BEAUTIFUL THEOREMS OF GEOMETRY AS VAN AUBEL`S
... The intersection of P R with QS, denoted F , differs in general from M . Now, P R and QS have equal length and are perpendicular, but by what ratio does their intersection point F divide the line segments? In order to find out, the center points of each square may be connected to form P Q, QR, RS an ...
... The intersection of P R with QS, denoted F , differs in general from M . Now, P R and QS have equal length and are perpendicular, but by what ratio does their intersection point F divide the line segments? In order to find out, the center points of each square may be connected to form P Q, QR, RS an ...
Full version - Villanova Computer Science
... particles in the Universe! Deductive methods (the following slides) are more practical, even though, generally, we have: ...
... particles in the Universe! Deductive methods (the following slides) are more practical, even though, generally, we have: ...
Oh Yeah? Well, Prove It.
... Proof: If a is an even integer, then a can be written as an integer multiple of 2 (this is the definition of an even integer). Thus, we can write a = 2m for some integer m. Similarly, since b is even, we can write b = 2n for some integer n (not necessarily the same integer as we used to write a, of ...
... Proof: If a is an even integer, then a can be written as an integer multiple of 2 (this is the definition of an even integer). Thus, we can write a = 2m for some integer m. Similarly, since b is even, we can write b = 2n for some integer n (not necessarily the same integer as we used to write a, of ...
4-3: Alternating Series, and the Alternating Series Theorem
... • (∗) A remarkable theorem of Cauchy shows that if you have a conditionally convergent series, then I can add all the terms up in a different order to obtain any number I want in the limit. Obviously this contrasts to what happens in the case of summing a finite number of numbers. – The general proo ...
... • (∗) A remarkable theorem of Cauchy shows that if you have a conditionally convergent series, then I can add all the terms up in a different order to obtain any number I want in the limit. Obviously this contrasts to what happens in the case of summing a finite number of numbers. – The general proo ...
Arithmetic in Metamath, Case Study: Bertrand`s Postulate
... (in addition to increasing readability). The specific choice of 4 = 22 also works well with algorithms like exponentiation by squaring, used in the calculation of xk mod n in primality tests (see Section 4). Remark 1. It is important to recognize that “n is a numeral” is not a statement of the objec ...
... (in addition to increasing readability). The specific choice of 4 = 22 also works well with algorithms like exponentiation by squaring, used in the calculation of xk mod n in primality tests (see Section 4). Remark 1. It is important to recognize that “n is a numeral” is not a statement of the objec ...
Writing Proofs
... statement B is false.) Let’s call these finitely many prime numbers p1 , p2 , . . . , pN . Now consider the number m = p1 p2 · · · pN , the product of all of those prime numbers. This number m is evenly divisible by each of p1 , p2 , . . . , pN . Therefore, by Theorem 2, m + 1 is NOT divisible by an ...
... statement B is false.) Let’s call these finitely many prime numbers p1 , p2 , . . . , pN . Now consider the number m = p1 p2 · · · pN , the product of all of those prime numbers. This number m is evenly divisible by each of p1 , p2 , . . . , pN . Therefore, by Theorem 2, m + 1 is NOT divisible by an ...
Math 245 - Cuyamaca College
... Without the following skills, competencies and/or knowledge, students entering this course will be highly unlikely to succeed: 1) Identify the properties of the integers including commutativity, associativity, identities and inverses. 2) Understand simple mathematical proofs. 3) Solve polynomial equ ...
... Without the following skills, competencies and/or knowledge, students entering this course will be highly unlikely to succeed: 1) Identify the properties of the integers including commutativity, associativity, identities and inverses. 2) Understand simple mathematical proofs. 3) Solve polynomial equ ...
Review guide for Exam 2
... This guide will follow in a similar vein as the previous guide. As with the previous exam, material will be divided between “proofs” (all taken from problems on this sheet) and more explicit examples. Congruences are the largest part of this exam, so by the nature of the material there will be more ...
... This guide will follow in a similar vein as the previous guide. As with the previous exam, material will be divided between “proofs” (all taken from problems on this sheet) and more explicit examples. Congruences are the largest part of this exam, so by the nature of the material there will be more ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.