An Introduction to Surreal Numbers
... the numbers are formed in a somewhat recursive manner we can use induction on the day that a number, or set of numbers, is created to trace properties of arbitrary numbers back to the earliest created numbers, of which we have a solid understanding. In the previous proof, we saw how for any set of n ...
... the numbers are formed in a somewhat recursive manner we can use induction on the day that a number, or set of numbers, is created to trace properties of arbitrary numbers back to the earliest created numbers, of which we have a solid understanding. In the previous proof, we saw how for any set of n ...
The Circle Method
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
Sample Writing
... Euclid’s Algorithm, discovered by the ancient Greek mathematician, Euclid, provides a method for finding the greatest common divisor for a pair of integers. An algorithm is a step-by-step procedure for performing a calculation, so Euclid’s algorithm is a series of steps based on the idea of divisibi ...
... Euclid’s Algorithm, discovered by the ancient Greek mathematician, Euclid, provides a method for finding the greatest common divisor for a pair of integers. An algorithm is a step-by-step procedure for performing a calculation, so Euclid’s algorithm is a series of steps based on the idea of divisibi ...
Random Number Generator
... is obviously true, since x 1 = ax 0 + c ( mod m), 0 <= x 1 < m. Assume that the formula is valid for the k th term, so that x k = a k x 0 + c(a k - 1) / ( a -1 ) (mod m ), 0 <= x k < m. Since x k+1= a x k + c (mod m ), 0 <= x k+1 < m. we have x k+1 = a( a k x 0 + c(a k -1)/(a-1)) + c = a k+1 x 0 + c ...
... is obviously true, since x 1 = ax 0 + c ( mod m), 0 <= x 1 < m. Assume that the formula is valid for the k th term, so that x k = a k x 0 + c(a k - 1) / ( a -1 ) (mod m ), 0 <= x k < m. Since x k+1= a x k + c (mod m ), 0 <= x k+1 < m. we have x k+1 = a( a k x 0 + c(a k -1)/(a-1)) + c = a k+1 x 0 + c ...
Direct Proof More Examples Contraposition
... Since 3n + 2 = 2(3k + 1) we know that 3n + 2 is even. However, we are given that 3n + 2 is odd. This is a contradiction. Therefore our assumption that n is even must be incorrect. Therefore when 3n + 2 is odd, n must be odd. 2. Prove: No integer is both even and odd. Proof: Assume there exists some ...
... Since 3n + 2 = 2(3k + 1) we know that 3n + 2 is even. However, we are given that 3n + 2 is odd. This is a contradiction. Therefore our assumption that n is even must be incorrect. Therefore when 3n + 2 is odd, n must be odd. 2. Prove: No integer is both even and odd. Proof: Assume there exists some ...
a classification of gaussian primes
... The history of our number system is a well documented one; there are a plethora of books that describe that slow advancement we made as a species from using those most intuitive and yet amazingly abstract objects, the natural numbers, up to the 19th century where complex numbers were finally given a ...
... The history of our number system is a well documented one; there are a plethora of books that describe that slow advancement we made as a species from using those most intuitive and yet amazingly abstract objects, the natural numbers, up to the 19th century where complex numbers were finally given a ...
Inductive-Deductive Reasoning
... For example: An attorney states that his client is innocent because the crime victim was hit by a car. Since his client does not have a license. He can deduce that his client is innocent. ...
... For example: An attorney states that his client is innocent because the crime victim was hit by a car. Since his client does not have a license. He can deduce that his client is innocent. ...
Beginning Logic - University of Notre Dame
... We will define what it means for a statement in a propositional or predicate language to be true in an appropriate formal setting. To show that an argument is not valid, we will look for a “counter-example”, a setting in which the premises are all true and the conclusion is false. IV. Analysis of ar ...
... We will define what it means for a statement in a propositional or predicate language to be true in an appropriate formal setting. To show that an argument is not valid, we will look for a “counter-example”, a setting in which the premises are all true and the conclusion is false. IV. Analysis of ar ...
AN INVITATION TO ADDITIVE PRIME NUMBER THEORY A. V.
... of kth powers, the sequence of prime numbers, the values taken by a polynomial F (X) ∈ Z[X] at the positive integers or at the primes, etc.). In this survey, we discuss almost exclusively problems of the latter kind. The main focus will be on two questions, known as Goldbach’s problem and the Waring ...
... of kth powers, the sequence of prime numbers, the values taken by a polynomial F (X) ∈ Z[X] at the positive integers or at the primes, etc.). In this survey, we discuss almost exclusively problems of the latter kind. The main focus will be on two questions, known as Goldbach’s problem and the Waring ...
Section 5.2: GCF and LCM
... Thus, to find the GCF of two numbers, this theorem can be applied repeatedly until a remainder of zero is obtained. The final divisor that leads to the zero remainder is the GCF of the two numbers. ...
... Thus, to find the GCF of two numbers, this theorem can be applied repeatedly until a remainder of zero is obtained. The final divisor that leads to the zero remainder is the GCF of the two numbers. ...
Theorem
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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.