An integer is a number that can be written without a fractional component. For example, 21, 4, 0, −2048 are integers, while 9.75, 5+1/2, √2 are not. The set of integers consists of zero, the positive natural numbers called whole numbers or counting numbers, their additive inverses; the set of integers is denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen. Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite; the integers form the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the algebraic integers that are rational numbers; the symbol Z can be annotated to denote various sets, with varying usage amongst different authors: Z+, Z+ or Z> for the positive integers, Z≥ for non-negative integers, Z≠ for non-zero integers.

Some authors use Z * for non-zero integers, others use it for. Additionally, Zp is used to denote either the set of integers modulo p, i.e. a set of congruence classes of integers, or the set of p-adic integers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, 0, Z is closed under subtraction; the integers form a unital ring, the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring Z. Z is not closed under division. Although the natural numbers are closed under exponentiation, the integers are not; the following table lists some of the basic properties of addition and multiplication for any integers a, b and c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group.

It is a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + … + 1 or + + … +. In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z; the first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse; this means. All the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity, it is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in Z for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings; the lack of zero divisors in the integers means. The lack of multiplicative inverses, equivalent to the fact that Z is not closed under division, means that Z is not a field; the smallest field containing the integers as a subring is the field of rational numbers.

The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field, its ring of integers can be extracted, which includes Z as its subring. Although ordinary division is not defined on Z, the division "with remainder" is defined on them, it is called Euclidean division and possesses the following important property: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b. The integer q is called the quotient and r is called the remainder of the division of a by b; the Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain; this implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an unique way.

This is the fundamental theorem of arithmetic. Z is a ordered set without upper or lower bound; the ordering of Z is given by::… −3 < −2 < −1 < 0 < 1 < 2 < 3 < … An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither positive; the ordering of integers is compatible with the algebraic operations in the following way: if a < b and c < d a + c < b + d if a < b and 0 < c ac < bc. It follows; the integers are the only nontrivial ordered abelian group whose positive elements are well-ordered. This is equivalent to the statement that any Noetherian valuation ring is either a field or a discrete valuation ring. In elementary school teaching, integers are intuitively defined as the natural numbers and the negations of the natural numbers. However, this style of definition leads to many different cases (each


Limacidae known by their common name the keelback slugs, are a taxonomic family of medium-sized to large, air-breathing land slugs, terrestrial pulmonate gastropod molluscs in the superfamily Limacoidea. The distribution of the family Limacidae is the western Palearctic. There are 28 species of Limacidae in adjacent countries. In this family, the number of haploid chromosomes lies between 21 and 25 and lies between 31 and 35. Zhiltsov & Schileyko elevated the subfamily Bielziinae to family level, based on the morphology of the reproductive system of Bielzia coerulans; the following two subfamilies were recognized in the taxonomy of Bouchet & Rocroi: subfamily Limacinae Lamarck, 1801 - synonyms: Limacopsidae Gerhardt, 1935. Some authors, for example Russian malacologists, classify genus Bielzia within separate family Limacopsidae. Caspilimax P. Hesse, 1926 Caucasolimax Likharev et Wiktor, 1980 Gigantomilax O. Boettger, 1883Gigantomilax csikii Soós, 1924 Gigantomilax lederi Gigantomilax majoricensis Ambigolimax Pollonera, 1887, Lehmannia Heynemann, 1862 Limacopsis Simroth, 1888 Malacolimax Malm, 1868 Turcomilax Simroth, 1901subfamily Eumilacinae Eumilax O. Boettger, 1881 - type genus of the subfamily Eumilacinae Eumilax brandti Metalimax Simroth, 1896 A cladogram showing the phylogenic relationships of this family to other families within the limacoid clade: Parasites of slugs in this family include larvae of the marsh flies Sciomyzidae, others.

Slugs of Florida on the UF / IFAS Featured Creatures Web site

Ezekiel 21

Ezekiel 21 is the twenty-first chapter of the Book of Ezekiel in the Hebrew Bible or the Old Testament of the Christian Bible. This book contains the prophecies attributed to the prophet/priest Ezekiel, is one of the Books of the Prophets. In chapters 20 to 24 there are "further predictions regarding the fall of Jerusalem", this chapter includes a prophecy against the Ammonites; the original text of this chapter is written in the Hebrew language. This chapter is divided into 32 verses; some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis, the Petersburg Codex of the Prophets, Aleppo Codex, Codex Leningradensis. There is a translation into Greek known as the Septuagint, made in the last few centuries BC. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus, Codex Alexandrinus and Codex Marchalianus. Sigh therefore, son of man, with a breaking heart, sigh with bitterness before their eyes.

And it shall be when they say to you, ‘Why are you sighing?’ that you shall answer, ‘Because of the news. As in Ezekiel 12:17-20, "Ezekiel is instructed in to act out the response to YHWH's actions, in this case moaning pitifully in order to provoke the people's curiosity and so provide further opportunity to warn them of the coming disaster". Son of man and say, ‘Thus says the Lord!’ Say: ‘A sword, a sword is sharpened And polished!’"Son of man": this phrase is used 93 times to address Ezekiel, including seven times in this chapter. "Sword": it is first "polished" "engaged." Make a sign. Appoint a road for the sword to go to Rabbah, of the Ammonites, to Judah, into fortified Jerusalem; the signpost represented the decision faced by Nebuchadrezzar, king of Babylon, regarding which of these two capitals to attack. For the king of Babylon stands at the parting of the road, at the fork of the two roads, to use divination: he shakes the arrows, he consults the images, he looks at the liver. Three methods of divination are noted: Biblical scholar Julie Galambush comments that "the arrows functioned like lots, first labelled and shaken together in a quiver, after which one was drawn out".

Ammonites Babylon Israel Jerusalem Judah Rabbah Related Bible parts: Ezekiel 18, Hebrews 12 Bromiley, Geoffrey W.. International Standard Bible Encyclopedia: vol. iv, Q-Z. Eerdmans. Brown, Francis; the Brown-Driver-Briggs Hebrew and English Lexicon. Hendrickson Publishers. ISBN 978-1565632066. Clements, Ronald E. Ezekiel. Westminster John Knox Press. ISBN 9780664252724. Gesenius, H. W. F.. Gesenius' Hebrew and Chaldee Lexicon to the Old Testament Scriptures: Numerically Coded to Strong's Exhaustive Concordance, with an English Index. Translated by Tregelles, Samuel Prideaux. Baker Book House. Joyce, Paul M.. Ezekiel: A Commentary. Continuum. ISBN 9780567483614. Würthwein, Ernst; the Text of the Old Testament. Translated by Rhodes, Erroll F. Grand Rapids, MI: Wm. B. Eerdmans. ISBN 0-8028-0788-7. Retrieved January 26, 2019. Ezekiel 21 Hebrew with Parallel English Ezekiel 21 Hebrew with Rashi's Commentary Ezekiel 21 English Translation with Parallel Latin Vulgate