Essays on numbers and figures

Using Numbers

Unary numerals used in the neural circuits responsible for birdsong production. The command signals for different notes in the birdsong emanate from different points in the HVC.


This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness. The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the geometric numerals 1, 10, , , The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals.

A sign-value system does not need arithmetic numerals because they are made by repetition except for the Ionic system , and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses both arithmetic and geometric numerals. In certain areas of computer science, a modified base k positional system is used, called bijective numeration , with digits 1, 2, This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros.

Bijective base- k numeration is also called k -adic notation, not to be confused with p -adic numbers. In a positional base b numeral system with b a natural number greater than 1 known as the radix , b basic symbols or digits corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b. Unless specified by context, numbers without subscript are considered to be decimal. By using a dot to divide the digits into two groups, one can also write fractions in the positional system.

The highest used position is close to the order of magnitude of the number. The number of tally marks required in the unary numeral system for describing the weight would have been w. A number has a terminating or repeating expansion if and only if it is rational ; this does not depend on the base. A number that terminates in one base may repeat in another thus 0.

Essays on Numbers and Figures

An irrational number stays aperiodic with an infinite number of non-repeating digits in all integral bases. This is used in punycode , one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of a—z and 0—9, representing 0—25 and 26—35 respectively.

A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b i. Unlike a regular based numeral system, there are numbers like 9b where 9 and b each represents 35; yet the representation is unique because ac and aca are not allowed — the a would terminate the number.

The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes. The case with all threshold values equal to 1 corresponds to bijective numeration , where the zeros correspond to separators of numbers with digits which are non-zero. Not all languages have numeral systems.

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Specifically, there is not much need for numeral systems among hunter-gatherers who do not engage in commerce. Many languages around the world have no numerals above two to four—or at least did not before contact with the colonial societies—and speakers of these languages may have no tradition of using the numerals they did have for counting.

Indeed, several languages from the Amazon have been independently reported to have no specific number words other than 'one'. Such languages do not have a word class of 'numeral'. Most languages with both numerals and counting use base 8, 10, 12, or Base 10 appears to come from counting one's fingers, base 20 from the fingers and toes, base 8 from counting the spaces between the fingers attested in California , and base 12 from counting the knuckles 3 each for the four fingers.

For very large and very small numbers, traditional systems have been superseded by the use of scientific notation and the system of SI prefixes. Traditional systems continue to be used in everyday life. Many languages of Melanesia have or once had counting systems based on parts of the body which do not have a numeric base; there are or were no numerals, but rather nouns for relevant parts of the body—or simply pointing to the relevant spots—were used for quantities.

For example, 1—4 may be the fingers, 5 'thumb', 6 'wrist', 7 'elbow', 8 'shoulder', etc. For numbers beyond this, the torso, legs and toes may be used, or one might count back up the other arm and back down the first, depending on the people.

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Binary systems are base 2, often using zeros and ones. With only two symbols binary is useful for logical systems like computers. Base 3 counting has practical usage in some analog logic, in baseball scoring and in self—similar mathematical structures.

Essays on the Theory of Numbers, by Richard Dedekind, 1.1

Some Austronesian and Melanesian ethnic groups, some Sulawesi and some Papua New Guineans , count with the base number four, using the term asu and aso , the word for dog , as the ubiquitous village dog has four legs. As an example of the system's ease a realistic scenario could include a farmer returning from the market with fifty asu heads of pig , less 30 asu of pig bartered for 10 asu 40 of goats noting his new pig count total as twenty asu : 80 pigs remaining. The system has a correlation to the dozen counting system and is still in common use in these areas as a natural and easy method of simple arithmetic.

Quinary systems are based on the number 5. It is almost certain the quinary system developed from counting by fingers five fingers per hand. Aztec was a vigesimal base system with sub-base 5. Kanum is one these languages. Septenary systems are very rare, as few natural objects consistently have seven distinctive features. Traditionally, it occurs in week-related timing. It has been suggested that the Palikur language may have a base-seven system, but this is dubious.

Octal counting systems are based on the number 8. Examples can be found in the Yuki language of California and in the Pamean languages of Mexico , because the Yuki and Pame keep count by using the four spaces between their fingers rather than the fingers themselves. It has been suggested that Nenets has a base-nine system. A majority of traditional number systems are decimal. This dates back at least to the ancient Egyptians , who used a wholly decimal system. Anthropologists hypothesize this may be due to humans having five digits per hand, ten in total.

Duodecimal numeric systems have some practical advantages over decimal.

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It is much easier to divide the base digit twelve which is a highly composite number by many important divisors in market and trade settings, such as the numbers 2 , 3 , 4 and 6. Because of several measurements based on twelve, [13] many Western languages have words for base-twelve units such as dozen , gross and great gross , which allow for rudimentary duodecimal nomenclature , such as "two gross six dozen" for Ancient Romans used a decimal system for integers , but switched to duodecimal for fractions , and correspondingly Latin developed a rich vocabulary for duodecimal-based fractions see Roman numerals.

A notable fictional duodecimal system was that of J. Tolkien 's Elvish languages , which used duodecimal as well as decimal. The traditional Chinese units of measurement were base The suanpan Chinese abacus can be used to perform hexadecimal calculations such as additions and subtractions. South Asian monetary systems were base One rupee in Pakistan and India was divided into 16 annay. A single anna was subdivided into four paisa or twelve pies thus there were 64 paise or pies in a rupee. The anna was demonetised as a currency unit when India decimalised its currency in , followed by Pakistan in Vigesimal numbers use the number 20 as the base number for counting.

Anthropologists are convinced the system originated from digit counting, as did bases five and ten, twenty being the number of human fingers and toes combined. Some include the classical Mesoamerican cultures, still in use today in the modern indigenous languages of their descendants, namely the Nahuatl and Mayan languages see Maya numerals. A modern national language which uses a full vigesimal system is Dzongkha in Bhutan.

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Notes serve two purposes: to provide bibliographic information and to provide additional context for information in the text. When it comes to citing illustrations, using notes allows for the bibliographic information as close to the illustration as possible. Note: Use semicolons to denote entry sections when long series of commas make these sections difficult to ascertain as being like or separate see examples below. The following examples provide information on how a note might look following an illustration. If an illustration requires more than one note, label additional notes with lowercase letters, starting with a see the note underneath the example table above.

United States; Dept. In this example, the commas in Manufacturing, Mining, and Construction Statistics prompt the need for semicolons in order for the series information to be read easily.

Even if Manufacturing, Mining, and Construction Statistics had not appeared in the entry, the multiple "author names" of United States, Dept. Furthermore, the publisher and date in a standard entry are separated by a comma and belong together; thus, their inclusion here US Dept. Do not embed illustrations tables, figures, or examples in manuscripts for publication. Put placeholders in the text to show where the illustrations will go.

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Type these placeholders on their own line, flush left, and bracketed e. At the end of the document, provide label, number, caption, and source information in an organized list. Send files for illustrations in the appropriate format to your editor separately. If you provide source information with your illustrations, you do not need to provide this information on the Works Cited page.

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