Odd number

“ And we attribute indeed things that are more excellent and more simple to the nature of the odd number, but things that are less excellent and more various, to the nature of the even number. For every where indeed, the odd number is the leader of impartible, simple, and unical goods; but the even number is the cause of divided, various, and generative progressions. And thus we may see all the orders of beings woven together according to divine number which is most ancient, intellectual, and exempt from all the dinumerated genera. ”

“ Socrates: Now if a number is to be twice as great as another, it must be two times that number?
Boy: Certainly.
Socrates: And if a number is two times any whole number, it must then be an even number, must it not?
Boy: Yes, Socrates.
Socrates: So, in our ratio we want to square to get two, the top number cannot be odd, can it?
Boy: No, Socrates. Therefore, the group of odd over even rational numbers cannot have the square root of two in it! Nor can the group ratios of odd numbers over odd numbers. ”

“ The one which puts first all the odd numbers and then all the even ones will be defined by: " and are finite integers, and either is odd and is even or is less than and both are odd or both are even." We shall not trouble, as a rule, to give these formulæ in future; but the fact that they could be given is essential.
The number which we have called , namely, the number of a series consisting of two progressions, is sometimes called . Multiplication, like addition, depends upon the order of the factors: a progression of couples gives a series such as which is itself a progression ”

“ For ages, people have had great faith in odd numbers. They have often been used as charms and for medicine. Some one says: "Some philosophers are of opinion that all things are composed of number, prefer the odd before the other, and attribute to it a great efficacy and perfection, especially in matters of physic: wherefore it is that many doctors prescribed always an odd pill, an odd draught, or drop to be taken by their patients. For the perfection thereof they allege these following numbers: as 7 Planets, 7 wonders of the World, 9 Muses, 3 Graces, God is 3 in 1, &c." ”

“ When we square an even over odd ratio, the first number becomes even times even, which is two times two times some other whole number, which means it is four times the whole number, and this number must be double the second number, which is odd, as it was made of odd times odd. ”

“ It is such a Number, as divided by the rest of the Divisors, or their Product, the Remainder is 0; but divided by its own Divisor, the Remainder is an Unit. We require the Divisors proposed to be Primitive each to other, i.e. that no two or more of them can be reduced to lesser terms by any common Divisor: For if so, the Question may be possible in it self, but not revolvable by help of such Multipliers, such being impossible to be found. The reason is, because the Product of an Odd and Even Number is always Even, and that divided by an Even Number, leaves either Nothing, or an Even Number. ”

“ In the even tens it is quite regularly decimal, unless 80 is of the structure suggested above. On the other hand, the odd tens are formed in the ordinary vigesimal manner. The reason for this anomaly is not obvious. I know of no other number system that presents the same peculiarity, and cannot give any hypothesis which will satisfactorily account for its presence here. ”

“ This is due to his habit of counting by pairs, which enables him to discover whether any number within reasonable limit is odd or even. Some of the negro tribes of Africa, and of the Indian tribes of America, have the same habit. Progression by pairs may seem to some tribes as natural as progression by single units. It certainly is not at all rare; and in Australia its influence on spoken number systems is most apparent.
Any number system which passes the limit 10 is reasonably sure to have either a quinary, a decimal, or a vigesimal structure. ”

“ The Assiniboines call 7, shak ko we, or u she nah, the odd number.150 The Crow 1, hamat, signifies “the least”;151 the Mississaga 1, pecik, a very small thing.152 In Javanese, Malay, and Manadu, the words for 1, which are respectively siji, satu, and sabuah, signify 1 seed, 1 pebble, and 1 fruit respectively153—words as natural and as much to be expected at the beginning of a number scale as any finger name could possibly be. ”

“ Evidently, then, these thinkers also consider that number is the principle both as matter for things and as forming their modifications and their permanent states, and hold that the elements of number are the even and the odd, and of these the former is unlimited, and the latter limited; ”

“ We are enabled, then, as a result of the study of these guesses, to say that under the conditions offered, aside from a preference of 0 over 1 to end the numbers selected, digits representing odd numbers are conspicuously preferred to those representing even numbers. ”

“ The large number of online offerings is bewildering. To be successful, you must have a sound search strategy. Your first task is to locate useful SOURCES of information. The next, to decide how best to find that specific piece of information online. You must PLAN your search. Although one source of information, like an online database, is supposed to cover your area of interest, it may still be unable to give you what you want. ”

“ They need new newspapers, magazines, information sources, and services. Trial and error are required to find online gold mines. As you get more experience, focus tends to shift from getting information to digesting. Getting the data gets 'into your fingers', and doesn't bother much anymore. The number of retrieved lines increases. The only bad news is that your reading speed remains at the same old level. In our time, people tend to talk more than they listen, and you usually find more information than knowledge. Therefore, say NO to irrelevant information. It is seldom worth keeping. ”

“ A few people must be responsible for getting the meetings fired up and keep the discussion rolling. The meeting's organization may depend on the number of participants, where they come from, the exclusivity of the forum, and the purpose of the "meeting." In large meetings, with free access for outsiders, the best strategy may be to appoint a Moderator-Editor. ”

“ If I select "k" from a menu provided by my communications program's command scripts, then my system will retrieve today's business news from Tokyo and put them up on my screen. In the process, my system tells the modem to do several things, including "call a number," "speed 2400 bps," "redial if busy," "go on-hook when done." The only thing that I have to do, is press "k". The communications program and the modem will do the rest. Automatic communication is impossible without an intelligent modem. ”

“ This is the third time; I hope good luck lies in odd numbers. Away! go. They say there is divinity in odd numbers, either in nativity, chance, or death. ”

“ Asynchronous or synchronous modems? —————————————————- Formerly, data communication was done by sending job commands to a mainframe computer, and having the result returned in one batch. The modems were called synchronous. Such modems (and computers) are still in use in some large corporations. Most of today's online services are based on an interactive dialog between the user and the remote computer. The user enters a command, for example a letter or a number in a menu, and the result is returned almost immediately. ”

“ Others refer to "conferences" by using the name of the software used to administer the discussions, like LISTSERV, PortaCom, News, Usenet, Caucus, or PARTIcipate. On the bottom line, we're still talking email. However, while private mail is usually read by one recipient only, 'conference mail' may be read by thousands of people from the whole world. All of them can talk and discuss SIMULTANEOUSLY. It is almost impossible for one single individual to dominate. The number of active participants can therefore be far larger than in 'face-to- face' conferences. ”

“ The Hayes standard ————————— The U.S. company Hayes Microcomputer Products, Inc. pioneered command-driven modems. Their Smartmodem became a success, and "Hayes compatibility" a standard for intelligent modems. Today, it is as unimportant to buy a Hayes modem to get access to Hayes commands, as to buy an IBM PC to run PC software. Automatic dialing (autodial) was one of Smartmodem's important features. The modem could call a number and prepare for data communication, once a connection had been set up. ”

“ This list may suggest lack of connectivity between networks that do indeed have connections. For example, Internet email may easily be sent to ATT Mail, Alternex, BIX, BITNET, FredsNaetet, GeoNet, GreenNet, and many others on this list. Connectivity changes constantly. Check to see if you really need it, as this service is far from free. DASnet also lets you send email as telex, fax and by ordinary mail. They charge you by the number of characters transferred, and the destination address. ”

“ The services ———————- The most popular online services are electronic mail, chat, file transfers, conferences and discussion forums, news, reading of online journals and grassroots publications, database searching, entertainment. The online world has an infinite number of niches, things that people are interested in and have fun doing. ”

“ Any such mode of counting, whether involving the use of the fingers or not, is to be regarded simply as an extraneous aid in the expression or comprehension of an idea which the mind cannot grasp, or cannot retain, without assistance. The German student scores his reckoning with chalk marks because he might otherwise forget; while the Andaman Islander counts on his fingers because he has no other method of counting,—or, in other words, of grasping the idea of number. A single illustration may be given which typifies all practical methods of numeration. ”

“ Another and much more common oddity is the tendency to visualize numerals in a peculiar manner, which characterizes, as I have roughly reckoned, about one woman in every fifteen, and one man in every thirty. Those who do so are never able to dissociate any single number from its own particular place in the field of their mental view, so that when they think of a series of numbers they always visualize them in a certain form. ”

“ That the mind distinguish carefully two ideas, which are different one from another only by the addition or subtraction of one unit : (2) That it retain in memory the names or marks of the several combinations, from an unit to that number ; and that not confusedly, and at random, but in that exact order that the numbers follow one another. In either of which, if it trips, the whole business of numbering will be disturbed, and there will remain only the confused idea of multitude, but the ideas necessary to distinct numeration will not be attained to. ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ But the grand agent for transforming experimental into deductive sciences, is the science of number. The properties of numbers, alone among all known phenomena, are, in the most rigorous sense, properties of all things whatever. All things are not coloured, or ponderable, or even extended; but all things are numerable. And if we consider this science in its whole extent, from common arithmetic up to the calculus of variations, the truths already ascertained seem all but infinite, and admit of indefinite extension. ”

“ The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds further, the quinary base may be retained, or it may be supplanted by a decimal or a vigesimal base. How readily the one or the other may predominate is seen by a glance at the following numerals:
By the time 20 is reached the savage has probably allowed his conception of any aggregate to be so far modified that this number does not present itself to his mind as 4 fives. ”

“ Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. ”

“ A fresh mental effort is required of the ignorant child of nature; and the result is “all the fingers and one more,” “both hands and one more,” “one on another count,” or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. ”

“ A most inviting field for research is, however, furnished by the primitive methods of counting and of giving visible expression to the idea of number. Our starting-point must, of course, be the sign language, which always precedes intelligible speech; and which is so convenient and so expressive a method of communication that the human family, even in its most highly developed branches, never wholly lays it aside. It may, indeed, be stated as a universal law, that some practical method of numeration has, in the childhood of every nation or tribe, preceded the formation of numeral words. ”