Fermat`s two square theorem for rationals
... (provided they exist). Don’t we all know it! An explanation for the widespread inability to see the obvious that we are witnessing here is perhaps provided by the introduction to another (very long) answer: In his early days, Fermat realized that a natural number that can be written as a sum of two ...
... (provided they exist). Don’t we all know it! An explanation for the widespread inability to see the obvious that we are witnessing here is perhaps provided by the introduction to another (very long) answer: In his early days, Fermat realized that a natural number that can be written as a sum of two ...
INTRODUCTION TO LOGIC Lecture 6 Natural Deduction Proofs in
... P a → Qa ∀z (Qz → Rz) Qa Qa → Ra ...
... P a → Qa ∀z (Qz → Rz) Qa Qa → Ra ...
equivalents of the compactness theorem for locally finite sets of
... The above could be seen as a variant of Fn ↔ F, n ≥ 3, proved by Levy [3] and Mycielski [5]. Let us, however, note that the arguments are quite different. As an immediate corollary of the above considerations we also obtain ACn → | Fnf in , for n ≥ 3. In case of AC2 → F2f in and ACf in → Ff in the p ...
... The above could be seen as a variant of Fn ↔ F, n ≥ 3, proved by Levy [3] and Mycielski [5]. Let us, however, note that the arguments are quite different. As an immediate corollary of the above considerations we also obtain ACn → | Fnf in , for n ≥ 3. In case of AC2 → F2f in and ACf in → Ff in the p ...
Continued fractions in p-adic numbers
... Suppose that there exist infinitely many n such that (qn , bn ) 6= (p − 1, 1). Then the continued fraction (5.1) converges to an irrational p-adic number. Conversely, for any irrational p-adic number α, there exist unique sequences {qn } and {bn } with qn ∈ S for n ≥ 1, b1 ∈ Z, bn ∈ N for n ≥ 2 and ...
... Suppose that there exist infinitely many n such that (qn , bn ) 6= (p − 1, 1). Then the continued fraction (5.1) converges to an irrational p-adic number. Conversely, for any irrational p-adic number α, there exist unique sequences {qn } and {bn } with qn ∈ S for n ≥ 1, b1 ∈ Z, bn ∈ N for n ≥ 2 and ...
BSc Chemistry - e
... population mean µ with probability 1. Thus law of large numbers are said to be strong because it guarantees stable long term results of averages of random events. This is certainly a strong mathematical statement. ...
... population mean µ with probability 1. Thus law of large numbers are said to be strong because it guarantees stable long term results of averages of random events. This is certainly a strong mathematical statement. ...
Predicate Calculus - SIUE Computer Science
... In the predicate calculus, it is not possible to use truth tables to prove most results since statements depend on one or more variables. This makes the job of proving results quite a bit more difficult. Would it be possible to use truth tables if the domain(s) of the variable(s) are finite? Logical ...
... In the predicate calculus, it is not possible to use truth tables to prove most results since statements depend on one or more variables. This makes the job of proving results quite a bit more difficult. Would it be possible to use truth tables if the domain(s) of the variable(s) are finite? Logical ...
Irrationality of Square Roots - Mathematical Association of America
... where c0 , . . . , cn−1 are integers, would have to be ≥ 1/q n−1 . We can construct arbitrarily small positive expressions of the form (3) by expanding (α − α)k in powers of α and eliminating terms with exponents ≥ n by repeated use of α n = −an−1 α n−1 − · · · − a0 . We conclude that α is irratio ...
... where c0 , . . . , cn−1 are integers, would have to be ≥ 1/q n−1 . We can construct arbitrarily small positive expressions of the form (3) by expanding (α − α)k in powers of α and eliminating terms with exponents ≥ n by repeated use of α n = −an−1 α n−1 − · · · − a0 . We conclude that α is irratio ...
p q
... Axioms (公設) An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: Example: In Euclidean geometry the following are axioms Given two distinct points, there is exactly one line that contains them. Given a line and a ...
... Axioms (公設) An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: Example: In Euclidean geometry the following are axioms Given two distinct points, there is exactly one line that contains them. Given a line and a ...
report
... A sequence of real numbers is said to be equidistributed if the quantity of terms which fall within an interval is proportional only to the length of the interval. Weyl’s Equidistribution theorem defines a class of such sequences: the fractional parts of integer multiples of irrational numbers. Equi ...
... A sequence of real numbers is said to be equidistributed if the quantity of terms which fall within an interval is proportional only to the length of the interval. Weyl’s Equidistribution theorem defines a class of such sequences: the fractional parts of integer multiples of irrational numbers. Equi ...
8-4 Similarity in Right Triangles M11.C.1 2.2.11.A
... 8-4 Similarity in Right Triangles M11.C.1 2.2.11.A OBJECTIVES: 1) TO FIND AND USE RELATIONSHIPS IN SIMILAR RIGHT TRIANGLES. ...
... 8-4 Similarity in Right Triangles M11.C.1 2.2.11.A OBJECTIVES: 1) TO FIND AND USE RELATIONSHIPS IN SIMILAR RIGHT TRIANGLES. ...
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.