8.3 polar form and demoivre`s theorem
... The polar form adapts nicely to multiplication and division of complex numbers. Suppose we are given two complex numbers in polar form z1 5 r1 scosu1 1 i sinu1d and z 2 5 r2 scosu2 1 i sinu2d. Then the product of z1 and z 2 is given by z1 z 2 5 r1 r2 scosu1 1 i sinu1dscosu2 1 i sinu2d 5 r1r2 fscos u ...
... The polar form adapts nicely to multiplication and division of complex numbers. Suppose we are given two complex numbers in polar form z1 5 r1 scosu1 1 i sinu1d and z 2 5 r2 scosu2 1 i sinu2d. Then the product of z1 and z 2 is given by z1 z 2 5 r1 r2 scosu1 1 i sinu1dscosu2 1 i sinu2d 5 r1r2 fscos u ...
generatingfunctionology - Penn Math
... arise from recurrence formulas. Sometimes, however, from the generating function you will find a new recurrence formula, not the one you started with, that gives new insights into the nature of your sequence. (c) Find averages and other statistical properties of your sequence. Generating functions c ...
... arise from recurrence formulas. Sometimes, however, from the generating function you will find a new recurrence formula, not the one you started with, that gives new insights into the nature of your sequence. (c) Find averages and other statistical properties of your sequence. Generating functions c ...
Writing a Mizar article in nine easy steps
... generally Mizar errors are quite obvious. Exercise 1.2.1 The minimal Mizar article that we showed is 2 lines long. What is the shortest article in the MML? What is the longest? What is the average number of lines in the articles in the MML? As a rule of thumb an article less than a 1000 lines is con ...
... generally Mizar errors are quite obvious. Exercise 1.2.1 The minimal Mizar article that we showed is 2 lines long. What is the shortest article in the MML? What is the longest? What is the average number of lines in the articles in the MML? As a rule of thumb an article less than a 1000 lines is con ...
Lectures on Number Theory
... latter case is excluded, we conclude that r − r0 = 0, that is r = r0 . Therefore a(q − q 0 ) = 0, which implies q − q 0 = 0, i.e. q = q 0 . More generally, we say that r0 is a remainder when b is divided by a whenever there is an integer q 0 such that b = aq 0 + r0 without any further restriction on ...
... latter case is excluded, we conclude that r − r0 = 0, that is r = r0 . Therefore a(q − q 0 ) = 0, which implies q − q 0 = 0, i.e. q = q 0 . More generally, we say that r0 is a remainder when b is divided by a whenever there is an integer q 0 such that b = aq 0 + r0 without any further restriction on ...
Elementary Number Theory, A Computational Approach
... For us an algorithm is a finite sequence of instructions that can be followed to perform a specific task, such as a sequence of instructions in a computer program, which must terminate on any valid input. The word “algorithm” is sometimes used more loosely (and sometimes more precisely) than defined ...
... For us an algorithm is a finite sequence of instructions that can be followed to perform a specific task, such as a sequence of instructions in a computer program, which must terminate on any valid input. The word “algorithm” is sometimes used more loosely (and sometimes more precisely) than defined ...
n - Washington University in St. Louis
... These slides are partly based on Lawrie Brown’s slides supplied with William Stallings’s ...
... These slides are partly based on Lawrie Brown’s slides supplied with William Stallings’s ...
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.