text:
# Real number explained

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials.[1] The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as

\sqrt{2}

(1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as (3.14159265...).[2] Real numbers can be used to measure (approximately) physical observables such as time, mass, energy; and in one dimension, distance, velocity, acceleration, force, momentum, etc. The set of real numbers is denoted using the symbol R or

R

[3] and is sometimes called "the reals".[4]

Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one.[5] The real line can be thought of as a part of the complex plane, and the real numbers can be thought of as a part of the complex numbers.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field up to an isomorphism, whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. In fact, the cardinality of the set of all real numbers, denoted by

akc

and called the cardinality of the continuum, is strictly greater than the cardinality of the set of all natural numbers (denoted

\aleph0

, 'aleph-naught').

The statement that there is no subset of the reals with cardinality strictly greater than

\aleph0

and strictly smaller than

akc

is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.[6]

## History

Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" ("The rules of chords") in include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava, who were aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined.[7] Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2.

The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots and fourth roots).[8]

In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.

In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that cannot be rational; Adrien-Marie Legendre (1794) completed the proof,[9] and showed that is not the square root of a rational number.[10] Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.

Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof. Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitz[11] and Paul Gordan.[12]

The development of calculus in the 18th century used the entire set of real numbers without having defined them rigorously. The first rigorous definition was published by Georg Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. For more, see Cantor's first uncountability proof.

## Definition

See main article: Construction of the real numbers.

The real number system

(R;{}+{};{}{};{}<{})

can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers.[13] Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of Tarski), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.

### Axiomatic approach

Let

R

denote the set of all real numbers, then:
• The set

R

is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field

R

is ordered, meaning that there is a total order ≥ such that for all real numbers x, y and z:
• if xy, then x + zy + z;
• if x ≥ 0 and y ≥ 0, then xy ≥ 0.
• The order is Dedekind-complete, meaning that every non-empty subset S of

R

with an upper bound in

R

has a least upper bound (a.k.a., supremum) in

R

.

The last property is what differentiates the real numbers from the rational numbers (and from other more exotic ordered fields). For example,

\{x\inQ:x2<2\}

has a rational upper bound (e.g., 1.42), but no least rational upper bound, because

\sqrt{2}

is not rational.

These properties imply the Archimedean property (which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that, and thus, which is a contradiction with the upper-bound property of N.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields

R1

and

R2

, there exists a unique field isomorphism from

R1

to
 R2
. This uniqueness allows us to think of them as essentially the same mathematical object.

For another axiomatization of

R

, see Tarski's axiomatization of the reals.

### Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number—in this case . For details and other constructions of real numbers, see construction of the real numbers.

## Properties

### Basic properties

• Any non-zero real number is either negative or positive.
• The sum and the product of two non-negative real numbers is again a non-negative real number, i.e., they are closed under these operations, and form a positive cone, thereby giving rise to a linear order of the real numbers along a number line.
• The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called countably infinite. This establishes that in some sense, there are more real numbers than there are elements in any countable set.
• There is a hierarchy of countably infinite subsets of the real numbers, e.g., the integers, the rational numbers, the algebraic numbers and the computable numbers, each set being a proper subset of the next in the sequence. The complements of all these sets (irrational, transcendental, and non-computable real numbers) in the reals are all uncountably infinite sets.
• Real numbers can be used to express measurements of continuous quantities. They may be expressed by decimal representations, most of them having an infinite sequence of digits to the right of the decimal point; these are often represented like 324.823122147..., where the ellipsis (three dots) indicates that there would still be more digits to come. This hints at the fact that we can precisely denote only a few, selected real numbers with finitely many symbols.

More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.

### Completeness

See main article: Completeness of the real numbers.

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):

A sequence (xn) of real numbers is called a Cauchy sequence if for any there exists an integer N (possibly depending on ε) such that the distance is less than ε for all n and m that are both greater than N. This definition, originally provided by Cauchy, formalizes the fact that the xn eventually come and remain arbitrarily close to each other.

A sequence (xn) converges to the limit x if its elements eventually come and remain arbitrarily close to x, that is, if for any there exists an integer N (possibly depending on ε) such that the distance is less than ε for n greater than N.

Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.

The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).

The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.

For example, the standard series of the exponential function

ex=

 infty \sum n=0
 xn n!

converges to a real number for every x, because the sums

 M \sum n=N
 xn n!

can be made arbitrarily small (independently of M) by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that

ex

is well defined for every x.

### "The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, is larger).

Additionally, an order can be Dedekind-complete, see . The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that

R

is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of

R

. Thus

R

is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of

N

. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory.

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

The real numbers form a metric space: the distance between x and y is defined as the absolute value . By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

Every nonnegative real number has a square root in

R

, although no negative number does. This shows that the order on

R

is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make

R

the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as

R

. Ordered fields that satisfy the same first-order sentences as

R

are called nonstandard models of

R

. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in

R

), we know that the same statement must also be true of

R

.

R

of real numbers is an extension field of the field

Q

of rational numbers, and

R

can therefore be seen as a vector space over

Q

. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.

The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on

R

with the property that every non-empty subset of

R

has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.[14]

A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.

## Applications and connections to other areas

### Real numbers and logic

The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.[15]

The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.

Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).

The continuum hypothesis posits that the cardinality of the set of the real numbers is

\aleph1

; i.e. the smallest infinite cardinal number after

\aleph0

, the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.

### In physics

In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.

Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.[16]

### In computation

With some exceptions, most calculators do not operate on real numbers. Instead, they work with finite-precision approximations called floating-point numbers. In fact, most scientific computation uses floating-point arithmetic. Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not.

Computers cannot directly store arbitrary real numbers with infinitely many digits. The achievable precision is limited by the number of bits allocated to store a number, whether as floating-point numbers or arbitrary-precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas for them (such as

\sqrt{2},

\arcsin(2/23),

or
 1 style\int 0

xxdx

) rather than their rational or decimal approximation. It is not in general possible to determine whether two such expressions are equal (the constant problem).

A real number is called computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

### "Reals" in set theory

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

## Vocabulary and notation

Mathematicians use mainly the symbol R to represent the set of all real numbers. Alternatively, it may be used

R

, the letter "R" in blackboard bold, which may be encoded in Unicode (and HTML) as . As this set is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.

The sets of positive real numbers and negative real numbers are often noted

R+

and

R-

, respectively;

R+

and

R-

are also used.[17] The non-negative real numbers can be noted

R\ge

but one often sees this set noted

R+\cup\{0\}.

In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively
 R+
and
 R-.
[17] In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted
 * R +
and
 *. R -
[17]

The notation

Rn

refers to the set of the -tuples of elements of

\R

(real coordinate space), which can be identified to the Cartesian product of copies of

R.

It is an -dimensional vector space over the field of the real numbers, often called the coordinate space of dimension ; this space may be identified to the -dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a point of the Euclidean space is identified with the tuple of its Cartesian coordinates.

In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").

## Generalizations and extensions

The real numbers can be generalized and extended in several different directions:

• The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field.
• The affinely extended real number system adds two elements and . It is a compact space. It is no longer a field, or even an additive group, but it still has a total order; moreover, it is a complete lattice.
• The real projective line adds only one value . It is also a compact space. Again, it is no longer a field, or even an additive group. However, it allows division of a non-zero element by zero. It has cyclic order described by a separation relation.
• The long real line pastes together copies of the real line plus a single point (here denotes the reversed ordering of) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.
• Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and are therefore non-Archimedean ordered fields.
• Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

## References

### Sources

• Cantor, Georg (1874). "German: Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen". German: Journal für die Reine und Angewandte Mathematik, volume 77, pp. 258–62.
• Feferman, Solomon (1989). The Number Systems: Foundations of Algebra and Analysis, AMS Chelsea, .
• Katz, Robert (1964). Axiomatic Analysis, D.C. Heath and Company.
• Landau, Edmund (2001). Foundations of Analysis. American Mathematical Society,.
• Howie, John M. Real Analysis. Springer, 2005, .
• .

## Notes and References

1. Web site: real number | Definition, Examples, & Facts | Britannica. www.britannica.com.
2. Web site: Real number mathematics. 2020-08-11. Encyclopedia Britannica. en.
3. Web site: Weisstein. Eric W.. Real Number . 2020-08-11 . mathworld.wolfram.com.
4. Oxford English Dictionary, 3rd edition, 2008, s.v. 'real', n.2, B.4: "Mathematics. A real number. Usually in plural."
5. Web site: Real number . Oxford Reference . 2011-08-03.
6. Book: Koellner, Peter. The Stanford Encyclopedia of Philosophy. Edward N.. Zalta. June 16, 2019. Metaphysics Research Lab, Stanford University. Stanford Encyclopedia of Philosophy.
7. T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 410–11. In: .
8. Jacques Sesiano, "Islamic mathematics", p. 148, in
9. .
10. .
11. Beweis der Transendenz der Zahl e . Adolf . Hurwitz . Mathematische Annalen . 43 . 134–35 . 1893 .
12. Transcendenz von e und π . Paul . Gordan . . 43 . 2–3 . 222–224 . 1893 . 10.1007/bf01443647. 123203471 .
13. Web site: 18.095 Lecture Series in Mathematics . Lecture #1 . 2015-01-05 .
14. , chapter V.
15. , chapter 2.
16. Wheeler. John Archibald. John Archibald Wheeler. 1986. Hermann Weyl and the Unity of Knowledge: In the linkage of four mysteries—the "how come" of existence, time, the mathematical continuum, and the discontinuous yes-or-no of quantum physics—may lie the key to deep new insight. 27854250. American Scientist. 74. 4. 366–75. 1986AmSci..74..366W.
Ingemar. Bengtsson. The Number Behind the Simplest SIC-POVM. Foundations of Physics. 2017. 47. 8. 1031–41. 10.1007/s10701-017-0078-3. 1611.09087. 2017FoPh...47.1031B. 118954904.
17. [École Normale Supérieure]


text:
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Arithmetic operations on the complex plane. need asap If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Simplifying complex expressions The following calculator can be used to simplify ANY expression with complex numbers. Two plus negative five over two, over two, and it's imaginary part is going to be the mean of these two numbers so plus, plus three minus one. If you need support with math and in particular with letter algebra calculator or linear algebra come visit us at Sofsource.com. Because an ordered pair of numbers represents coordinates of a point in the two-dimensional Cartesian plane, the midpoint calculator is most often used in analytical geometry. This topic covers: - Adding, subtracting, multiplying, & dividing complex numbers - Complex plane - Absolute value & angle of complex numbers - Polar coordinates of complex numbers Our mission is to provide a free, world-class education to anyone, anywhere. The complex number calculator is able to calculate complex numbers when they are in … 7. Thanks for the feedback. Enter coordinates (x_A,y_A) and (x_B,y_B) of two points A and B in the box. Math Calculator. If you're seeing this message, it means we're having trouble loading external resources on our website. Natural log: ln. It's an online Geometry tool requires 2 endpoints in the two-dimensional Cartesian coordinate plane. ... pre-calculus-coordinate-geometry-calculator. It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus and inverse of the complex number. For instance, the midpoint of a ruler is its point of balance. Nope, this is serious stuff; it’s about finding the slope of a line, finding the equation of a line... EN: pre-calculus-coordinate-geometry-calculator menu, EN: pre-calculus-midpoint-calculator menu, midpoint\:(5\sqrt{3},\:7\sqrt{5}),\:(\sqrt{3},\:-\sqrt{5}). The complex number online calculator, allows to perform many operations on complex numbers. Arithmetic operations on complex numbers. Complex numbers and complex planes. The midpoint also has applications in physics. person_outlineAntonschedule 2018-10-09 12:39:46. 9. midpoint (0, 0), (6, 8) en. Distance and midpoint of numbers in the complex plane. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Practice: Distance of complex numbers. How to find a midpoint 2. The calculator will find the approximate solution of the first-order differential equation using the Euler's method, with steps shown. The complex symbol notes i. This is the currently selected item. Each segment between the midpoint M and an endpoint have the equal Input Data : The distance formula is essentially the Pythagorean theorem applied to the distances between the and coordinates. Midpoint of a Line Segment. Practice Problem 1: Belgrade is located at (44.5^o, 20.3^o) and Paris is located at (48.9^o,2.35^o), which represent north latitude and west longitude. Complex conjugates. Distance & midpoint of complex numbers. Introduction to imaginary numbers. 5. Polar and Cartesian Coordinates. Distance and midpoint of complex numbers. Adding and subtracting complex numbers. If we have coordinates of two points A(x_A,y_A) and B(x_B,y_B), then the midpoint of the line segment overline{AB} is determined by the formula, By continuing with ncalculators.com, you acknowledge & agree to our. Distance and midpoint calculator This online calculator will compute and plot the distance and midpoint of a line segment. The distance is to be rounded to the tenths place, one decimal place. Practice: Midpoint of complex numbers. It means that the midpoint of the segment connecting z_1 = a + ib and z_2 = c + id in the complex plane is the point (z_1+z_2)/2 with the coordinates: The sum of a number y and −3 is −8. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Distance Between 2 Points. Find what is a midpoint of a line segment? Complex conjugates. ... Complex numbers and complex planes (free lessons) Adding and subtracting complex numbers . Part II. Angle and absolute value of complex numbers. Distance between two points calculator uses coordinates of two points A(x_A,y_A) and B(x_B,y_B) in the two-dimensional Cartesian coordinate plane and find the length of the line segment \overline{AB}. Multiplying and dividing complex numbers. Show all work for full credit. ... in polar form. 3. Related Symbolab blog posts. Objective : Distance and midpoint of complex numbers. Choose from a variety of common mathematical and trigonometry formula solvers. Multiplying and dividing complex numbers. Graphing lines calculator Distance and midpoint calculator Triangle area, altitudes, medians, centroid, circumcenter, orthocenter Intersection of two lines calculator Equation of a line passing through the two given points Distance between a line and a point The midpoint calculator will take two coordinates in the Cartesian coordinate system and find the point directly in-between both of them. Download mobile versions. 6. So it's going to be two plus negative five. Given two complex numbers (either algebraically or graphically), find their midpoint. Complex Numbers Topics: 1. - 8019672 ChrisH811 ChrisH811 01/02/2018 Mathematics High School Please help ASAP! Point 1(x_A, y_A) = (0, 2) Practice Problem 2: Find coordinates of the midpoint M of \overline{PQ}. For any line segment, the midpoint is its point of balance, center of mass or center of gravity. Common logarithms. It is often said that the point M bisects the segment overline{AB}. We offer an algebra calculator to solve your algebra problems step by step, as well as lessons and practice to help you master algebra. Given two complex numbers (either algebraically or graphically), find their distance. Type in any equation to get the solution, steps and graph This website uses cookies to ensure you get the best experience. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Precalculus (10th Edition) Edit edition. Polar Form of Complex Numbers; Convert polar to rectangular using hand-held calculator; Polar to Rectangular Online Calculator; 5. Evaluating logarithms without a calculator. To create your new password, just click the link in the email we sent you. Find the midpoint between the complex numbers. Distance and Midpoint Formula in the Complex Plane The modulus of the complex number a+biis ˜a+bi˜=˚a2+b2. ... Equations Inequalities System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & … The calculator will generate a step by step explanation for each operation. It is necessary to follow the next steps: M(x_M,y_M)\equiv M(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}), Area & Perimeter of a Rectangle Calculator, =(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}). Please try again using a different payment method. Free midpoint calculator - calculate the midpoint between two points using the Midpoint Formula step-by-step. Exponential Form of Complex Numbers; Euler Formula and Euler Identity interactive graph; 6. Fundamental Theorem of Algebra. The work with steps shows the complete step-by-step calculation for how to find the coordinates of center point of line segment having 2 end points A at coordinates (5,8) and B at coordinates (3,2). Any line segment has a unique midpoint. The point M is the midpoint of the line segment overline{AB} if it is an element of the segment and divides it into two congruent segments, overline{AM}\congoverline{MB}. It's an alternate method to finding the midpoint of a line segment without compass and ruler. This point is often useful in geometry. Distance and midpoint of complex numbers. Math Calculator. So if we had some, let's say that some complex number, let's just call it a, is the midpoint, it's real part is going to be the mean of these two numbers. Similarly, the y-coordinate of the midpoint M of the segment overline{AB} is the arithmetic mean of the y-coordinates of the endpoints of the segment overline{AB}. It's an online Geometry tool requires 2 2 endpoints in the two-dimensional Cartesian coordinate plane. The x-coordinate of the midpoint M of the segment overline{AB} is the arithmetic mean of the x-coordinates of the endpoints of the segment overline{AB}. How to enter numbers: Enter any integer, decimal or fraction. length. In other words, is its point of balance. This website uses cookies to ensure you get the best experience. Find the distance using the modulus of the difference between z 1 = -8 + 3i and z 2 = 7 - 4i. Show all work for full credit. The calculator will generate a step-by-step explanation on how to obtain the results. Conic Sections Trigonometry. Point 2(x_B, y_B) = (8, 8) 2. It also demonstrates elementary operations on complex numbers. Step 1: Enter the expression you want to evaluate. Add or subtract complex numbers, and plot the result in the complex plane. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. By … Fractions should be entered with a forward such as '3/4' for the fraction $$\frac{3}{4}$$. Complex Number Calculator. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Works across all devices. This is the distance between the origin (0, 0) and the point (a, b) in the complex plane. There are 9 calculators in this category . Math Calculators, Lessons and Formulas. Popular Formula Solvers. It is time to solve your math problem The calculator will simplify any complex expression, with steps shown. Find the midpoint between the complex numbers. z1 = 9 - 9i z2 = 10 - 9i (9.5, 9) ... Then solve. Equations Inequalities System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & Comp. Midpoint calculator uses coordinates of two points A(xA, yA) A (x A, y A) and B(xB, yB) B (x B, y B) in the two-dimensional Cartesian coordinate plane and find the halfway point between two given points A A and B B on a line segment. Graphical Representation of Complex Numbers; 4. Free equations calculator - solve linear, quadratic, polynomial, radical, exponential and logarithmic equations with all the steps. Add or subtract complex numbers, and plot the result in the complex plane. As a supplement to this calculator, we have written an article below that discusses how to find the midpoint and what the midpoint formula is. Operations with one complex number This calculator extracts the square root , calculate the modulus , finds inverse , finds conjugate and transform complex number to polar form . Find the midpoint between the complex numbers. For two points in the complex plane, the distance between the points is the modulus of the difference of the two complex numbers. 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midpoint of complex numbers calculator 2021

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# Volcano Watch — Rootless Shields are Not a Gang of Nomadic Warriors

Release Date:

Until last November, the current eruption was characterized by a perched lava channel. The fissure system that opened up just east of Puu Oo on July 21, 2007 issued lava into an open channel that frequently overflowed. These lava overflows, over time, built confining levees that elevated, or perched, the lava stream more than 30 m (100 feet) above the pre-July 21 ground level.

Closeup of a small dome fountain from the top of the new rootless shield.

(Public domain.)

The formation of this perched lava channel is unique in the 25-year history of the current eruption. Among Kīlauea eruptions, this feature would be enough to distinguish the current activity. But not to be outdone, the eruption is producing a new class of geological features that is no less unique and visually stunning. These features are known as rootless shields.

Shield-like features are familiar formations of Hawaiian volcanoes, but rootless shields are less familiar. Indeed, Hawaiian volcanoes are generally referred to as shield volcanoes due to their broad, gently-sloping sides. The shield-like nature is the result of frequently erupting sheets of very fluid lava that, over time, cover extensive areas in a radial pattern. If you've been to the southern part of the Big Island on a clear day, you've seen the beautiful, sloping shield of Mauna Loa Volcano that completely dominates the landscape. Mauna Loa's shape was acquired in just such a way.

Rootless shields are smaller, shield-shaped mounds that form on active lava flows. But, in contrast to shield volcanoes like Mauna Loa, which have deep roots into magma chambers below, rootless shields are fed by shallow lava tubes that flow just below the surface. Thus, they are "rootless," because the source of their lava is not a deep conduit.

Once the Thanksgiving Eve breakout started, a shield began to form directly over the breakout point on Fissure D of the July 21st eruption-the ultimate source of lava flows since July 21, 2007. It was fed directly from fissure D and was, thus, not rootless. Within weeks, a second, third, and fourth shield formed, each at or near the downhill base of the previous shield. This resulted in a chain of shields that step downslope and trace the lava tubes formed by the Thanksgiving Eve breakout. Because these shields are built over a lava tube, and not a deep magma source, they are considered rootless shields. To date, there are six well-defined rootless shields-four on the main branch of the tube which extends towards the Royal Gardens subdivision, and two more on a tube that branches off and from the main tube.

The shields are low-lying and can be difficult to spot from the air, but are unmistakable from the ground. The largest is about 40 m (approximately 130 feet) high, and 400 m (approximately 1300 feet) wide. Amazingly, shields like this can be built in a week!

The rootless shields are not entirely stable. Like a dam, they may rupture or collapse, disgorging lava from their interiors. For example, it was the rupture of the southeastern-most rootless shield that sent lava flowing toward the Royal Gardens subdivision earlier in January.

Why rootless shields are forming in this area is a bit of a mystery. We suspect that their formation is controlled by a combination of land-slope and rate of lava supply. These factors are known to contribute to the formation of perched lava ponds, which often develop on the flanks or summit of shields.

Despite their impressive growth, an important implication of these rootless shields is that their formation prevents the flow from heading downhill. In other words, more lava is consumed in shield building and less in extending the flow. This is good news for communities downslope.

————————————————————————————————————————————————————————————————

### Volcano Activity Update

Kīlauea summit and Puu Oo continued to deflate. There have been two deflation-inflation tilt events in the past week. Seismic tremor levels at the summit are elevated to nearly moderate levels. Summit sulfur dioxide emissions have nearly tripled. Earthquakes were located mostly beneath the general summit area and the south flank faults.

On July 21, 2007, lava began erupting from a set of fissures on the east flank of Puu Oo. Eruptive activity soon stabilized at fissure D, 2.3 kilometers (1.4 mi) northeast of Puu Oo. For most of last fall, this lava was directed entirely into a perched lava channel, consisting of separate pools of molten lava separated by bridges of cooled lava. At dawn on November 21, lava began to erupt directly from fissure D, outside of the perched channel, creating the Thanksgiving Eve breakout (TEB) flow. From December 27 until last weekend, the lava supply to the original perched channel was completely redirected through the TEB outlet. By Monday, January 21, though, lava had reentered the perched channel and partially refilled pool 1. That lava has apparently stagnated, but could certainly resume again.

The TEB flow has continued to build itself vertically and laterally, as a series of low shields, over the last several weeks. Two weeks ago, the front of the southeast-most shield collapsed and a large volume of lava surged out to form a rapidly moving aa flow. The flow advanced about 3.4 km (2.1 miles) reaching to within 180 m (~590 ft) of the top of the Royal Gardens subdivision before stagnating. The eruption then resumed its construction of low shields within about 2 km (1.2 miles) of the TEB outlet. In the past week, a pahoehoe flow fed from the lower end of the southeast-most shield has been creeping slowly downslope.

Weak incandescence in Puu Oo was observed at night by Webcam a few times in December-for the first time since August 31-but has otherwise been absent since. As in years past, Puu Oo likely is serving as a large chimney, beneath which lava is stored briefly and degassed substantially enroute to the erupting fissure.

Vent areas are hazardous. Access to the eruption site, in the Puu Kahaualea Natural Area Reserve, is closed (http://www.state.hi.us/dlnr/chair/pio/HtmlNR/07-N076.htm).

One earthquake beneath Hawaii Island was reported felt within the past week. A magnitude-3.1 earthquake occurred at 5:25 p.m. H.s.t on Sunday, January 20, 2008, and was located 7 km (4 miles) northwest of Mauna Kea summit at a depth of 24 km (15 miles).

Mauna Loa is not erupting. Four earthquakes were located beneath the summit. Extension between locations spanning the summit, indicating inflation, continues at steady, slow rates.

Visit our Web site for daily Kīlauea eruption updates and nearly real-time Hawaii earthquake information. Kīlauea daily update summaries are also available by phone at (808) 967-8862. skip past bottom navigational bar.


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# [Tex/LaTex] Why I can’t insert image with the following code

graphics

I tried to insert an image with the following line, but it doesn't work, keeps saying the image doesn't exist. Why is that?

    \includegraphics{{C:/Users/XYZ/Pictures/Rplot01.jpg}}

Here's an example of an includegraphics line of code:

\includegraphics[width=10cm,height=5cm,keepaspectratio]{kSED7}

To look at this a little more carefully, after the backslash and command, there are the optional brackets to specify width, height, and ratio, and then (the important part) there are curly braces. You should only have one set of curly braces.

Then, I don't know what you are using for your LaTeX editor, but I use an online LaTeX editor. The way you enter the image name on this particular editor is not like you have specified. In the online editor the image should appear off to the left, like in the screenshot below:

Then, you simply type in kSED7 or YKjf6 instead of the full line you did.

The main thing I would say to try is to use only one set of curly braces instead of two.

Finally, also be sure to use the line

\usepackage{graphicx}

at the very beginning of your LaTeX document.

Update:

In light of your comment, you can center a graphic by using the command

\centerline{}

with your \includegraphics{} function inside the curly braces. An example of this would be

\centerline{\includegraphics[width=10cm,height=5cm,keepaspectratio]{kSED7}}

Hope this helps!


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# In Exam

Let f:R..........s be a ring homomorphism .Prove that the inverse image of an ideal of S is an ideal of R

Thank you


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# zbMATH — the first resource for mathematics

Sequence transformations and their applications. (English) Zbl 0566.47018
Mathematics in Science and Engineering, Vol. 154. New York - London etc.: Academic Press. XIX, 257 p. (1981).
In this book a sequence transformation T is a mapping defined on sequences s in a Banach space B into sequences in B, with the following conditions:
i) $$T(\alpha s)=\alpha T(s)$$, $$\alpha\in {\mathbb{C}}$$; ii) $$T(s+c)=T(s)+T(c)$$, c is a constant sequence; iii) T is regular; iv)T(s) converges more rapidly than s.
In the classical (Toeplitz) theory mostly linear T are considered but, as the author indicates, the classical methods restrict the field of applications considerably. Those who want to study the abstract theory of sequence transformations will not find what they look for because it is especially in the applications to numerical problems that the author is interested. (Therefore mostly $$B={\mathbb{C}})$$. But in this direction the book is very complete. In fact it provides an abundance of information for those who are interested in the concrete evaluation of limits of any kind (sums of series, integrals,...). The author touches virtually every area of analysis, including interpolation and approximation, special functions, continued fractions and optimization methods. Also lots of examples are worked out. It is difficult to list here the algorithms described in the book. Not only because of the enormous variety of methods but also because of the difficulty in notation (this fact is the only - minor - disadvantage). The book is completed with an extended list of references.

##### MSC:
 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46A45 Sequence spaces (including Köthe sequence spaces) 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 41A05 Interpolation in approximation theory 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
##### Keywords:
sequence transformation


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lilypond-user
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## Re: Microtonal Helmholtz-Ellis notation in Lilypond: fine-tuning

 From: Torsten Anders Subject: Re: Microtonal Helmholtz-Ellis notation in Lilypond: fine-tuning Date: Tue, 8 Sep 2009 00:26:56 +0100

```On 07.09.2009, at 19:57, Robin Bannister wrote:
```
```Torsten Anders wrote:
```
```However, I failed to do the same for chord tones.
```
```Something like this?
```
```
Great, thank you very much!!

```
Now my remaining problem is the spacing. For example, can anyone explain to me how to use the following snippet from the documentation. It appears to do nothing for me.
```
\once \override Score.SeparationItem #'padding = #1

Are there other ways to add some horizontal spacing before a note?

Thank you!

Best
Torsten

```


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# Scales 101 – part IV: leaving a gap

So far we’ve characterised the scaled pointclasses among the projective hierarchy as well as establishing Steel’s result that $\Sigma_1^{J_\alpha(\mathbb R)}$ is scaled for all $\alpha>0$ such that $\text{Det}(J_\alpha(\mathbb R)$. We now move on to boldface territory, finishing off this series on scales.

We’re going to need the notion of a gap, where we say that an interval $[\alpha,\beta]$ is a gap if $J_\alpha(\mathbb R)\prec_1^{\mathbb R} J_\beta(\mathbb R)$ and this stops being the case when we extend the interval. Here $\prec_1^{\mathbb R}$ means that the left-hand side is a $\Sigma_1$-elementary substructure with parameters allowed from $\mathbb R\cup\{V_{\omega+1}\}$. This is also sometimes called a $\Sigma_1$-gap, but since we’re not going to need the $\Sigma_n$-gaps for anything we stick to the ‘gap’ terminology.

For convenience we also say that $[(\bf\delta^2_1)^{L(\mathbb R)},\Theta^{L(\mathbb R)}]$ is a gap, where we note that $(\bf\delta^2_1)^{L(\mathbb R)}$ is least such that $J_{(\bf\delta^2_1)^{L(\mathbb R)}}(\mathbb R)\prec_1^{\mathbb R} L(\mathbb R)$. Our first lemma states that gaps can be used to characterise ordinals.

Lemma. The gaps partition the ordinals below $\Theta^{L(\mathbb R)}$.

Proof. Let $\gamma<\Theta^{L(\mathbb R)}$ and let $\alpha\leq\gamma$ be least such that $J_\alpha(\mathbb R)\prec_1^{\mathbb R}J_\gamma(\mathbb R)$ and $\beta\geq\alpha$ the supremum of all $\delta<\Theta^{L(\mathbb R)}$ such that $J_\gamma(\mathbb R)\prec_1^{\mathbb R}J_\delta(\mathbb R)$. Then $[\alpha,\beta]$ is clearly a gap with $\gamma\in[\alpha,\beta]$.

Now say that two gaps $[\alpha,\beta]$ and $[\alpha',\beta']$ overlapped at some $\gamma$. This means that $J_\alpha(\mathbb R)\prec_1^{\mathbb R}J_\gamma(\mathbb R)$ and $J_\gamma(\mathbb R)\prec_1^{\mathbb R}J_{\beta'}(\mathbb R)$ but $J_\alpha(\mathbb R)\not\prec_1^{\mathbb R}J_{\beta'}(\mathbb R)$. But this is impossible as we’re requiring that all parameters lie in $\mathbb R\cup\{V_{\omega+1}\}$QED

We can thus split an arbitrary $\alpha<\Theta^{L(\mathbb R)}$ into three cases:

1. $\alpha$ begins a gap;
2. $\alpha$ ends a gap;
3. $\alpha$ lies properly within a gap.

When it comes to finding scaled pointclasses, the following result shows that we can disregard the third possibility.

Theorem (Martin). Let $[\alpha,\beta]$ be a gap and assume $\text{Det}(J_{\alpha+1}(\mathbb R))$. Then there’s a $\Pi_1^{J_\alpha(\mathbb R)}$ subset of $\mathbb R\times\mathbb R$ with no $\bf\Sigma_1^{J_\beta(\mathbb R)}$ uniformisation. In particular, if $\gamma\in(\alpha,\beta)$ and $\text{Det}(J_\gamma(\mathbb R))$ then none of the classes $\bf\Sigma_n^{J_\gamma(\mathbb R)}$ or $\bf\Pi_1^{J_\gamma(\mathbb R)}$ are scaled, for any $n<\omega$.

We thus only have to analyse case 1 and 2. Let’s start off with the first case where $\alpha$ begins a gap. Again we may assume $\alpha>0$, since we already covered the $\alpha=0$ case. The case where $n=1$ turns out to just be a corollary of the lightface result that we covered last time.

Corollary. If $\alpha>0$ begins a gap and $\text{Det}(J_\alpha(\mathbb R))$ holds then $\bf\Sigma_1^{J_\alpha(\mathbb R)}$ has the scale property.

Proof. Note that since $\alpha$ begins a gap there’s a partial $\bf\Sigma_1^{J_\alpha(\mathbb R)}$ surjection $\mathbb R\to J_\alpha(\mathbb R)$. Indeed, by forming the hull $\mathcal H:=\text{cHull}_1^{J_\alpha(\mathbb R)}(\mathbb R\cup\{V_{\omega+1}\})$ we see that $\mathcal H=J_\alpha(\mathbb R)$ by condensation and minimality, as $\alpha$ begins a gap, so the associated $\bf\Sigma_1^{J_\alpha(\mathbb R)}$ Skolem function is then as desired. Steel’s lightface result then gives the desired scale. QED

When we move to $n>1$ then the existence of scales turn out to depend on the admissibility of $\alpha$, where we say that $\alpha$ is admissible if $J_\alpha(\mathbb R)\models\textsf{KP}$, or equivalently that there’s no $\Sigma_1^{J_\alpha(\mathbb R)}$ cofinal map $\gamma\to\omega\alpha$ for some $\gamma<\omega\alpha$. Technically speaking the terminology admissible is reserved for the $L$-hierarchy, so we could’ve called it something like $\mathbb R$-admissible — but as we won’t need to consider the $L$-variants of admissibility, there won’t be any confusion. Now, the key lemma is the following.

Lemma. Assume that $\alpha$ begins a gap and is inadmissible (not admissible). Then for all $n\geq 1$,

$\bf\Sigma_{n+1}^{J_\alpha(\mathbb R)}=\exists^{\mathbb R}\bf\Pi_n^{J_\alpha(\mathbb R)}$ and $\bf\Pi_{n+1}^{J_\alpha(\mathbb R)}=\forall^{\mathbb R}\bf\Sigma_n^{J_\alpha(\mathbb R)}$.

I won’t give the proof here (a full proof can be found in my note), but the idea is that the conjunction of $\alpha$ beginning a gap and being inadmissible implies that we get a cofinal map $\mathbb R\to\omega\alpha$. The above two results then give us that $\bf\Sigma_{2n+1}^{J_\alpha(\mathbb R)}$ and $\bf\Pi_{2n+2}^{J_\alpha(\mathbb R)}$ have the scale property, whenever $\alpha>0$ begins a gap, is inadmissible, $\text{Det}(J_{\alpha+1}(\mathbb R))$ holds and $n<\omega$. To see that inadmissibility is necessary, Martin comes to the rescue again.

Theorem (Martin). Assume $\alpha$ begins a gap, is admissible and $\text{Det}(J_{\alpha+1}(\mathbb R))$ holds. Then there’s a $\Pi_1^{J_\alpha(\mathbb R)}$ subset of $\mathbb R\times\mathbb R$ with no uniformisation in $J_{\alpha+1}(\mathbb R)$. In particular, none of the classes $\bf\Sigma_n^{J_\alpha(\mathbb R)}$ or $\bf\Pi_n^{J_\alpha(\mathbb R)}$ are scaled for $n>1$.

This finishes the case where $\alpha$ begins a gap. We’ll need a definition for the gap endings. We say that a gap $[\alpha,\beta]$ is strong if for every $b\in J_\beta(\mathbb R)$ there’s a $\gamma<\beta$ and an $a\in J_\gamma(\mathbb R)$ such that for every $\varphi(v)\in\Sigma_1\cup\Pi_1$ it holds that

$J_\gamma(\mathbb R)\models\varphi[a]$ iff $J_\beta(\mathbb R)\models\varphi[b]$.

It turns out that for us to have any chance of finding scales at the end of gaps, we have to require our gaps to be weak. Martin returns with a counterexample.

Theorem (Martin). Let $[\alpha,\beta]$ be a strong gap and assume $\text{Det}(J_{\alpha+1}(\mathbb R))$. Then there’s a $\Pi_1^{J_\alpha(\mathbb R)}$ relation which has no uniformisation in $J_{\beta+1}(\mathbb R)$. In particular, none of the classes $\bf\Sigma_n^{J_\beta(\mathbb R)}$ or $\bf\Pi_n^{J_\beta(\mathbb R)}$ are scaled for $n<\omega$.

So we can focus solely on weak gaps then. Furthermore, if $\rho_k(J_\beta(\mathbb R))\neq\mathbb R$ (i.e. we’re not projecting to 0) for every $k\leq n$ then we don’t get any new subsets of reals of interest, meaning that

$\bf\Sigma_n^{J_\beta(\mathbb R)}=\bf\Sigma_n^{J_\alpha(\mathbb R)}$ and $\bf\Pi_n^{J_\beta(\mathbb R)}=\bf\Pi_n^{J_\alpha(\mathbb R)}$,

just reducing it to the begins-a-gap case. Taking this into account, we do have the following theorem, whose proof is a technical tour de force based on the same ideas as the lightface proof that I’ve written up here.

Theorem (Steel). Let $[\alpha,\beta]$ be a weak gap and assume $\text{Det}(J_{\alpha+1}(\mathbb R))$. If $n<\omega$ is least such that $\rho_n(J_\beta(\mathbb R))=\mathbb R$ then

$\bf\Sigma_{n+2k}^{J_\beta(\mathbb R)}$ and $\bf\Pi_{n+2k+1}^{J_\beta(\mathbb R)}$

are scaled, for every $k<\omega$.

Aaaand that finishes the complete characterisation, which is used to organise the cases in the core model induction. So if you’re ever wondering if $\bf\Sigma_n^{J_\alpha(\mathbb R)}$ is scaled, we can now use the following handy flowchart (pdf available here):

Here’s a link to the follow-up post


text:
## Finite model property of guarded fragments

#### Ian Hodkinson

The guarded fragment of first-order logic, GF, was introduced in [H. Andreka, J. van Benthem, and I. Nemeti, Modal languages and bounded fragments of predicate logic, Journal of Philosophical Logic, 27(3):217-274, 1998].  The link is to the Kluwer journal page where you may (if it lets you in) be able to retrieve a .pdf file of the article.

The idea came from algebraic logic, and the motivation was to find a `modal-style' fragment of first-order logic that shared the nice properties of modal logic (such as decidability with reasonable complexity, and the finite model property), for similar reasons.  The rough idea of the guarded fragment is that quantification must always be relativised to an atomic formula. Other more expressive fragments along the same lines were introduced later:
The lectures will discuss these fragments and their background, but will mainly concern recent proofs of the finite model property for them.  These proofs use combinatorial results on extending partial isomorphisms of a finite structure to full automorphisms of a larger but still finite structure, developed in model theory by Hrushovski, Herwig, and Lascar.  The first application of these results to guarded fragments was in [E. Grädel, On the restraining power of guards, Journal of Symbolic Logic, vol. 64, pp. 1719-1742, 1999], where it is shown among other things (such as complexity results) that the guarded fragment has the finite model property.  LGF, PF, and CGF were shown to have the finite model property as well in I. Hodkinson, Loosely guarded fragment of first-order logic has the finite model property, Studia Logica 70 (2002) 205-240.  A simpler and nicer proof is in I. Hodkinson and M. Otto, Finite Conformal Hypergraph Covers, with Two Applications, Bull. Symbolic Logic 9 (2003) 387-405, available from the Bulletin here.

### Extending partial isomorphisms

1. Hrushovski's original proof that any finite graph A embeds in a finite graph B such that any partial isomorphism of A extends to an automorphism of B is in  E. Hrushovski, Extending partial isomorphisms of graphs, Combinatorica 12 (1992), 411-416.
2. John Truss proved earlier in [J.K.Truss, Generic automorphisms of homogeneous structures, Proceedings of the London Mathematical Society 64, 121-141, 1992] that given a finite graph A and a single partial isomorphism p, there is a finite graph B extending A in which p is induced by an automorphism.
3. Hrushovski's result was used to prove the small index property for the random graph in W. A. Hodges, I. M. Hodkinson, D. Lascar and S. Shelah, The small index property for $\omega$-stable $\omega$-categorical structures and for the random graph, J. London Math. Soc. (2) 48 (1993), 204 218.
4. The first Herwig paper [B. Herwig, Extending partial isomorphisms on finite structures, Combinatorica 15 (1995) 365--371]  extended Hrusovski's result to arbitrary relational structures.  Here is a possibly early version of this paper, called Many $\omega$-categorical structures have the small index property.
5. The second Herwig paper [B. Herwig, Extending partial isomorphisms for the small index property of many $\omega$-categorical structures, Israel J. Math. 107 (1998) 93-124] extended this to structures omitting certain homomorphic images (eg if A is a graph with no complete subgraph of size n, then B can be found with the same property).
6. The Herwig-Lascar paper[B. Herwig and D. Lascar, Extending partial isomorphisms and the profinite topology on the free groups, Trans. Amer. Math. Soc. 352 (2000) 1985-2021] generalised this approach much further.  These two papers are both on Herwig's home page.
7. The Herwig-Lascar paper has references to related work in free groups by Almeida and Delgado (and Marshall Hall).

### Related work

The proof technique was used by Grohe in finite model theory - e.g., in [Martin Grohe, Equivalence in Finite-Variable Logics is Complete for Polynomial Time, Proc. IEEE Symposium on Foundations of Computer Science (LICS) 1996, pp. 264-273] and [Martin Grohe, Arity Hierarchies, Annals of Pure and Applied Logic 82 (1996) 103-163].

For other work see, e.g., [K. Auinger and B. Steinberg, On The Extension Problem For Partial Permutations, preprint I think].

A Google search will produce more.  It seems quite a big area now.


text:
Gaaljh

2022-06-25

I have a multivariable function that I have defined the cost function and its gradient with respect to the variable vector. Let's call the cost function $f\left(\stackrel{\to }{x}\right)$, and variable vector $\stackrel{\to }{x}$. I have been using nonlinear conjugate gradient descenet method for minimization. Algorithm is as follows:
$k=0\phantom{\rule{0ex}{0ex}}x=0\phantom{\rule{0ex}{0ex}}{g}_{0}={\mathrm{\nabla }}_{\stackrel{\to }{x}}f\left({x}_{0}\right)\phantom{\rule{0ex}{0ex}}\mathrm{\Delta }{x}_{0}=-{g}_{0}$
${x}_{k+1}←{x}_{k}+t\mathrm{\Delta }{x}_{k}\phantom{\rule{0ex}{0ex}}{g}_{k+1}←\mathrm{\nabla }f\left({x}_{k+1}\right)\phantom{\rule{0ex}{0ex}}\mathrm{\Delta }{x}_{k+1}←-{g}_{k+1}+\gamma \mathrm{\Delta }{x}_{k}$
I know that I need to update formula so the solution "ascends" instead of descension. How can I update the algorithm? Also I wonder how the wolfe condition changes for maximization problems. Thank you and have a nice day.

Myla Pierce

Expert

Convert your function from maximization problem to minimization by setting ${f}_{new}\left(x\right)=-f\left(x\right)$ and run default conjugate gradient for minimization.

Since you had specifically asked how the formulas and wolfe conditions update you can plug in $-f\left(x\right)$ for function values and $-\mathrm{\nabla }f\left(x\right)$ for gradients and see for yourself.

Do you have a similar question?


text:
# Introduction

## Welcome to the 🤗 Course!

This course will teach you about natural language processing (NLP) using libraries from the Hugging Face ecosystem — 🤗 Transformers, 🤗 Datasets, 🤗 Tokenizers, and 🤗 Accelerate — as well as the Hugging Face Hub. It’s completely free and without ads.

## What to expect?

Here is a brief overview of the course:

• Chapters 1 to 4 provide an introduction to the main concepts of the 🤗 Transformers library. By the end of this part of the course, you will be familiar with how Transformer models work and will know how to use a model from the Hugging Face Hub, fine-tune it on a dataset, and share your results on the Hub!
• Chapters 5 to 8 teach the basics of 🤗 Datasets and 🤗 Tokenizers before diving into classic NLP tasks. By the end of this part, you will be able to tackle the most common NLP problems by yourself.
• Chapters 9 to 12 dive even deeper, showcasing specialized architectures (memory efficiency, long sequences, etc.) and teaching you how to write custom objects for more exotic use cases. By the end of this part, you will be ready to solve complex NLP problems and make meaningful contributions to 🤗 Transformers.

This course:

## Who are we?

Matthew Carrigan is a Machine Learning Engineer at Hugging Face. He lives in Dublin, Ireland and previously worked as an ML engineer at Parse.ly and before that as a post-doctoral researcher at Trinity College Dublin. He does not believe we’re going to get to AGI by scaling existing architectures, but has high hopes for robot immortality regardless.

Lysandre Debut is a Machine Learning Engineer at Hugging Face and has been working on the 🤗 Transformers library since the very early development stages. His aim is to make NLP accessible for everyone by developing tools with a very simple API.

Sylvain Gugger is a Research Engineer at Hugging Face and one of the core maintainers of the 🤗 Transformers library. Previously he was a Research Scientist at fast.ai, and he co-wrote Deep Learning for Coders with fastai and PyTorch with Jeremy Howard. The main focus of his research is on making deep learning more accessible, by designing and improving techniques that allow models to train fast on limited resources.

Are you ready to roll? In this chapter, you will learn:

• How to use the pipeline function to solve NLP tasks such as text generation and classification