs (Euclidean geometry). Learners can construct geometric proofs at a secondary school level and understand their meaning. They understand the role of undefined terms, definitions, axioms and theorems in Euclidean ...
For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x 2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just ...