Scientific Notation - Revision history

Lila: /* Resources */

2021-10-22T18:53:44Z

Resources

Lila at 18:51, 22 October 2021

2021-10-22T18:51:43Z

Lila: /* E notation */

2021-10-22T18:50:54Z

E notation

Lila: /* E notation */

2021-10-22T18:50:16Z

E notation

Lila at 18:48, 22 October 2021

2021-10-22T18:48:57Z

Lila at 18:46, 22 October 2021

2021-10-22T18:46:45Z

Lila at 18:45, 22 October 2021

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← Older revision		Revision as of 18:53, 22 October 2021
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	* [https://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html Scientific Notation], Texas A&M University		* [https://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html Scientific Notation], Texas A&M University
	* [https://www.mathsisfun.com/numbers/scientific-notation.html Scientific Notation], Math Is Fun		* [https://www.mathsisfun.com/numbers/scientific-notation.html Scientific Notation], Math Is Fun
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Scientific_notation Scientific notation, Wikipedia] under a CC BY-SA license

← Older revision		Revision as of 18:51, 22 October 2021
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	==E notation==		==E notation==
−	~~<!-- Section title used as link target. -->~~
	[[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]]		[[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]]

@@ Line 52: / Line 52: @@
 [[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]]
-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''), '''EE''', '''EX''', '''E''', or '''<math>\times 10^x</math>''' 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 "× 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.
+Most calculators and many computer programs present very large and very small results in scientific notation, typically invoked by a key labelled '''EXP''' (for ''exponent''), '''EEX''' (for ''enter exponent''), '''EE''', '''EX''', '''E''', or '''<math>\times 10^x</math>''' 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 "× 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.
 ==Resources==

@@ Line 52: / Line 52: @@
 [[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]]
-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.
+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''), '''EE''', '''EX''', '''E''', or '''<math>\times 10^x</math>''' 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 "× 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.
 ==Resources==

@@ Line 31: / Line 31: @@
 |}
 In scientific notation, nonzero numbers are written in the form
-:''m'' × 10{{sup|''n''}}
+: <math> m\times 10^n </math>
 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.

@@ Line 7: / Line 7: @@
 |-
 |{{math|1=2}}
-|{{val|2e0}}
+|{{math|1=2e0}}
 |-
-|{{val|300}}
+|{{math|1=300}}
-|{{val|3e2}}
+|{{math|1=3e2}}
 |-
-|{{val|4321.768}}
+|{{math|1=4321.768}}
-|{{val|4.321768e3}}
+|{{math|1=4.321768e3}}
 |-
-|{{val|−53000}}
+|{{math|1=−53000}}
-|{{val|-5.3e4}}
+|{{math|1=-5.3e4}}
 |-
-|{{val|6720000000}}
+|{{math|1=6720000000}}
-|{{val|6.72e9}}
+|{{math|1=6.72e9}}
 |-
-|{{val|0.2}}
+|{{math|1=0.2}}
-|{{val|2e-1}}
+|{{math|1=2e-1}}
 |-
-|{{val|987}}
+|{{math|1=987}}
-|{{val|9.87e2}}
+|{{math|1=9.87e2}}
 |-
-|{{val|0.00000000751}}
+|{{math|1=0.00000000751}}
-|{{val|7.51e-9}}
+|{{math|1=7.51e-9}}
 |}
 In scientific notation, nonzero numbers are written in the form
@@ Line 39: / Line 39: @@
 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 {{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 {{math|1=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.
-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 {{val|1230400}} would become {{val|1.2304e6}} if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as {{val|1.23040e6}} or {{val|1.230400e6}}. Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.
+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 {{math|1=1230400}} would become {{math|1=1.2304e6}} if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as {{math|1=1.23040e6}} or {{math|1=1.230400e6}}. 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 {{val|1.67262192369e-27|(51)|u=kg}}, which is shorthand for {{val|1.67262192369e-27|0.00000000051|u=kg}}.
+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 {{math|1=1.67262192369e-27|(51)|u=kg}}, which is shorthand for {{math|1=1.67262192369e-27|0.00000000051|u=kg}}.
 ==E notation==

@@ Line 6: / Line 6: @@
 !Scientific notation
 |-
-|{{val|2}}
+|{{math|1=2}}
 |{{val|2e0}}
 |-

@@ Line 1: / Line 1: @@
-'''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.
+'''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;"
@@ Line 32: / Line 32: @@
 In scientific notation, nonzero numbers are written in the form
 :''m'' × 10{{sup|''n''}}
-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
+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.
+Decimal floating point is a computer arithmetic system closely related to scientific notation.
-==Normalized notation==
+==Significant figures==
-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.
+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 {{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.
@@ Line 59: / Line 46: @@
 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 {{val|1.67262192369e-27|(51)|u=kg}}, which is shorthand for {{val|1.67262192369e-27|0.00000000051|u=kg}}.
+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==
 <!-- Section title used as link target. -->
-[[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]]
+[[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]]
-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>
+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.

@@ Line 1: / Line 1: @@
-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. The form of a number in scientific notation is <math> m * 10^{k} </math>, where m is a number with 1 digit in front of the decimal point and k is an integer. For example, 65000000000 is written as <math> 6.5 * 10^{10} </math> in scientific notation, and 0.0000000009 is <math> 9 * 10^{-10} </math> in scientific notation. To write a number in scientific notation, move the decimal point to be directly after the first (AKA leftmost) nonzero digit. If the decimal was moved n digits to the left, multiply the new number by <math> 10^{n} </math>. If it was moved n digits to the right, multiply by <math> 10^{-n} </math>.
+'''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.
-Examples:
+{| class="wikitable" style="float:right; margin:5px;"
-* 700: 700 = 700., so we move the decimal point two digits to the left. Thus, 700. = <math> 7.00 * 10^2 </math> in scientific notation.
+|-
-* 0.00051: We need to move the decimal place 4 digits to the right to place it directly after 5, so 0.00051 = <math> 5.1 * 10^{-4} </math> in scientific notation.
+!Decimal notation
-* <math> 111\sqrt{7000} </math>: <math> 111\sqrt{7000} </math> is approximately 9287, so <math> 111\sqrt{7000} \approx 9.287 * 10^{3}</math>.
+!Scientific notation