Scientific notation - This class requires HEAVY Notes, this will help to get ahead in the game. Very PDF

Preview

Summary

This class requires HEAVY Notes, this will help to get ahead in the game. Very tough class if you have Ben Miller...

Description

Scientific notation - Wikipedia, the free encyclopedia

Page 1 of 8

Scientific notation From Wikipedia, the free encyclopedia

Scientific notation is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is commonly used in calculators, and by scientists, mathematicians, doctors, and engineers. In scientific notation all numbers are written like this:

("a times ten raised to the power of b"), where the exponent b is an integer, and the coefficient a is any real number (but see normalized notation below), called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).

Ordinary decimal notation Scientific notation (normalized) 300

3 × 102

4,000

4 × 103

5,720,000,000

5.72 × 109

0.000 000 0061

6.1 × 10−9

Scientific notation Wikipedia the free encyclopedia

Page 2 of 8

notation  5.1 Converting  5.2 Basic operations  



6 See also 7 Notes and references 8 External links

Normalized notation Any given number can be written in the form of a × 10b in many ways; for example 350 can be written as 3.5 × 102 or 35 × 101 or 350 × 100. In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1  | a| < 10). For example, 350 is written as 3.5 × 102. This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., minus one half is −5 × 10−1). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 itself cannot be written in normalised scientific notation since the mantissawould have to be zero and the exponent undefined. In many fields, scientific notation is normally in this way, except during intermediate calculations or when an unnormalised form, such as engineering notation, is desired. (Normalized) scientific notation is often called exponential notation—although the latter term is more general and also applies when a is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (as in 315 × 220).

E notation Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 107 cannot always be A calculator display conveniently represented on computers, typewriters and showing the Avogadro calculators, an alternative format is often used: the letter E or constant in E notation e represents times ten raised to the power of, thus replacing the × 10, followed by the value of the exponent. Note that the x character e is not related to the mathematical constant e or the exponential function e (a

Scientific notation Wikipedia the free encyclopedia

Page 3 of 8

confusion that is less likely with capital E); and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation (though the latter also occurs).

Examples and alternatives 





[1]

In the C++, FORTRAN, MATLAB, Perl, Java and Python programming languages, 6.0221418E23 or 6.0221418e23 is equivalent to 6.022 1418 × 1023. FORTRAN also uses "D" to signify double precision numbers.[2] The ALGOL 60 programming language uses a subscript ten "10" character instead of the letter E, for example: 6.02214151023 .[3] The ALGOL 68 programming language has the choice of 4 characters: e, E, \, or10. By examples: 6.0221415e23, 6.0221415E23, 6.0221415\23 or 6.02214151023 .[4]







Decimal Exponent Symbol is part of "The Unicode Standard 6.0" e.g. 6.0221415฀23 - it was included to accommodate usage in the programming languages Algol 60 and Algol 68. The TI-83 series and TI-84 Plus series of calculators use a stylized E character to display decimal exponent and the 10 character to denote an equivalent Operator[7]. The Simula programming language requires the use of & (or && for long), for example: 6.0221415&23 (or 6.0221415&&23). [5]

Engineering notation Main article: Engineering notation Engineering notation differs from normalized scientific notation in that the exponent b is restricted to multiples of 3. Consequently, the absolute value of a is in the range 1  |a| < 1000, rather than 1  |a| < 10. Though similar in concept, engineering notation is rarely called scientific notation. This allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. For example, 12.5 × 10−9 m can be read as "twelve-point-five nanometers" or written as 12.5 nm, while −8

its scientific notation counterpart 1.25 × 10 m would likely be read out as "one-pointtwo-five times ten-to-the-negative-eighth meters".

Use of spaces In normalized scientific notation, in E notation, and in engineering notation, the space (which in typesetting may be represented by a normal width space or a thin space) that is

Scientific notation Wikipedia the free encyclopedia

Page 4 of 8

allowed only before and after "×" or in front of "E" or "e" is sometimes omitted, though it [6]

is less common to do so before the alphabetical character.

Examples 







An electron's mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. In scientific notation, this is written 9.109 3822 × 10−31 kg. The Earth's mass is about 5 973 600 000 000 000 000 000 000 kg. In scientific notation, this is written 5.9736 × 1024 kg. The Earth's circumference is approximately 40 000 000 m. In scientific notation, this is 4 × 107 m. In engineering notation, this is written 40 × 106 m. In SI writing style, this may be written "40 Mm" (40 megameters). 4

An inch is 25 400 micrometers. Describing an inch as 2.5400 × 10 µm unambiguously states that this conversion is correct to the nearest micrometer. An 4

approximated value with only three significant digits would be 2.54 × 10 µm instead. In this example, the number of significant zeros is actually infinite (which is not the case with most scientific measurements, which have a limited degree of precision). It can be properly written with the minimum number of significant zeros used with other numbers in the application (no need to have more significant digits that other factors or addends). Or a bar can be written over a single zero, indicating that it repeats forever. The bar symbol is just as valid in scientific notation as it is in decimal notation.

Significant figures Ambiguity of the last digit in scientific notation It is customary in scientific measurements to record all the significant digits from the measurements, and to guess one additional digit if there is any information at all available to the observer to make a guess. The resulting number is considered more valuable than it would be without that extra digit, and it is considered a significant digit because it contains some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together). Additional information about precision can be conveyed through additional notations. In some cases, it may be useful to know how exact the final significant digit is. For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.602 176 487(40) × 10−19 C,[7] which is shorthand for −19 C. 1.602 176 487 ± 0.000 000 040 × 10

Scientific notation Wikipedia the free encyclopedia

Page 5 of 8

Order of magnitude Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is 0.000 000 000 000 000 000 000 000 001 6726 kg. If this is written as 1.6726 × 10−27 kg, it is easier to compare this mass with that of the electron, given above The order of magnitude of the ratio of the masses can be obtained by comparing the exponents instead of the more error-prone task of counting the leading zeros. Inthis case, −27 is larger than −31 and therefore the proton is roughly four orders of magnitude (about 10 000 times) more massive than the electron. Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as billion, which might indicate either 109 or 1012.

Using scientific notation Converting To convert from ordinary decimal notation to scientific notation, move the decimal separator the desired number of places to the left or right, so that the significand will be in the desired range (between 1 and 10 for the normalized form). If you moved the decimal point n places to the left then multiply by 10n ; if you moved the decimal point n places to the right then multiply by 10−n. For example, starting with 1 230 000, move the decimal point six places to the left yielding 1.23, and multiply by 106, to give the result 6

1.23 × 10 . Similarly, starting with 0.000 000 456, move the decimal point seven places to the right yielding 4.56, and multiply by 10−7, to give the result 4.56 × 10−7. 0

If the decimal separator did not move then the exponent multiplier is logically 10 , which is correct since 100 = 1. However, the exponent part "× 100" is normally omitted, so, for 0 example, 1.234 × 10 is just written as 1.234. To convert from scientific notation to ordinary decimal notation, take the significand and move the decimal separator by the number of places indicated by the exponent — left if the exponent is negative, or right if the exponent is positive. Add leading or trailing 10

zeroes as necessary. For example, given 9.5 × 10 , move the decimal point ten places to the right to yield 95 000 000 000. Conversion between different scientific notation representations of the same number is achieved by performing opposite operations of multiplication or division by a power of ten on the significand and the exponent parts. The decimal separator in the significand is shifted n places to the left (or right), corresponding todivision (multiplication) by 10n,

Scientific notation Wikipedia the free encyclopedia

Page 6 of 8

and n is added to (subtracted from) the exponent, corresponding to a canceling n multiplication (division) by 10 . For example: 1.234 × 103 = 12.34 × 102 = 123.4 × 10 1 = 1234

Basic operations Given two numbers in scientific notation,

and

Multiplication and division are performed using the rules for operation with exponential functions:

and

Some examples are:

and

Addition and subtraction require the numbers to be represented using the same exponential part, so that the significant can be simply added or subtracted. : Next, add or subtract the significants:

An example:

Scientific notation Wikipedia the free encyclopedia

Page 7 of 8

See also       

Engineering notation Binary prefix Floating point ISO 31-0 ISO 31-11 Significant figure Scientific pitch notation

Notes and references ^ http://download.oracle.com/javase/tutorial/java/nutsandbolts/datatypes.html ^ http://www.math.hawaii.edu/lab/197/fortran/fort3.htm#double ^ Report on the Algorithmic Language ALGOL 60, Ed. P. Naur, Copenhagen 1960 ^ "Revised Report on the Algorithmic Language Algol 68". September 1973. http://www.springerlink.com/content/k902506t443683p5/. Retrieved April 30, 2007. 5. ^ "SIMULA Standard As defined by the SIMULA Standards Group - 3.1 Numbers". August 1986. http://prosjekt.ring.hibu.no/simula/Standard/chap_1.htm. Retrieved October 6, 2009. 6. ^ Samples of usage of terminology and variants: [1], [2], [3], [4], [5], [6] 7. ^ NIST value for the elementary charge

1. 2. 3. 4.

External links     

 

Decimal to Scientific Notation Converter Scientific Notation to Decimal Converter Scientific Notation in Everyday Life An exercise in converting to and from scientific notation MathAce » Scientific Notation—Basic explanation and sample questions with solutions. [as of 20 Apr 2011, this link is bad (404 error)] Scientific Notation To Decimal Converter. Decimal To Scientific Notation Converter.

Retrieved from "http://en.wikipedia.org/w/index.php? title=Scientific_notation&oldid=454648842" Categories: Numeration Measurement Notation

 

This page was last modified on 9 October 2011 at 02:47. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of use for details. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit

Scientific notation Wikipedia the free encyclopedia

organization.

Page 8 of 8...