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Positional notation

Positional notation or place-value notation is a method of representing or encoding numbers. Indian mathematicians developed the Hindu-Arabic numeral system, the modern decimal positional notation in the 9th century. Positional notation is distinguished from other notations (such as Roman numerals) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This greatly simplified arithmetic and led to the quick spread of the notation across the world.

With the use of a radix point, the notation can be extended to include fractions and the numeric expansions of real numbers.

## History

Today, the base 10 (decimal) system is ubiquitous. It was likely motivated by counting with the ten fingers. However, other bases have been used. For example, the Babylonian numeral system, credited as the first positional number system, was base 60.

Counting rods and most abacuses in history represented numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation) were used such as Roman Numerals, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic.

With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (13th&ndash;16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.

Georges Ifrah concludes in his Universal History of Numbers:

Thus it would seem highly probable under the circumstances that the discovery of zero and the place-value system were inventions unique to the Indian civilization. As the Brahmi notation of the first nine whole numbers (incontestably the graphical origin of our present-day numerals and of all the decimal numeral systems in use in India, Southeast and Central Asia and the Near East) was autochthonous and free of any outside influence, there can be no doubt that our decimal place-value system was born in India and was the product of Indian civilization alone.|

Aryabhata stated "sthÄ�nam sthÄ�nam daÅ›a guá¹‡am" meaning "From place to place, ten times in value". His system lacked zero. The zero was added by Brahmagupta. Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras.

A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheques require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud.

## Mathematics

### Base of the numeral system

In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9.

The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.

The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.

(In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)

In base-10 (decimal) positional notation, there are 10 decimal digits and the number

2506 = 2 \times 10^3 + 5 \times 10^2 + 0 \times 10^1 + 6 \times 10^0 .

In base-16 (hexadecimal), there are 16 hexadecimal digits (0â€“9 and Aâ€“F) and the number

171\mathrm{B} = 1 \times 16^3 + 7 \times 16^2 + 1 \times 16^1 + \mathrm{B} \times 16^0 (where B represents the number eleven as a single symbol)

In general, in base-b, there are b digits and the number

a_3 a_2 a_1 a_0 = a_3 \times b^3 + a_2 \times b^2 + a_1 \times b^1 + a_0 \times b^0 (Note that a_3 a_2 a_1 a_0 represents a sequence of digits, not implicit multiplication)

### Digits and numerals

In order to discuss bases other than the decimal system (base ten), a distinction needs to be made between a number and the digit representing that number. Each digit may be represented by a unique symbol or by a limited set of symbols.

For example, in the decimal positional numeral system, there are ten possible digits in each position. These are "0", "1", "2", "3", "4", "5", "6", "7", "8" , and "9" (henceforth "0-9"). In other bases, the digits used may be unfamiliar or may be used to indicate numbers other than those they represent in the

Scientific calculator

A scientific calculator is a type of electroniccalculator, usually but not always handheld, designed to calculate problems in science (especially physics), engineering, and mathematics. They have almost completely replaced slide rules in almost all traditional applications, and are widely used in both education and professional settings.

In certain contexts such as higher education, scientific calculators have been superseded by graphing calculators, which offer a superset of scientific calculator functionality along with the ability to graph input data and write and store programs for the device. There is also some overlap with the financial calculator market.

## Functions

Modern scientific calculators generally have many more features than a standard four or five-function calculator, and the feature set differs between manufacturers and models; however, the defining features of a scientific calculator include:

In addition, high-end scientific calculators will include:

While most scientific models have traditionally used a single-line display similar to traditional pocket calculators, many of them have at the very least more digits (10 to 12), sometimes with extra digits for the floating point exponent. A few have multi-line displays, with some recent models from Hewlett-Packard, Texas Instruments, Casio, Sharp, and Canon using dot matrix displays similar to those found on graphing calculators.

## Uses

Scientific calculators are used widely in any situation where quick access to certain mathematical functions is needed, especially those such as trigonometric functions that were once traditionally looked up in tables; they are also used in situations requiring back-of-the-envelope calculations of very large numbers, as in some aspects of astronomy, physics, and chemistry.

They are very often required for math classes from the junior high school level through college, and are generally either permitted or required on many standardized tests covering math and science subjects; as a result, many are sold into educational markets to cover this demand, and some high-end models include features making it easier to translate the problem on a textbook page into calculator input, from allowing explicit operator precedence using parentheses to providing a method for the user to enter an entire problem in as it is written on the page using simple formatting tools.

## History

The first scientific calculator that included all of the basic features above was the programmable Hewlett-PackardHP-9100A, released in 1968, though the Wang LOCI-2 and the Mathatronics Mathatron had some features later identified with scientific calculator designs. The HP-9100 series was built entirely from discrete transistor logic with no integrated circuits, and was one of the first uses of the CORDIC algorithm for trigonometric computation in a personal computing device, as well as the first calculator based on reverse Polish notation entry. HP became closely identified with RPN calculators from then on, and even today some of their high-end calculators (particularly the long-lived HP-12C financial calculator and the HP-48 series of graphing calculators) still offer RPN as their

Egyptian fraction

An Egyptian fraction is the sum of distinct unit fractions, such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positiverational numbera/b; for instance the Egyptian fraction above sums to 43/48. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2/3 and 3/4 as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.

## Ancient Egypt

For more information on this subject, seeEgyptian numerals, Eye of Horus, and Egyptian mathematics.

Egyptian fraction notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. Five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period; it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations.

### Notation

To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph D21 (er, "[one] among" or possibly re, mouth) above a number to represent the reciprocal of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example:

The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size of numbers greater than 1/2 when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation.

The Egyptians also used an alternative notation modified from the Old Kingdom and based on the parts of the Eye of Horus to denote a special set of fractions of the form 1/2k (for k = 1, 2, ..., 6), that is, dyadic rational numbers. These "Horus-Eye fractions" were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, as described in the Akhmim Wooden Tablet. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the usual Egyptian fraction notation as multiples of a ro, a unit equal to 1/320 of a hekat.

### Calculation methods

Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, they do not match any single identity; rather, different methods were used for prime and for composite denominators, and more than one method was used for numbers of each type:

• For small odd prime denominators p, the expansion 2/p = 2/(p + 1) + 2/p(p + 1) was used.
• For larger prime denominators, an expansion of the form 2/p = 1/A + (2A-p)/Ap was used, where A is a number with many divisors (such as a practical number) in the range p/2 < A< p. The remaining term (2A-p)/Ap was expanded by representing the number 2A-p as a sum of divisors of A and forming a fraction d/Ap for each such divisor d

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as numbers 1 and 2, function symbols sin and +; conceptual symbols, such as lim, dy/dx, equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter-Dynkin diagrams.

## Definition

A mathematical notation is a writing system used for recording concepts in mathematics.

• The notation uses symbols or symbolic expressions which are intended to have a precise semantic meaning.
• In the history of mathematics, these symbols have denoted numbers, shapes, patterns, and change. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing mathematics as a language.

The media used for writing are recounted below, but common materials currently include paper and pencil, board and chalk (or dry-erase marker), and electronic media. Systematic adherence to mathematical concepts is a fundamental concept of mathematical notation. (See also some related concepts: Logical argument, Mathematical logic, and Model theory.)

## Expressions

A mathematical expression is a sequence of symbols which can be evaluated. For example, if the symbols represent numbers, the expressions are evaluated according to a conventional order of operations which provides for calculation, if possible, of any expressions within parentheses, followed by any exponents and roots, then multiplications and divisions and finally any additions or subtractions, all done from left to right. In a computer language, these rules are implemented by the compilers. For more on expression evaluation, see the computer science topics: eager evaluation, lazy evaluation, and evaluation operator.

## Precise semantic meaning

Modern mathematics needs to be precise, because ambiguous notations do not allow formal proofs. Suppose that we have statements, denoted by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the symbols refer to those denoted objects, perhaps in a model. The semantics of that object has a heuristic side and a deductive side. In either case, we might want to know the properties of that object, which we might then list in an intensional definition.

Those properties might then be expressed by some well-known and agreed-upon symbols from a table of mathematical symbols. This mathematical notation might include annotation such as

• "All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function"
• "A mapping from the real numbers to the complex numbers"

In different contexts, the same symbol or notation can be used to represent different concepts. Therefore, to fully understand a piece of mathematical writing, it is important to first check the definitions that an author gives for the notations that are being used. This may be problematic if the author assumes the reader is already familiar with the notation in use.

## History

### Counting

It is believed that a mathematical notation to represent counting was first developed at least 50,000 years ago â€” early mathematical ideas such as finger counting have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts.

The development of zero as a number is one of the most important developments in early mathematics. It was used as a placeholder by the Babylonians and Greek Egyptians, and then as an integer by the Mayans, Indians and Arabs. (See From Yahoo Answers

Question:i understand the basics like three to the third power equals 27. i dont know how to do decimals, such as 2 to the 1.6th power. i need steps on how to write out the answer or how to get the answer from a scientific calculator. also any websites with related material is welcome, thanx.

Answers:The best way to go about it is think of decimals as fractions and applying it as such: 1.6 = 8/5 for example: x^1/2 = sqrt(x) x^3/2 = (sqrt(x))^3 so x^1.6 can be treated as (x^1/5)^8

Question:The instructions are- enter each of the following functions into the "Y=" screen one at a time. After the function is entered, go to 2nd "graph" to look at the table of values generated by your function. Use the up and down arrow keys to scroll through the values. Remember the domain is input or x-values and range is output or y-values. Write your answers in Algebraic, Interval, and Set Notation. Note: all answers will be intergers. 1. Y=X I don't want to put my whole homework sheet on here; because i want to learn how to do it. But I just don't get it! It comes up on my graphing calculator as a diagonal line. Bleh. Does anyone know how to explain? I'll put another problem up so you can explain more if you get it. Thank you! 2. y= -7

Answers:Here's your problem. You are at the "y=" screen, you enter the right side your equation (so for your first problem, just put in x). Now push the 2nd button, it's just below the "y=" key, it should be yellow or blue depending on what calculator you have. After you have pressed the 2nd button, push the graph button. That will take you to a table of values. You can then use your up and down arrow keys to look up and down at the y values and the corresponding x values that go with them. Then you can write them down. NOTE: For the first one, all real numbers will work, you will never come to end if you keep scrolling on the table-so don't try to go on forever. Hope this helps some. Algebraic notation would be something like this (this is just an example): 0 < x <4. Interval notation would be something like this (again, just an example): (- , 6] U [8,12). Remember with set notation a bracket means the number IS included, meaning it can be used, a parenthesis means it is NOT included and cannot be used. Set notation would be something like this (again, just an example): {x such that x is inbetween 0 and 4}

Question:what are these numbers scientific notation 56,000 70,000 20,000 Now, assume each number for task 1 doubles. Show how each number is multiplied by 2 and find the new numbers. Be sure each number is in scientific notation!

Answers:In scientific notation you write a number with a long string of zeros in a shorthand fashion. 56000 Look at the first non-zero part and rewrite as a number between 1 and 10 56 -> 5.6 Compare where the decimal is now (after the 5) and where it was originally (4 places to the right or positive 4). This tells you the power of 10 to use. 56000 = 5.6x10^4 or 5.6E4 in calculator speak. The others work the same 70000 = 7.0x10^4 = 7.0E4 20000 = 2.0x10^4 = 2.0E4 What happens if you double a number in scientific notation? 2x(5.6x10^4) = 11.2x10^4, but this breaks the rule about the significant part being between 1 and 10. 11.2 = 1.12x10 11.2x10^4 = 1.12x10x10^4 = 1.12x10^5 or 1.12E5 Or 2x56000 = 112000 = 1.12x10^5 (because the decimal should be 5 to the right from its new location) 2x70000 = 140000 = 1.4x10^5 2x20000 = 40000 = 4.0x10^4 (Notice the power of ten doesn't change here.) Just in passing... .00000645 = 6.45x10^-6 = 6.45E-6 because the decimal belongs 6 steps to the left of its new location.

Question:

Answers:17 325/1000 Hope this helps....R (the numbers after the decimal are tenths , hundreths, and thousanths)