Difference between revisions of "Scientific Notation"

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'''Scientific notation''' is a way of expressing [[real numbers|numbers]] that are too large or too small (usually would result in a long string of digits) to be conveniently written in [[decimal form]]. It may be referred to as '''scientific form''' or '''standard index form''', or '''standard form''' in the UK. This [[base ten]] notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain [[arithmetic operations]]. On scientific calculators it is usually known as "SCI" display mode.
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'''Scientific notation''' is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as '''scientific form''' or '''standard index form''', or '''standard form''' in the UK. This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.
  
 
{| class="wikitable" style="float:right; margin:5px;"
 
{| class="wikitable" style="float:right; margin:5px;"
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In scientific notation, nonzero numbers are written in the form  
 
In scientific notation, nonzero numbers are written in the form  
 
:''m'' × 10{{sup|''n''}}
 
:''m'' × 10{{sup|''n''}}
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or ''m'' times ten raised to the power of ''n'', where ''n'' is an [[integer]], and the [[coefficient]] ''m'' is a nonzero [[real number]] (usually between 1 and 10 in absolute value, and nearly always written as a [[decimal|terminating decimal]]). The integer ''n'' is called the [[exponent]] and the real number ''m'' is called the ''[[significand]]'' or ''mantissa''.<ref name="Calio2017">{{cite book | last1 = Caliò | first1 = Franca | first2 = Lazzari | last2 = Alessandro | title = Elements of Mathematics with Numerical Applications | publisher = Società Editrice Esculapio
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or ''m'' times ten raised to the power of ''n'', where ''n'' is an integer, and the coefficient ''m'' is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal). The integer ''n'' is called the exponent and the real number ''m'' is called the ''significand'' or ''mantissa''. The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes ''m'', as in ordinary decimal notation. In normalized notation, the exponent is chosen so that the absolute value (modulus) of the significand ''m'' is at least 1 but less than 10.
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| pages = 30–32 | date = September 2017 | isbn = 978-8893850520 }}</ref> The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the [[fractional part]] of the [[common logarithm]]. <!-- These are actually floating-point terms, not scientific notation. --> If the number is negative then a minus sign precedes ''m'', as in ordinary decimal notation. In [[#Normalized notation|normalized notation]], the exponent is chosen so that the [[absolute value]] (modulus) of the significand ''m'' is at least 1 but less than 10.
 
  
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[[Decimal floating point]] is a computer arithmetic system closely related to scientific notation.
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Decimal floating point is a computer arithmetic system closely related to scientific notation.
  
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==Normalized notation==
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==Significant figures==
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{{main|Normalized number}}
 
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Any given real number can be written in the form {{gaps|''m''|e=''n''}} in many ways: for example, 350 can be written as {{val|3.5e2}} or {{val|35e1}} or {{val|350e0}}.<!-- “Any given real number” includes irrational numbers, which can't be represented exactly. -->
 
  
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In ''normalized'' scientific notation (called "standard form" in the UK), the exponent ''n'' is chosen so that the [[absolute value]] of ''m'' remains at least one but less than ten ({{nowrap begin}}1 ≤ |''m''| < 10{{nowrap end}}). Thus 350 is written as {{val|3.5e2}}. This form allows easy comparison of numbers: numbers with bigger exponents are (due to the normalization) larger than those with smaller exponents, and subtraction of exponents gives an estimate of the number of [[orders of magnitude]] separating the numbers.  It is also the form that is required when using tables of [[common logarithm]]s. In normalized notation, the exponent ''n'' is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as {{val|5e-1}}). The 10 and exponent are often omitted when the exponent is 0.
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A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant.
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Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized or differently normalized form, such as [[engineering notation]], is desired. Normalized scientific notation is often called '''[[exponentiation|exponential]] notation'''—although the latter term is more general and also applies when ''m'' is not restricted to the range 1 to 10 (as in engineering notation for instance) and to [[base (exponentiation)|base]]s other than 10 (for example, {{gaps|3.15|base=2|e=20}}).
 
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==Engineering notation==
 
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{{Main|Engineering notation}}
 
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Engineering notation (often named "ENG" on scientific calculators) differs from normalized scientific notation in that the exponent ''n'' is restricted to [[multiple (mathematics)|multiples]] of 3. Consequently, the absolute value of ''m'' is in the range 1 ≤ |''m''| < 1000, rather than 1 ≤ |''m''| < 10. Though similar in concept, engineering notation is rarely called scientific notation. Engineering notation allows the numbers to explicitly match their corresponding [[SI prefixes]], which facilitates reading and oral communication. For example, {{val|12.5e-9|u=m}} can be read as "twelve-point-five nanometres" and written as {{val|12.5|u=nm}}, while its scientific notation equivalent {{val|1.25e-8|u=m}} would likely be read out as "one-point-two-five times ten-to-the-negative-eight metres".
 
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==Significant figures==
 
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{{Main|Significant figures}}
 
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A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes [[Significant figures#Identifying significant digits|indicated to be significant]].
 
 
Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number. Unfortunately, this leads to ambiguity. The number {{val|1230400}} is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision. The same number, however, would be used if the last two digits were also measured precisely and found to equal 0— seven significant figures.  
 
Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number. Unfortunately, this leads to ambiguity. The number {{val|1230400}} is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision. The same number, however, would be used if the last two digits were also measured precisely and found to equal 0— seven significant figures.  
  
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It is customary in scientific measurement to record all the definitely known digits from the measurement and to estimate at least one additional digit if there is any information at all available on its value. The resulting number contains more information than it would without the extra digit, which may be considered a significant digit because it conveys some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).
 
It is customary in scientific measurement to record all the definitely known digits from the measurement and to estimate at least one additional digit if there is any information at all available on its value. The resulting number contains more information than it would without the extra digit, which may be considered a significant digit because it conveys some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).
  
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Additional information about precision can be conveyed through additional notation. It is often useful to know how exact the final digit is. For instance, the accepted value of the mass of the [[proton]] can properly be expressed as {{val|1.67262192369e-27|(51)|u=kg}}, which is shorthand for {{val|1.67262192369e-27|0.00000000051|u=kg}}.
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Additional information about precision can be conveyed through additional notation. It is often useful to know how exact the final digit is. For instance, the accepted value of the mass of the proton can properly be expressed as {{val|1.67262192369e-27|(51)|u=kg}}, which is shorthand for {{val|1.67262192369e-27|0.00000000051|u=kg}}.
  
 
==E notation==
 
==E notation==
 
<!-- Section title used as link target. -->
 
<!-- Section title used as link target. -->
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[[Image:Avogadro's number in e notation.jpg|thumb|upright|A [[Texas Instruments]] [[TI-84 Plus series|TI-84 Plus]] calculator display showing the [[Avogadro constant]] in E notation]]
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[[Image:Avogadro's number in e notation.jpg|thumb|upright|A Texas Instruments TI-84 Plus calculator display showing the Avogadro constant in E notation]]
  
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Most [[calculator]]s and many [[computer program]]s present very large and very small results in scientific notation, typically invoked by a key labelled {{button|EXP}} (for ''exponent''), {{button|EEX}} (for ''enter exponent''), {{button|EE}}, {{button|EX}}, {{button|E}}, or {{button|×10<sup>''x''</sup>}} depending on vendor and model. Because [[subscript and superscript|superscripted]] exponents like 10<sup>7</sup> cannot always be conveniently displayed, the letter ''E'' (or ''e'') is often used to represent "times ten raised to the power of" (which would be written as {{nowrap|"× 10<sup>''n''</sup>"}}) and is followed by the value of the exponent; in other words, for any two real numbers ''m'' and ''n'', the usage of "''m''E''n''" would indicate a value of ''m'' × 10<sup>''n''</sup>. In this usage the character ''e'' is not related to the [[e (mathematical constant)|mathematical constant ''e'']] or the [[exponential function]] ''e''<sup>''x''</sup> (a confusion that is unlikely if scientific notation is represented by a capital ''E''). Although the ''E'' stands for ''exponent'', the notation is usually referred to as ''(scientific) E notation'' rather than ''(scientific) exponential notation''. The use of E notation facilitates data entry and readability in textual communication since it minimizes keystrokes, avoids reduced font sizes and provides a simpler and more concise display, but it is not encouraged in some publications.<ref>{{Citation |author-last=Edwards |author-first=John |date=2009 |title=Submission Guidelines for Authors: HPS 2010 Midyear Proceedings |publisher=Health Physics Society |location=[[McLean, Virginia]] |page=5 |url=http://hps.org/documents/2010_midyear_author-submission-guidelines.pdf |access-date=2013-03-30 |url-status=live |archive-url=https://web.archive.org/web/20130515112227/http://hps.org/documents/2010_midyear_author-submission-guidelines.pdf |archive-date=2013-05-15}}</ref>
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Most calculators and many computer programs present very large and very small results in scientific notation, typically invoked by a key labelled {{button|EXP}} (for ''exponent''), {{button|EEX}} (for ''enter exponent''), {{button|EE}}, {{button|EX}}, {{button|E}}, or {{button|×10<sup>''x''</sup>}} depending on vendor and model. Because superscripted exponents like 10<sup>7</sup> cannot always be conveniently displayed, the letter ''E'' (or ''e'') is often used to represent "times ten raised to the power of" (which would be written as {{nowrap|"× 10<sup>''n''</sup>"}}) and is followed by the value of the exponent; in other words, for any two real numbers ''m'' and ''n'', the usage of "''m''E''n''" would indicate a value of ''m'' × 10<sup>''n''</sup>. In this usage the character ''e'' is not related to the mathematical constant ''e'' or the exponential function ''e''<sup>''x''</sup> (a confusion that is unlikely if scientific notation is represented by a capital ''E''). Although the ''E'' stands for ''exponent'', the notation is usually referred to as ''(scientific) E notation'' rather than ''(scientific) exponential notation''. The use of E notation facilitates data entry and readability in textual communication since it minimizes keystrokes, avoids reduced font sizes and provides a simpler and more concise display, but it is not encouraged in some publications.
  
  

Revision as of 12:44, 22 October 2021

Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, or standard form in the UK. This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.

Decimal notation Scientific notation
Template:Val Template:Val
Template:Val Template:Val
Template:Val Template:Val
Template:Val Template:Val
Template:Val Template:Val
Template:Val Template:Val
Template:Val Template:Val
Template:Val Template:Val

In scientific notation, nonzero numbers are written in the form

m × 10Template:Sup

or m times ten raised to the power of n, where n is an integer, and the coefficient m is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal). The integer n is called the exponent and the real number m is called the significand or mantissa. The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m, as in ordinary decimal notation. In normalized notation, the exponent is chosen so that the absolute value (modulus) of the significand m is at least 1 but less than 10.

Decimal floating point is a computer arithmetic system closely related to scientific notation.

Significant figures

A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number. Unfortunately, this leads to ambiguity. The number Template:Val is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision. The same number, however, would be used if the last two digits were also measured precisely and found to equal 0— seven significant figures.

When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the placeholding zeroes are no longer required. Thus Template:Val would become Template:Val if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as Template:Val or Template:Val. Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.

Estimated final digits

It is customary in scientific measurement to record all the definitely known digits from the measurement and to estimate at least one additional digit if there is any information at all available on its value. The resulting number contains more information than it would without the extra digit, which may be considered a significant digit because it conveys 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 notation. It is often useful to know how exact the final digit is. For instance, the accepted value of the mass of the proton can properly be expressed as Template:Val, which is shorthand for Template:Val.

E notation

A Texas Instruments TI-84 Plus calculator display showing the Avogadro constant in E notation

Most calculators and many computer programs present very large and very small results in scientific notation, typically invoked by a key labelled Template:Button (for exponent), Template:Button (for enter exponent), Template:Button, Template:Button, Template:Button, or Template:Button depending on vendor and model. Because superscripted exponents like 107 cannot always be conveniently displayed, the letter E (or e) is often used to represent "times ten raised to the power of" (which would be written as Template:Nowrap) and is followed by the value of the exponent; in other words, for any two real numbers m and n, the usage of "mEn" would indicate a value of m × 10n. In this usage the character e is not related to the mathematical constant e or the exponential function ex (a confusion that is unlikely if scientific notation is represented by a capital E). Although the E stands for exponent, the notation is usually referred to as (scientific) E notation rather than (scientific) exponential notation. The use of E notation facilitates data entry and readability in textual communication since it minimizes keystrokes, avoids reduced font sizes and provides a simpler and more concise display, but it is not encouraged in some publications.


Resources

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