What is a rational number?

A fraction where the numerator and denominator are both integers.

How are positive integers defined?

Positive integers are whole numbers greater than zero.

What characterizes a proper fraction?

A proper fraction is a fraction where the numerator is less than the denominator.

A fraction with a numerator greater than its denominator

What does it mean for a number to be irrational?

It cannot be expressed as a fraction of two integers.

Which statement is true about integers and whole numbers?

All whole numbers are integers.

All integers are whole numbers.

Some integers are not whole numbers.

Representing Number-1

1.Linguistics is the science that studies language. 2.Linguist:Someone who studies linguistics. 3.The Subfields of Linguistics Phonetics deals with the sounds of language. Phonology deals with how the sounds are organized. Morphology deals with how sounds are put together to form words. Syntax deals with how sentences are formed. Semantics deals with the meaning of words, sentences, and texts. Pragmatics deals with how sentences and texts are used in the world (i.e., in context) Text Linguistics deals with units larger than sentences, such as paragraphs and texts. 4.Prescriptive: This approach consists basically of stating what is considered right and wrong in language. 5.Descriptive: This approach, on the other hand, consists of describing the facts. Descriptive linguistics is dedicated to describing the rules of the language, and the language is seen as essentially rule governed. 6.Language is rule-governed, creative, universal, innate, and learned, all at the same time. 7.Linguists understand language as a system of arbitrary vocal signs. 8.Linguistic signs: involve sequences of sounds which represent concrete objects and events as well as abstractions.Signs may be related to the things they represent in a number of ways. 9.Iconic: which resemble the things they represent (as do, for example, photographs, diagrams, star charts, or chemical models). 10.Indexical: which point to or have a necessary connection with the things they represent (as do, for example, smoke to fire, a weathercock to the direction of the wind, a symptom to an illness, a smile to happiness, or a frown to anger). 11.Describe the characteristics of human language: Creative: (The structural elements of human language can be combined to produce new utterances, which neither the speaker nor his hearers may ever have made or heard before.) Rule-governed: (Language is made of rules.) Universal: (There are some aspects that are present in all languages of the world.) Innate:(all humans possess an innate capacity for language, activated in infancy by minimal environmental stimuli. Chomsky) Uniquely human: (Language is what sets us apart from other species. It is what makes us human.) Learned:(Children acquire language from their natural setting.) 12.Differentiate between iconic, indexical and symbolic signs. A. iconic, which resemble the things they represent (as do, for example, photographs, diagrams, star charts, or chemical models) B. indexical, which point to or have a necessary connection with the things they represent (as do, for example, smoke to fire, a weathercock to the direction of the wind, a symptom to an illness, a smile to happiness, or a frown to anger). c. symbolic, which are only conventionally related to the thing they represent (as do, for example, a flag to a nation, a rose to love, a wedding ring to marriage). 12. Distinguish between different senses of the grammar word. The prescriptivist´s grammar (Grammar is a set of rules that label the different utterances as either right or wrong.) The descriptivist´s grammar (Grammar is a set of rules that govern the langauge spoken by people. ) The linguist´s grammar (Grammar is the subconscious knowledge of the set of rules that enables speakers to use the language) The speaker´s grammar (Grammar is the intrinsic linguistic knowledge within a native speaker) 13.Describe common fallacies about language and grammar: ►One type of grammar is simpler than another. ►Changes in grammar involve deterioration in a language ►Grammars should be logical and analogical (that is, regular) ►People must be taught the grammatical rules of their language. ►Only some languages have grammar. ►Grammars differ from each other in unpredictable ways. 14.Generality: All Languages Have a Grammar 15. Equality: All Grammars Are Equal 16.Changeability: Grammars Change Over Time 17. Universality: Grammars Are Alike in Basic Ways 18.Tacitness: Grammatical Knowledge Is Subconscious 19.Linguistics is defined as the study of language systems. It is the scientific study of language. 20.Historical approach:It is the study of language change. 21.Linguistic Competence: is the unconscious knowledge speakers of a language have about the system that enables them to create and understand novel utterances. 22.Performance: is the use of it. Performance is “the actual use of language in concrete situations.” 23.I-Language (internal language): which is the intrinsic linguistic knowledge within a native speaker. 24.E-Language (external language): which is the observable language—the output from a speaker. 25.Parole ('speech') refers to the concrete instances of the use of langue, including texts which provide the ordinary research material for linguistics. 26.Langue: 27.Language: is a system of communication that is non-stereotyped and non-finite; it is unlimited in its scope. 28.Grammar: to refer to a subconscious linguistic system of a particular type. Grammar makes possible the production and comprehension of a potentially unlimited number of utterances. 29.Communication and animals: Selecting a mode of communication (speech,writing, gesture). Delivering the symbols through a medium, a physical basis for communication, light, air, or ink. Decoding of the symbols to obtain the information. 30.SIGNS: Communication relies on using something to stand for something else. Words are an obvious example of this: You do not have to have a car, a sandwich, or your cousin present in order to talk about them—the words car, sandwich, and cousin stand for them instead. This same phenomenon is found in animal communication as well. 31.The signifier: A signifier is that part of a sign that stimulates at least one sense organ of the receiver of a message.A signifier can also be a picture, a photograph, a sign language gesture, or one of the many other words for tree in different languages. 32.The signified: The signified component of the sign refers to both the real world object it represents and its conceptual content. The first of these is the real world content of the sign, its extension or referent within a system of signs such as English, avian communication, or sign language. 33.Iconic signs or icons: always bear some resemblance to their referent. A photograph is an iconic sign; so too is a stylized silhouette of a female or a male on a restroom door. 34.Some iconic tokens: a. open-mouth threat by a Japanese macaque; b. park recreation signs; c. onomatopoeic words in English. 35.An indexical sign, or index, fulfils its function by pointing out its referent, typically by being a partial or representative sample of it. Indexes are not arbitrary, since their presence has in some sense been caused by their referent. For this reason it is sometimes said that there is a causal link between an indexical sign and its referent.The track of an animal, for example, points to the existence of the animal by representing a part of it. The presence of smoke is an index of fire. 36.Symbolic signs: bear an arbitrary relationship to their referents and in this way are distinct from both icons and indexes. Human language is highly symbolic in that the vast majority of its signs bear no inherent resemblance or causal connection to their referents, as the following words show. 37.Mixed signs Signs: are not always exclusively of one type or another. Symptomatic signs, for example, may have iconic properties, as when a dog opens its mouth in a threat to bite. Symbolic signs such as traffic lights are symptomatic in that they reflect the internal state of the mechanism that causes them to change color. 38.Signals: All signs can act as signals when they trigger a specific action on the part of the receiver, as do traffic lights, words in human language such as the race starter's "Go!", or the warning calls of birds. 39.SIGN STRUCTURE: No matter what their type, signs show different kinds of structure. A basic distinction is made between graded and discrete sign structure. 40.Graded signs convey their meaning by changes in degree. A good example of a gradation in communication is voice volume. The more you want to be heard, the louder you speak along an increasing scale of loudness. There are no steps or jumps from one level to the next that can be associated with a specific change in meaning. 41.Discrete signs are distinguished from each other by categorical (stepwise) differences. There is no gradual transition from one sign to the next. The words of human language are good examples of discrete signs. 42.A VIEW OF ANIMAL COMMUNICATION ►Largely iconic ►Largely symptomatic ►Little arbitrary ►Not deliberate ►Not conscious ►Not symbolic ►Stimulus bound

Covalent Molecules and Compounds Just as an atom is the simplest unit that has the fundamental chemical properties of an element, a molecule is the simplest unit that has the fundamental chemical properties of a covalent compound. Some pure elements exist as covalent molecules. Hydrogen, nitrogen, oxygen, and the halogens occur naturally as the diatomic (“two atoms”) molecules H2, N2, O2, F2, Cl2, Br2, and I2 (part (a) in Figure 3.1.1). Similarly, a few pure elements exist as polyatomic (“many atoms”) molecules, such as elemental phosphorus and sulfur, which occur as P4 and S8 (part (b) in Figure 3.1.1). Each covalent compound is represented by a molecular formula, which gives the atomic symbol for each component element, in a prescribed order, accompanied by a subscript indicating the number of atoms of that element in the molecule. The subscript is written only if the number of atoms is greater than 1. For example, water, with two hydrogen atoms and one oxygen atom per molecule, is written as H2O. Similarly, carbon dioxide, which contains one carbon atom and two oxygen atoms in each molecule, is written as CO2. Covalent compounds that predominantly contain carbon and hydrogen are called organic compounds. The convention for representing the formulas of organic compounds is to write carbon first, followed by hydrogen and then any other elements in alphabetical order (e.g., CH4O is methyl alcohol, a fuel). Compounds that consist primarily of elements other than carbon and hydrogen are called inorganic compounds; they include both covalent and ionic compounds. In inorganic compounds, the component elements are listed beginning with the one farthest to the left in the periodic table, as in CO2 or SF6. Those in the same group are listed beginning with the lower element and working up, as in ClF. By convention, however, when an inorganic compound contains both hydrogen and an element from groups 13–15, hydrogen is usually listed last in the formula. Examples are ammonia (NH3) and silane (SiH4). Compounds such as water, whose compositions were established long before this convention was adopted, are always written with hydrogen first: Water is always written as H2O, not OH2. The conventions for inorganic acids, such as hydrochloric acid (HCl) and sulfuric acid (H2SO4), are described elswhere. Note! For organic compounds: write C first, then H, and then the other elements in alphabetical order. For molecular inorganic compounds: start with the element at far left in the periodic table; list elements in same group beginning with the lower element and working up. Write the molecular formula of each compound. a. The phosphorus-sulfur compound that is responsible for the ignition of so-called strike anywhere matches has 4 phosphorus atoms and 3 sulfur atoms per molecule. b. Ethyl alcohol, the alcohol of alcoholic beverages, has 1 oxygen atom, 2 carbon atoms, and 6 hydrogen atoms per molecule. c. Freon-11, once widely used in automobile air conditioners and implicated in damage to the ozone layer, has 1 carbon atom, 3 chlorine atoms, and 1 fluorine atom per molecule. Solution: a. • A The molecule has 4 phosphorus atoms and 3 sulfur atoms. Because the compound does not contain mostly carbon and hydrogen, it is inorganic. • B Phosphorus is in group 15, and sulfur is in group 16. Because phosphorus is to the left of sulfur, it is written first. • C Writing the number of each kind of atom as a right-hand subscript gives P4S3 as the molecular formula. b. • A Ethyl alcohol contains predominantly carbon and hydrogen, so it is an organic compound. • B The formula for an organic compound is written with the number of carbon atoms first, the number of hydrogen atoms next, and the other atoms in alphabetical order: CHO. • C Adding subscripts gives the molecular formula C2H6O. c. • A Freon-11 contains carbon, chlorine, and fluorine. It can be viewed as either an inorganic compound or an organic compound (in which fluorine has replaced hydrogen). The formula for Freon-11 can therefore be written using either of the two conventions. • B According to the convention for inorganic compounds, carbon is written first because it is farther left in the periodic table. Fluorine and chlorine are in the same group, so they are listed beginning with the lower element and working up: CClF. Adding subscripts gives the molecular formula CCl3F. • C We obtain the same formula for Freon-11 using the convention for organic compounds. The number of carbon atoms is written first, followed by the number of hydrogen atoms (zero) and then the other elements in alphabetical order, also giving CCl3F. Write the molecular formula for each compound. a. Nitrous oxide, also called “laughing gas,” has 2 nitrogen atoms and 1 oxygen atom per molecule. Nitrous oxide is used as a mild anesthetic for minor surgery and as the propellant in cans of whipped cream. b. Sucrose, also known as cane sugar, has 12 carbon atoms, 11 oxygen atoms, and 22 hydrogen atoms. c. Sulfur hexafluoride, a gas used to pressurize “unpressurized” tennis balls and as a coolant in nuclear reactors, has 6 fluorine atoms and 1 sulfur atom per molecule. Answer: a. N2O b. C12H22O11 c. SF6. Ionic Compounds The substances described in the preceding discussion are composed of molecules that are electrically neutral; that is, the number of positively-charged protons in the nucleus is equal to the number of negatively-charged electrons. In contrast, ions are atoms or assemblies of atoms that have a net electrical charge. Ions that contain fewer electrons than protons have a net positive charge and are called cations. Conversely, ions that contain more electrons than protons have a net negative charge and are called anions. Ionic compounds contain both cations and anions in a ratio that results in no net electrical charge. Note! Ionic compounds contain both cations and anions in a ratio that results in zero electrical charge.An ionic compound that contains only two elements, one present as a cation and one as an anion, is called a binary ionic compound. One example is MgCl2, a coagulant used in the preparation of tofu from soybeans. For binary ionic compounds, the subscripts in the empirical formula can also be obtained by crossing charges: use the absolute value of the charge on one ion as the subscript for the other ion. This method is shown schematically as follows: Crossing charges. One method for obtaining subscripts in the empirical formula is by crossing charges. When crossing charges, it is sometimes necessary to reduce the subscripts to their simplest ratio to write the empirical formula. Consider, for example, the compound formed by Mg2+ and O2−. Using the absolute values of the charges on the ions as subscripts gives the formula Mg2O2:Polyatomic Ions Polyatomic ions are groups of atoms that bear net electrical charges, although the atoms in a polyatomic ion are held together by the same covalent bonds that hold atoms together in molecules. Just as there are many more kinds of molecules than simple elements, there are many more kinds of polyatomic ions than monatomic ions. Two examples of polyatomic cations are the ammonium (NH4+) and the methylammonium (CH3NH3+) ions. P. The method used to predict the empirical formulas for ionic compounds that contain monatomic ions can also be used for compounds that contain polyatomic ions. The overall charge on the cations must balance the overall charge on the anions in the formula unit. Thus, K+ and NO3− ions combine in a 1:1 ratio to form KNO3 (potassium nitrate or saltpeter), a major ingredient in black gunpowder. Similarly, Ca2+ and SO42− form CaSO4 (calcium sulfate), which combines with varying amounts of water to form gypsum and plaster of Paris. The polyatomic ions NH4+ and NO3− form NH4NO3 (ammonium nitrate), a widely used fertilizer and, in the wrong hands, an explosive. One example of a compound in which the ions have charges of different magnitudes is calcium phosphate, which is composed of Ca2+ and PO43− ions; it is a major component of bones. The compound is electrically neutral because the ions combine in a ratio of three Ca2+ ions [3(+2) = +6] for every two ions [2(−3) = −6], giving an empirical formula of Ca3(PO4)2; the parentheses around PO4 in the empirical formula indicate that it is a polyatomic ion. Writing the formula for calcium phosphate as Ca3P2O8 gives the correct number of each atom in the formula unit, but it obscures the fact that the compound contains readily identifiable PO43− ions.Summary • There are two fundamentally different kinds of chemical bonds (covalent and ionic) that cause substances to have very different properties. • The composition of a compound is represented by an empirical or molecular formula, each consisting of at least one formula unit.Contributors The atoms in chemical compounds are held together by attractive electrostatic interactions known as chemical bonds. Ionic compounds contain positively and negatively charged ions in a ratio that results in an overall charge of zero. The ions are held together in a regular spatial arrangement by electrostatic forces. Most covalent compounds consist of molecules, groups of atoms in which one or more pairs of electrons are shared by at least two atoms to form a covalent bond. The atoms in molecules are held together by the electrostatic attraction between the positively charged nuclei of the bonded atoms and the negatively charged electrons shared by the nuclei. The molecular formula of a covalent compound gives the types and numbers of atoms present. Compounds that contain predominantly carbon and hydrogen are called organic compounds, whereas compounds that consist primarily of elements other than carbon and hydrogen are inorganic compounds. Diatomic molecules contain two atoms, and polyatomic molecules contain more than two. A structural formula indicates the composition and approximate structure and shape of a molecule. Single bonds, double bonds, and triple bonds are covalent bonds in which one, two, and three pairs of electrons, respectively, are shared between two bonded atoms. Atoms or groups of atoms that possess a net electrical charge are called ions; they can have either a positive charge (cations) or a negative charge (anions). Ions can consist of one atom (monatomic ions) or several (polyatomic ions). The charges on monatomic ions of most main group elements can be predicted from the location of the element in the periodic table. Ionic compounds usually form hard crystalline solids with high melting points. Covalent molecular compounds, in contrast, consist of discrete molecules held together by weak intermolecular forces and can be gases, liquids, or solids at room temperature and pressure. An empirical formula gives the relative numbers of atoms of the elements in a compound, reduced to the lowest whole numbers. The formula unit is the absolute grouping represented by the empirical formula of a compound, either ionic or covalent. Empirical formulas are particularly useful for describing the composition of ionic compounds, which do not contain readily identifiable molecules. Some ionic compounds occur as hydrates, which contain specific ratios of loosely bound water molecules called waters of hydration.

Representing Numbers 1 - 15

Gauss 7: Representing Numbers 1

Representing Numbers: Ice Breaker 1.2

Representing Numbers: Ice Breaker 1.3

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