StewartCalc7e_01_07
... Intuitively, it is clear that when x is close to 3 but x ≠ 3, then f(x) is close to 5, and so limx 3f(x) = 5. To obtain more detailed information about how f(x) varies when x is close to 3, we ask the following question: How close to 3 does x have to be so that f(x) differs from 5 by less than 0.1 ...
... Intuitively, it is clear that when x is close to 3 but x ≠ 3, then f(x) is close to 5, and so limx 3f(x) = 5. To obtain more detailed information about how f(x) varies when x is close to 3, we ask the following question: How close to 3 does x have to be so that f(x) differs from 5 by less than 0.1 ...
Preliminaries()
... product of prime numbers. This fact implies that, if there is no prime number greater than 2, any natural number n should be equal to 2i for some integer i. However, we know that there exist natural numbers that are not power of 2, which contradicts to a know fact. It follows that there exists a pri ...
... product of prime numbers. This fact implies that, if there is no prime number greater than 2, any natural number n should be equal to 2i for some integer i. However, we know that there exist natural numbers that are not power of 2, which contradicts to a know fact. It follows that there exists a pri ...
Section 2.2 Subsets
... • Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number. • Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number א0 read “aleph-null”. • Equal sets: Set A is equal t ...
... • Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number. • Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number א0 read “aleph-null”. • Equal sets: Set A is equal t ...
Fractions, Percentages, Ratios, Rates
... If the index is 2, this is the special case we have met before, called squaring. We also have a special term for when the index is 3: it is called cubing the number. ...
... If the index is 2, this is the special case we have met before, called squaring. We also have a special term for when the index is 3: it is called cubing the number. ...
Arizona Study Guide
... This NES Profile provides information about the test, including the approximate percentage of the total test score derived from each content domain. The complete set of the content domains, the test framework, is provided here and contains all of the competencies and descriptive statements that defi ...
... This NES Profile provides information about the test, including the approximate percentage of the total test score derived from each content domain. The complete set of the content domains, the test framework, is provided here and contains all of the competencies and descriptive statements that defi ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.