Difference between revisions of "Scientific notation" - New World Encyclopedia

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'''Scientific notation''' is a compact form of writing numbers that are very large or very small, to simplify calculations and comparisons of these numbers. This type of notation is especially useful when expressing measurements such as distances to the [[star]]s, eons of [[time]], or sizes of [[molecule]]s, [[atom]]s, and [[subatomic particle]]s.
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'''Scientific notation''' is a compact form of writing numbers that are very large or very small, to simplify calculations and comparisons of these numbers. This type of notation is especially useful when expressing measurements such as distances to the [[star]]s, eons of [[time]], or the sizes of objects that are extremely large or small.
  
Each number is expressed in the form of a power of 10. For example, the speed of light, which is approximately 300 million meters per second (m/s), is written as 3&times;10<sup>8</sup> m/s.
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Each number is expressed in the form of a power of 10. For example, the [[Earth]]'s [[mass]] is about 5,973,600,000,000,000,000,000,000&nbsp;kilograms (kg). In scientific notation, this number is written as 5.9736&times;10<sup>24</sup>&nbsp;kg. An [[electron]]'s mass, which is about 0.00000000000000000000000000000091093826&nbsp;kg, is written as 9.1093826&times;10<sup>&minus;31</sup>&nbsp;kg. The speed of light, which is approximately 300,000,000 meters per second (m/s), is written as 3&times;10<sup>8</sup> m/s.
  
 
== General form for scientific notation ==
 
== General form for scientific notation ==
  
The general form of writing a number in scientific notation is ''a''&times;10<sup>''b''</sup>, where ''b'' is an [[integer]] and is called the ''[[exponent]]'', and ''a'' is any [[real number]], called the ''[[significand]]*'' or ''[[mantissa]]*''<ref>Using "mantissa" may cause confusion because it can also refer to the fractional part of the common [[logarithm]].</ref>
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The general form of writing a number in scientific notation is ''a''&times;10<sup>''b''</sup>, where ''b'' is an [[integer]] and is called the ''[[exponent]]'', and ''a'' is any [[real number]], called the ''[[significand]]*'' or ''[[mantissa]]*''.<ref>Using "mantissa" may cause confusion because it can also refer to the fractional part of the common [[logarithm]].</ref>
  
 
In "normalized form," ''a'' is chosen such that 1 ≤ ''a'' < 10. In other words, ''a'' can have any value ranging from 1 to less than 10. (In normalized form, the value of ''a'' can be 9.99 but not 10.) Also, the exponent ''b'' gives the ''[[order of magnitude]]'' of the number. The greater the value of ''b'', the larger the overall number; and conversely, the smaller the value of ''b'', the smaller the number. It is assumed that scientific notation should be expressed in normalized form, except during calculations or when an unnormalized form is desired (such as for engineering applications).
 
In "normalized form," ''a'' is chosen such that 1 ≤ ''a'' < 10. In other words, ''a'' can have any value ranging from 1 to less than 10. (In normalized form, the value of ''a'' can be 9.99 but not 10.) Also, the exponent ''b'' gives the ''[[order of magnitude]]'' of the number. The greater the value of ''b'', the larger the overall number; and conversely, the smaller the value of ''b'', the smaller the number. It is assumed that scientific notation should be expressed in normalized form, except during calculations or when an unnormalized form is desired (such as for engineering applications).
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Most [[calculator]]s and many [[computer program]]s present very large and very small results in scientific notation. Because exponents like 10<sup>7</sup> can't always be conveniently represented on computers, typewriters, and calculators, an alternate notation is often used: the "&times;10" is omitted and replaced by the letter ''E'' or ''e'' (short for exponent). Note that this is not related to the [[e (mathematical constant)|mathematical constant ''e'']]. In this case, the exponent is not [[superscript|superscripted]] but is left on the same level with the significand (e.g. E-6 ''not'' E<sup>-6</sup>).  The sign is often given even if positive (e.g. E+11 rather than E11).  For example, 1.56234&nbsp;E+29 is the same as 1.56234&times;10<sup>29</sup>.
 
Most [[calculator]]s and many [[computer program]]s present very large and very small results in scientific notation. Because exponents like 10<sup>7</sup> can't always be conveniently represented on computers, typewriters, and calculators, an alternate notation is often used: the "&times;10" is omitted and replaced by the letter ''E'' or ''e'' (short for exponent). Note that this is not related to the [[e (mathematical constant)|mathematical constant ''e'']]. In this case, the exponent is not [[superscript|superscripted]] but is left on the same level with the significand (e.g. E-6 ''not'' E<sup>-6</sup>).  The sign is often given even if positive (e.g. E+11 rather than E11).  For example, 1.56234&nbsp;E+29 is the same as 1.56234&times;10<sup>29</sup>.
  
==Motivation==
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== Usefulness ==
Scientific notation is a very convenient way to write large or small numbers and do calculations with them.  It also quickly conveys two properties of a measurement that are useful to scientists&mdash;[[significant figure]]s and order of magnitude. 
 
  
===Examples===
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As mentioned above, scientific notation is a convenient way to write large and small numbers and perform calculations with them. In addition, this notation helps avoid misinterpretation of terminology such as "billion" or "trillion", which have different meanings in different parts of the world. Moreover, scientific notation quickly conveys two properties of a measurement that are useful to scientists: [[significant figure]]*s and order of magnitude, as explained below.
*An [[electron]]'s mass is about 0.00000000000000000000000000000091093826&nbsp;kg. In scientific notation, this is written 9.1093826&times;10<sup>&minus;31</sup>&nbsp;kg.
 
*The [[Earth]]'s [[mass]] is about 5,973,600,000,000,000,000,000,000&nbsp;kg. In scientific notation, this is written 5.9736&times;10<sup>24</sup>&nbsp;kg.
 
  
 
===Significant digits===
 
===Significant digits===
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Scientific notation is useful for indicating the [[Accuracy and precision|precision]] with which a quantity was measured.  Including only the [[significant figures]] (the digits which are known to be reliable, plus one uncertain digit) in the coefficient conveys the precison of the value.  In the absence of any statement otherwise, the value of a [[physical quantity]] in scientific notation is assumed to have been measured to at least the quoted number of digits of precision with the last potentially in doubt by half a unit.  
 
Scientific notation is useful for indicating the [[Accuracy and precision|precision]] with which a quantity was measured.  Including only the [[significant figures]] (the digits which are known to be reliable, plus one uncertain digit) in the coefficient conveys the precison of the value.  In the absence of any statement otherwise, the value of a [[physical quantity]] in scientific notation is assumed to have been measured to at least the quoted number of digits of precision with the last potentially in doubt by half a unit.  
  
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===Order of magnitude===
 
===Order of magnitude===
Scientific notation also enables simple order of magnitude comparisons. A [[proton]]'s mass is 0.0000000000000000000000000016726&nbsp;kg. If this is written as 1.6726&times;10<sup>&minus;27</sup>&nbsp;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 simply comparing the exponents rather than counting all the leading zeros.  In this case, '&minus;27' is larger than '&minus;31' and therefore the proton is 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 '[[Long and short scales|billion]]', which might indicate either 10<sup>9</sup> or 10<sup>12</sup>.
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Scientific notation also enables simple comparisons of orders of magnitude. For example, a [[proton]]'s mass, which is 0.0000000000000000000000000016726&nbsp;kg, can be written as 1.6726&times;10<sup>&minus;27</sup>&nbsp;kg. As noted above, the electron's mass is 9.1093826&times;10<sup>&minus;31</sup>&nbsp;kg. To compare the [[orders of magnitude]]* of the masses, one can simply compare the exponents rather than counting all the leading zeros. In this case, "&minus;27" is larger than "&minus;31", and therefore the proton is four orders of magnitude (about 10,000 times) more massive than the electron.
  
 
==Using scientific notation==
 
==Using scientific notation==
 
===Converting===
 
===Converting===
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Multiplication and division by 10 are easy to perform in scientific notation.
 
Multiplication and division by 10 are easy to perform in scientific notation.
  
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==See also==
 
==See also==
*[[SI prefixes]]
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*International standard [[ISO 31-0]]
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* [[integer]]
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* [[SI prefixes]]
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* International standard [[ISO 31-0]]
  
 
== Footnotes ==
 
== Footnotes ==

Revision as of 16:38, 2 December 2006

Scientific notation is a compact form of writing numbers that are very large or very small, to simplify calculations and comparisons of these numbers. This type of notation is especially useful when expressing measurements such as distances to the stars, eons of time, or the sizes of objects that are extremely large or small.

Each number is expressed in the form of a power of 10. For example, the Earth's mass is about 5,973,600,000,000,000,000,000,000 kilograms (kg). In scientific notation, this number is written as 5.9736×1024 kg. An electron's mass, which is about 0.00000000000000000000000000000091093826 kg, is written as 9.1093826×10−31 kg. The speed of light, which is approximately 300,000,000 meters per second (m/s), is written as 3×108 m/s.

General form for scientific notation

The general form of writing a number in scientific notation is a×10b, where b is an integer and is called the exponent, and a is any real number, called the significand or mantissa.[1]

In "normalized form," a is chosen such that 1 ≤ a < 10. In other words, a can have any value ranging from 1 to less than 10. (In normalized form, the value of a can be 9.99 but not 10.) Also, the exponent b gives the order of magnitude of the number. The greater the value of b, the larger the overall number; and conversely, the smaller the value of b, the smaller the number. It is assumed that scientific notation should be expressed in normalized form, except during calculations or when an unnormalized form is desired (such as for engineering applications).

Variations

Engineering notation

Engineering notation involves restricting the exponent b to multiples of 3. Engineering notation is therefore not always normalized. Numbers in this form are easily read out using magnitude prefixes like mega or nano. For example, 12.5×10-9 meters might be read or written as "twelve point five nanometers" or 12.5nm.

Exponential notation

Most calculators and many computer programs present very large and very small results in scientific notation. Because exponents like 107 can't always be conveniently represented on computers, typewriters, and calculators, an alternate notation is often used: the "×10" is omitted and replaced by the letter E or e (short for exponent). Note that this is not related to the mathematical constant e. In this case, the exponent is not superscripted but is left on the same level with the significand (e.g. E-6 not E-6). The sign is often given even if positive (e.g. E+11 rather than E11). For example, 1.56234 E+29 is the same as 1.56234×1029.

Usefulness

As mentioned above, scientific notation is a convenient way to write large and small numbers and perform calculations with them. In addition, this notation helps avoid misinterpretation of terminology such as "billion" or "trillion", which have different meanings in different parts of the world. Moreover, scientific notation quickly conveys two properties of a measurement that are useful to scientists: significant figures and order of magnitude, as explained below.

Significant digits

Scientific notation is useful for indicating the precision with which a quantity was measured. Including only the significant figures (the digits which are known to be reliable, plus one uncertain digit) in the coefficient conveys the precison of the value. In the absence of any statement otherwise, the value of a physical quantity in scientific notation is assumed to have been measured to at least the quoted number of digits of precision with the last potentially in doubt by half a unit.

As an example, consider the Earth's mass as presented above in conventional notation. Since that representation gives no indication of the accuracy of the reported value, a reader could incorrectly assume from the twenty-five digits shown that it is known right down to the last kilogram! The scientific notation indicates that it is known with a precision of ± 0.00005×1024 kg, or ± 5×1019 kg.

Where precision in such measurements is crucial more sophisticated expressions of measurement error must be used.

Order of magnitude

Scientific notation also enables simple comparisons of orders of magnitude. For example, a proton's mass, which is 0.0000000000000000000000000016726 kg, can be written as 1.6726×10−27 kg. As noted above, the electron's mass is 9.1093826×10−31 kg. To compare the orders of magnitude of the masses, one can simply compare the exponents rather than counting all the leading zeros. In this case, "−27" is larger than "−31", and therefore the proton is four orders of magnitude (about 10,000 times) more massive than the electron.

Using scientific notation

Converting

Multiplication and division by 10 are easy to perform in scientific notation.

At the mantissa, multiplication by 10 may be seen as shifting the decimal point one position to the right (adding a zero if needed): 12.34×10=123.4. Division may be seen as shifting it to the left: 12.34/10=1.234

In the exponential part multiplication by 10 results in adding 1 to the exponent: 102×10=103. Division by 10 results in subtracting 1 from the exponent: 102/10=101.

Also notice that 1 is multiplication's neutral element and that 100=1.

To convert between different representations of the same number, all that is needed is to perform the opposite operations to each part. Thus multiplying the mantissa by 10, n times is done by shifting the decimal point n times to the right. Dividing by 10 the same number of times is done by adding −n to the exponent. Some examples:

Basic operations

Given two numbers in scientific notation,

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

some examples are:


Addition and subtraction require the numbers to be represented using the same exponential part, in order to simply add, or subtract, the mantissas, so it may take two steps to perform. First, if needed, convert one number to a representation with the same exponential part as the other. This is usually done with the one with the smaller exponent. Second, add or subtract the mantissas.

an example:

See also

  • integer
  • SI prefixes
  • International standard ISO 31-0

Footnotes

  1. Using "mantissa" may cause confusion because it can also refer to the fractional part of the common logarithm.

References
ISBN links support NWE through referral fees

  • Fearon, Globe. Access to Math: Exponents and Scientific Notation 1996.

External links

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