User:Dendodge/Sandbox/scientific notation

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Below is my attempt to translate Scientific notation from Portugese (where it is a featured article) to English. Babelfish did the translation and I just need to reword and reformat it. Any help would be greatly appreciated. Please place a horizontal line where you get up to and remove the previous one.

Scientific notation is a concise way to represent numbers, usually either particularly large (e.g. 100000000000) or very small (e.g. 0.00000000001). It is based on the use of exponentiation of 10 (the cases above, in scientific notation, would be: 1·10¹¹ and 1·10^-11, respectively).

Contents 1 Introduction 2 History 3 Description 3.1 Standardized scientific Notation 4 Operations 4.1 Addition and subtraction

[edit] Introduction

It observes the numbers below:

• 600 000
• 30 000 000
• 500 000 000 000 000
• 7 000 000 000 000 000 000 000 000 000 000 000
• 0.0004
• 0.00000001
• 0.0000000000000006
• 0.0000000000000000000000000000000000000000000000008

The representation of these numbers in the form established in the memorandum of understanding becomes difficult. A main factor of difficulty is the extremely high amount of zeros and its effect on the speed of reading of the numbers.

It can be thought that these values are not very useful, and therefore of almost non-existent use in daily life. But this is incorrect. In areas such as Physics and Chemistry these values are frequently used. For example, the furthest observable point of the universe is about 740 000 000 000 000 000 000 000 000 metres from Earth, and the mass of one proton is approximately 0.00000000000000000000000000167 grams.

For values such as these, scientific notation is a more compact form of the number. Another advantage of the scientific notation is that it can always represent the amount of significant figures. For example, in the aforementioned observable point of the universe's distance, the way that is written suggests 30 significant numbers. But this is not true (it would be an incredible coincidence if 25 zeros followed in a gauging).

[edit] History

The first known attempt to represent very extensive numbers was undertaken by Greek mathematician and philosopher, Archimedes, and described in his workmanship; the Accountant of Areia^[1], in 3rd century B.C.. It developed a method of numerical representation of how many grains of sand existed in the universe. The number for it in scientific notation was 1·10⁶³ grains. It was through scientific notation that the model of representation of real numbers through floating-point was conceived. This idea was proposed solely for the Leonardo Torres y Quevedo (1914), Konrad Zuse (1936) and George Robert Stibitz (1939). The codification in floating-point of modern computers is basically a scientific notation of base 2. Programming with the use of numbers in scientific notation consecrated a representation without number subscripts. 1.785·10⁵ and 2.36·10^-14 are respectively represented by 1.785E5 and 2.36E-14.

[edit] Description

A number written in scientific notation follows the following model:

m·10^e

Number m is called mantissa and e is the order of magnitude.

[edit] Standardized scientific Notation

The basic definition of scientific notation allows an infinity of representations for each value. However, the standardized scientific notation includes a restriction: mantissa must be bigger than or equal to 1 and lower than 10. In this way each number can be represented in only one way. To transform any number into standardized scientific notation we must dislocate the decimal point, obeying the balance principle. See the example below:
253,756.42
Standardized scientific notation demands that the mantissa is between 1 and 10. In this situation, the adequate value would be 2.5375642 (it observes that the sequence of numbers is the same, all that was changed was the position of the decimal point). For the exponent, valley the balance principle:(Not sure about that bit) "Each house decimal that diminishes the value of the mantissa increases the exponent in a unit and vice versa". In the case that the illustrious representative is 5, it observes the transformation step by step:

253,756.42=25,375.642·10¹=2,537.5642·10²=253.75642·10³=25.375642·10⁴=2.5375642·10⁵

Another example, with a lesser value of 1:

0.0000000475=0.000000475·10^-1=0.00000475·10^-2=0.0000475·10^-3=0.000475·10^-4=0.00475·10^-5=0.0475·10^-6=0.475·10^-7=4.75·10^-8

In this way, the examples above will be:

• 6x10⁵
• 3 x 10⁷
• 5 x 10¹⁴
• 7 x 10³³
• 4 x 10^-4
• 1 x 10^-8
• 6 x 10^-16
• 8 x 10^-49

[edit] Operations

[edit] Addition and subtraction

For somar (I'm not sure about that bit) or to deduct two numbers in scientific notation, it is necessary to represent both in the same way or one of the values must be transformed so that its exponent is equal to the exponent of the other. The transformation follows principle of equilibrium. The result will possibly not be in standardized scientific notation, being converted later.

Examples: 4.2·10⁷+3.5·10⁵=4.2·10⁷+0.035·10⁷=4.235·10⁷

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6,32 · 10 9 - 6,25 · 10 9 = 0,07 · 10 9 ' ' (non-standard) ' ' = 7 · 10 7 ' ' (standardized) ' ' === Multiplication === We multiply [ [ mantissa]]s and we add the exponents of each value. The result will not possibly be standardized, but it can be converted: Examples: (6,5 · 10 8). (3,2 · 10 5) = (6,5 · 3,2) · 10 8+5 = 20,8 · 10 13 ' ' (non-standard) ' ' = 2,08 · 10 14 (converted for the standardized notation) ' ' (4 · 10 6) · (1,6 · 10 -15) = (4 · 1,6) · 10 6+(-15) = 6,4 · 10 -9 ' ' (already standardized without conversion necessity) ' ' === Division === We divide the mantissas and we deduct the exponents from each value. The result will not possibly be standardized, but it can be converted: Examples: (8 · 10 17)/(2 · 10 9) = (8 /2). 10 17-9 = 4 · 10 8 ' ' (standardized) ' ' (2,4 · 10 -7)/(6,2 · 10 -11) = (2,4 /6,2) · 10 -7-(-11)? 0,3871 · 10 4 ' ' (non-standard) ' ' = 3.871 · 10³ ' ' (standardized) ' ' === Exponentiation === The mantissa is raised to the external exponent and the exponent of base ten is multiplied by the external exponent. (2 · 10 6) 4 = (2 4) · 10 6 · 4 = 16 · 10 24 = 1,6 · 10 25 ' ' (standardized) ' ' === Radiciação === Before making the radiciação she is necessary to transform an exponent for a multiple value of the index. After made this, the result is the radiciação of the mantissa multiplied for ten raised to the reason between the exponent and the index of the radical. \sqrt{1,6\cdot 10^{27}} = \sqrt{16\cdot 10^{26}} = \sqrt{16}\cdot 10^{26/2} = 4\cdot 10^{13} \sqrt[5]{6,7\cdot 10^{17}} = \sqrt[5]{670\cdot 10^{15}} = \sqrt[5]{670}\cdot 10^{15/5} \approx 3,674\cdot 10^{3} == References == # Template:Nota [ http://www.educ.fc.ul.pt/docentes/opombo/seminario/contadorareia/ordensvelhas.htm Description of the notation of Archimedes ] == Ver also == * [ [ Floating-point ] ] [ [ Categoria:Números ] ] [ [ mathematical Categoria:Notação ] ] [ [ af:Wetenskaplike notasie ] ] [ [ ca:Notació scientific ] ] [ [ de:Wissenschaftliche scientific Notation ] ] [ [ en:Scientific notation ] ] [ [ es:Notación ] ] [ [ fi:Tieteellinen merkintätapa ] ] [ [ fr:Notation scientifique ] ] [ [ it:Notazione scientifica ] ] [ [ nl:Wetenschappelijke notatie ] ] [ [ no:Vitenskapelig notasjon ] ] [ [ pl:Notacja naukowa ] ] [ [ simple:Scientific notation ] ] [ [ sv:Grundpotens ] ] [ [ th:??????????????????]] [ [ zh:?????]]