The Real Numbers - Laurel County Schools
... 3-7 The Real Numbers The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3. ...
... 3-7 The Real Numbers The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3. ...
Definition: lim f(x) = L means: (1) f is defined on an open interval
... Proof. The function x 7→ 4x − 5 is a polynomial function, so defined for all real numbers. [(This satisfies the first condition of limits.)] Let ǫ > 0. [(We broke up the part: For all ǫ > 0 something-or-other holds.)] Set δ = ǫ/4. [(The somethingor-other above said that there exists δ > 0 such that ...
... Proof. The function x 7→ 4x − 5 is a polynomial function, so defined for all real numbers. [(This satisfies the first condition of limits.)] Let ǫ > 0. [(We broke up the part: For all ǫ > 0 something-or-other holds.)] Set δ = ǫ/4. [(The somethingor-other above said that there exists δ > 0 such that ...
Prime Numbers are Infinitely Many: Four Proofs from
... numbers”). Euler and Erdös prove that the infinite series Σ1/p, being p primes, diverges; and this is sufficient (but not necessary) in order to state that prime numbers are infinitely many: it is interesting to underline that in ...
... numbers”). Euler and Erdös prove that the infinite series Σ1/p, being p primes, diverges; and this is sufficient (but not necessary) in order to state that prime numbers are infinitely many: it is interesting to underline that in ...
Full text
... The references that follow include mainly papers that have been referred to above8 Vandiver in his expository paper [22] remarks that some 1500 papers on Bernoulli numbers have been published! For Fermat's last theorem, the reader is referred to Vandiver 1 s expository paper [21] as well as Dickson ...
... The references that follow include mainly papers that have been referred to above8 Vandiver in his expository paper [22] remarks that some 1500 papers on Bernoulli numbers have been published! For Fermat's last theorem, the reader is referred to Vandiver 1 s expository paper [21] as well as Dickson ...
full text (.pdf)
... The last preliminary fact is the result of Chandra, Kozen, and Stockmeyer (1981) that PSPACE is equivalent to APTIME, so it suffices to give an alternating PTIME Turing machine to decide membership of sentences in R A N D O M (a). It is convenient to describe first an alternating PTIME algorithm whi ...
... The last preliminary fact is the result of Chandra, Kozen, and Stockmeyer (1981) that PSPACE is equivalent to APTIME, so it suffices to give an alternating PTIME Turing machine to decide membership of sentences in R A N D O M (a). It is convenient to describe first an alternating PTIME algorithm whi ...
6.1 Logic Statements or Propositions Negation
... To understand the logic behind the truth table for the conditional statement, consider the following statement. “If you get an A in the class, I will give you five bucks.” Let p = statement “ You get an A in the class” Let q = statement “ I will give you five bucks.” Now, if p is true (you got an A) ...
... To understand the logic behind the truth table for the conditional statement, consider the following statement. “If you get an A in the class, I will give you five bucks.” Let p = statement “ You get an A in the class” Let q = statement “ I will give you five bucks.” Now, if p is true (you got an A) ...
The mystery of the number 1089 – how
... runs through the positive numbers with 3 digits such that a1 > a3 . It is not true, however, that also for larger n all φn (a1 · · · an ) coincide. But we will be able to show that there are always surprisingly few different numbers in the range of φn and that – completely unexpectedly – Fibonacci n ...
... runs through the positive numbers with 3 digits such that a1 > a3 . It is not true, however, that also for larger n all φn (a1 · · · an ) coincide. But we will be able to show that there are always surprisingly few different numbers in the range of φn and that – completely unexpectedly – Fibonacci n ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".