China
Africa
Explore chapters and articles related to this topic
Natural Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
Definition 1.61: A field F with the total order ≤ is an ordered field if it satisfies the following for every a,b,c∈FIf a ≤ b then a + c ≤ b + c If 0 ≤ a and 0 ≤ b then 0 ≤ a × b, where 0 is the additive identity
Metric Spaces
Published in James K. Peterson, Basic Analysis III, 2020
Thus, we have shown ℚ˜ is a totally ordered field which is complete: i.e. satisfies the completeness axiom (also called the least upper bound axiom). We identify ℚ˜ with what we normally call the real numbers ℜ.
Real Numbers
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Next, we turn our attention to the following question: if b is a fixed real number, is there a positive integer N such that N > b? Now we can answer not only that, but even a more general form of the question: if a, b are positive real numbers and a < b, is there a positive integer N such that aN > b? In fact, the generalization can go one step further. The question can be asked in any ordered field, and when the answer is in the affirmative, we say that such a field is Archimedean, or that it has the Archimedean property.
Logics to formalise p-adic valued probability and their applications
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Angelina Ilić Stepić, Zoran Ognjanović
In the last two decades many logics that formalise real-valued [0, 1] probability have been developed [17–37]. In some of these formalisms one is able to make a statement of the form which is interpreted as ‘The probability of is at least s’. These operators depend on the ordering of the codomain, but is not an ordered field. Thus, we cannot use them to formalise p-adic valued probability. Therefore formalisms presented in [14–16] use other operators specific for the topology of . Moreover, the lack of ordering of requires the development of completely new techniques in order to prove statements about the logic, e.g. strong completeness. Other p-adic logics with p-adic valued measures are discussed in [38,39].
Improvement of constructing real numbers by Dedekind cuts
Published in International Journal of Mathematical Education in Science and Technology, 2023
In fact, it is well known that any two ordered fields with least-upper-bound property are isomorphic, and any ordered field contains a ‘rational field’ as its subfield. This reminds us that properties of real numbers can be yielded from those of rational numbers and the least-upper-bound property directly, that is, can be yielded independent of how real numbers are constructed.
Subsets of fields whose nth-root functions are rational functions
Published in International Journal of Mathematical Education in Science and Technology, 2018
We will generalize part of [1] from working over the field to working over more general ordered fields. Recall that a field F is called an ordered field if there is a set (called the set of positive elements ofF), with there exists such that u = −v} (called the set of negative elements ofF), such that is closed under addition and multiplication and F is the disjoint union of , {0} and . It follows that . Consequently, any ordered field has characteristic 0; that is, 1 + ⋅⋅⋅ + 1 ≠ 0 if the number of summands is k for any . Hence, any ordered field must be infinite and, up to isomorphism of fields, is a subfield of any ordered field. While and are ordered fields, cannot be given the structure of an ordered field. If F is an ordered field with its set of positive elements, one defines a binary ‘order’ relation < on F as follows: if a, b ∈ F, then a < b if . If a, b, c, … ∈ F, notations such as b > a, a ≤ b and a < c < b are given their usual meanings; and ‘<’ has the properties that one would expect from the behaviour of the usual ordering on (such as a < b implies a + c < b + c for all c ∈ F and da < db for all d > 0 in F).