Punctuation: Comma

The  Comma  shows the slightest separation which calls for punctuation at all. It should be omitted whenever possible. It is used to mark the least divisions of a sentence.

(1) A series of words or phrases has its parts separated by commas:-- "Lying, trickery, chicanery, perjury, were natural to him." "The brave, daring, faithful soldier died facing the foe." If the series is in pairs, commas separate the pairs: "Rich and poor, learned and unlearned, black and white, Christian and Jew, Muslim and Buddhist must pass through the same gate."

(2) A comma is used before a short quotation: "It was Patrick Henry who said, 'Give me liberty or give me death.'"

(3) When the subject of the sentence is a clause or a long phrase, a comma is used after such subject: "That he has no reverence for the God I love, proves his insincerity." "Simulated piety, with a black coat and a sanctimonious look, does not proclaim a Christian."

(4) An expression used parenthetically should be inclosed by commas: "The old man, as a general rule, takes a morning walk."

(5) Words in apposition are set off by commas: "McKinley, the President, was assassinated."

(6) Relative clauses, if not restrictive, require commas: "The book, which is the simplest, is often the most profound."

(7) In continued sentences each should be followed by a comma: "Electricity lights our dwellings and streets, pulls cars, trains, drives the engines of our mills and factories."

(8) When a verb is omitted a comma takes its place: "Lincoln was a great statesman; Grant, a great soldier."

(9) The subject of address is followed by a comma: "John, you are a good man."

(10) In numeration, commas are used to express periods of three figures: "Mountains 25,000 feet high; 1,000,000 dollars."

SUBJECT-VERB AGREEMENT

A singular subject calls for a singular verb, a plural subject demands a verb in the plural; as, "The boy writes," "The boys write."

The agreement of a verb and its subject is often destroyed by confusing (1) collective and common nouns; (2) foreign and English nouns; (3) compound and simple subjects; (4) real and apparent subjects.

  (1) A collective noun is a number of individuals or things   regarded as a whole; as, class regiment. When the individuals or things are prominently brought forward, use a plural verb; as The class were distinguished for ability. When the idea of the whole as a unit is under consideration employ a singular verb; as The regiment was in camp. (2) It is sometimes hard for the ordinary individual to distinguish the plural from the singular in foreign nouns, therefore, he should be careful in the selection of the verb. He should look up the word and be guided accordingly. "He was an alumnus of Harvard." "They were alumni of Harvard." (3) When a sentence with one verb has two or more subjects denoting different things, connected by and, the verb should be plural; as, "Snow and rain are   disagreeable." When the subjects denote the same thing and are connected by or the verb should be singular; as, "The man or the woman is to blame." (4) When the same verb has more than one subject of different persons or numbers, it agrees with the most prominent in thought; as, "He, and not you, is wrong."

  "Whether he or I am to be blamed."

(2) Never use the past participle for the past tense nor vice versa. This mistake is a very common one. At every turn we hear "He done it" for "He did it." "The jar was broke" instead of broken. "He would have went" for "He would have gone," etc.

(3) Take special care to distinguish between the nominative and objective case. The pronouns are the only words which retain the ancient distinctive case ending for the objective. Remember that the objective case follows transitive verbs and prepositions. Don't say "The boy who I sent to see you," but "The boy whom I sent to see you." Whom is here the object of the transitive verb sent. Don't say "She bowed to him and I" but "She bowed to him and me" since me is the objective case following the preposition to understood. "Between you and I" is a very common expression. It should be "Between you and me" since between is a preposition calling for the objective case.

(4) Be careful in the use of the relative pronouns who, which and that. Who refers only to persons; which only to things; as, "The boy who was drowned," "The umbrella which I lost." The relative that may refer to both persons and things; as, "The man that I saw." "The hat that I bought."

(5) Don't use the superlative degree of the adjective for the comparative; as "He is the richest of the two" for "He is the richer of the two."

Other mistakes often made in this connection are (1) Using the double comparative and superlative; as, "These apples are much more preferable."

"The most universal motive to business is gain." (2) Comparing objects which belong to dissimilar classes; as "There is no nicer life than a teacher." (3) Including objects in class to which they do not belong; as, "The fairest of her daughters, Eve." (4) Excluding an object from a class to which it does belong; as, "Caesar was braver than any ancient warrior."

(6) Don't use an adjective for an adverb or an adverb for an adjective. Don't say, "He acted nice towards me" but "He acted nicely toward me," and instead of saying "She looked beautifully" say "She looked beautiful."

(7) Place the adverb as near as possible to the word it modifies. Instead of saying, "He walked to the door quickly," say "He walked quickly to the door."

(8) Not alone be careful to distinguish between the nominative and objective cases of the pronouns, but try to avoid ambiguity in their use.

The amusing effect of disregarding the reference of pronouns is well illustrated by Burton in the following story of Billy Williams, a comic actor who thus narrates his experience in riding a horse owned by Hamblin, the manager:

"So down I goes to the stable with Tom Flynn, and told the man to put the saddle on him."

"On Tom Flynn?"

"No, on the horse. So after talking with Tom Flynn awhile I mounted him."

"What! mounted Tom Flynn?"

"No, the horse; and then I shook hands with him and rode off."

"Shook hands with the horse, Billy?"

"No, with Tom Flynn; and then I rode off up the Bowery, and who should I meet but Tom Hamblin; so I got off and told the boy to hold him by the head."

"What! hold Hamblin by the head?"

"No, the horse; and then we went and had a drink together."

"What! you and the horse?"

"No, me and Hamblin; and after that I mounted him again and went out of town."

"What! mounted Hamblin again?"

"No, the horse; and when I got to Burnham, who should be there but Tom Flynn,--he'd taken another horse and rode out ahead of me; so I told the hostler to tie him up."

"Tie Tom Flynn up?"

"No, the horse; and we had a drink there."

"What! you and the horse?"

"No, me and Tom Flynn."

Finding his auditors by this time in a horse laugh, Billy wound up with: "Now, look here,--every time I say horse, you say Hamblin, and every time I say Hamblin you say horse: I'll be hanged if I tell you any more about it."

Five ways ancient India changed the world – with maths

Christian Yates  

Senior Lecturer in Mathematical Biology, University of Bath

It should come as no surprise that the first recorded use of the number zero, recently discovered to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history going back over 3,000 years and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.

As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of trigonometry, algebra, arithmetic and negative numbers among other areas. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.

The number system

As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as the Vedas. In these texts, numbers were commonly expressed as combinations of powers of ten. For example, 365 might be expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name rather than a set of symbols. It is reasonable to believe that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.

From the third century BC, we also have written evidence of the Brahmi numerals, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics.

The concept of zero

Zero itself has a much longer history. The recently dated first recorded zeros, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the Babylonian and Mayan cultures in the early centuries AD and arguably in Sumerian mathematics as early as 3000-2000 BC.

But only in India did the placeholder symbol for nothing progress to become a number in its own right. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the democratisation of mathematics.

These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though Fibonnacci’s book liber abaci.

Solutions of quadratic equations

In the seventh century, the first written evidence of the rules for working with zero were formalised in the Brahmasputha Siddhanta. In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.

Rules for negative numbers

Brahmagupta also demonstrated rules for working with negative numbers. He referred to positive numbers as fortunes and negative numbers as debts. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.

This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative.

For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that negative numbers were absurd. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.

For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total.

Basis for calculus

This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his development of calculus in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.

But Indian mathematician Bhāskara had already discovered many of Leibniz’s ideas over 500 years earlier. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that would not be rediscovered in Europe for centuries.

The Kerala school of astronomy and mathematics, founded by Madhava of Sangamagrama in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would later be repeated in Europe including Taylor series expansions, infinitessimals and differentiation.

The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation suffers from the Eurocentric bias, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by providing key players at the forefront of every branch of mathematics.

(Adapted from www.theconversation.com`)