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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. It is especially useful when expressing measurements such as distances to [[star]]s, eons of [[time]], or the sizes of objects that are extremely large or small. This type of notation helps simplify calculations and comparisons of these numbers, and it helps express the precision with which quantities have been measured.
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]]'' (using "mantissa" may cause confusion because it can also refer to the fractional part of the common [[logarithm]]).
  
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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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==
 
==Variations==
 
===Engineering notation===
 
===Engineering notation===
  
[[Engineering notation]] involves restricting the exponent ''b'' to [[multiple]]s of 3. Engineering notation is therefore not always normalized. Numbers in this form are easily read out using magnitude [[SI prefix|prefixes]] like ''mega'' or ''nano''For example, 12.5&times;10<sup>-9</sup> meters might be read or written as "twelve point five nanometers" or 12.5nm.
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In [[engineering notation]], the exponent ''b'' is restricted to [[multiple]]s of 3. Engineering notation is therefore not always normalized. Numbers in this form are easily read out using magnitude [[SI prefix|prefixes]] such as ''mega'' or ''nano.'' For example, 12.5&times;10<sup>-9</sup> meters might be read or written as "twelve point five nanometers," or 12.5 nm.
  
===Exponential notation===
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=== Notation using E ===
  
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>.
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For devices such as [[calculator]]s, [[typewriter]]s, and some [[computer]]s, it may be difficult to show a number with an exponent, such as 10<sup>7</sup>. In such cases, an alternate notation is often used: "&times;10" is replaced by the letter ''E'' or ''e'' (for exponent), followed by the exponent. (Note that this ''e'' is not related to the [[e (mathematical constant)|mathematical constant ''e'']].) The sign of the exponent, whether positive or negative, is often given. For example, 1.56234&nbsp;E+29 is a representation of the number 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===
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.
 
  
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 &plusmn; 0.00005&times;10<sup>24</sup>&nbsp;kg, or &plusmn; 5&times;10<sup>19</sup>&nbsp;kg.
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Scientific notation is useful for indicating the [[Accuracy and precision|precision]] with which a quantity was measured. By including only the [[significant figure]]s (the digits that are known to be reliable, plus one uncertain digit) in the coefficient, one can convey the precision 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 digit potentially in doubt by half a unit.  
  
Where precision in such measurements is crucial more sophisticated expressions of [[measurement error]] must be used.
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Consider, for example, the Earth's mass given above in conventional notation (5,973,600,000,000,000,000,000,000&nbsp;kg). That representation gives no indication of the accuracy of the reported value, so a reader could incorrectly assume from the twenty-five digits shown that the mass is known right down to the last kilogram. By writing the number in scientific notation, one indicates that the Earth's mass is known with a precision of &plusmn; 0.00005&times;10<sup>24</sup>&nbsp;kg, or &plusmn; 5&times;10<sup>19</sup>&nbsp;kg.
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In cases where precision in such measurements is crucial, more sophisticated expressions of [[measurement error]] must be used.
  
 
===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" by the number 4, and therefore the proton is roughly four orders of magnitude (about 10,000 times) more massive than the electron.
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== Converting numbers to scientific notation ==
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 +
A number can be readily converted to scientific notation in a few simple steps. Consider the following examples.
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'''Example A'''
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To convert the number 123.4 to scientific notation, one can use the steps given below.
 +
 
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1. Write 123.4 in a form that shows it multiplied by 10<sup>0</sup>. (Note that 10<sup>0</sup> = 1, so this step is the same as multiplying the number by 1.)
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:<math>123.4 = 123.4\times10^0</math>
  
==Using scientific notation==
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2. Divide the mantissa (123.4) by 100, by shifting the decimal point two places to the left. Also multiply 10<sup>0</sup> by 100, by adding 2 to the exponent.
===Converting===
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:<math>123.4\times10^0 = (123.4/10^2) \times (10^0\times10^2) = 1.234\times10^2</math>
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&times;10=123.4. Division may be seen as shifting it to the left: 12.34/10=1.234
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Thus the scientific notation for 123.4 is 1.234&times;10<sup>2</sup>.
  
In the exponential part multiplication by 10 results in adding 1 to the exponent: 10<sup>2</sup>&times;10=10<sup>3</sup>. Division by 10 results in subtracting 1 from the exponent: 10<sup>2</sup>/10=10<sup>1</sup>.
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'''Example B'''
  
Also notice that 1 is multiplication's [[Identity element|neutral element]] and that 10<sup>0</sup>=1.
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To convert the number 0.001234 to scientific notation, one can use the following steps.
  
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 ''&minus;n'' to the exponent. Some examples:
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1. Write 0.001234 in a form that shows it multiplied by 10<sup>0</sup>.
 +
:<math>0.001234 = 0.001234\times10^0</math>
  
<math>123.4 = 123.4\times10^0 = (123.4/10^2) \times (10^0\times10^2) = 1.234\times10^2</math>
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2. Multiply the mantissa (0.001234) by 1,000, by shifting the decimal point three places to the right. Also divide 10<sup>0</sup> by 1,000, by adding &minus;3 to the exponent.
 +
:<math>.001234\times10^0 = (.001234\times 10^3) \times (10^0 / 10^3) = 1.234\times10^{-3}</math>
  
<math>.001234 = .001234\times10^0 = (.001234\times 10^3) \times (10^0 / 10^3) = 1.234\times10^{-3}</math>
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Thus the scientific notation for 0.001234 is 1.234&times;10<sup>&minus;3</sup>.
  
===Basic operations===
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== Basic operations with numbers in scientific notation ==
Given two numbers in scientific notation,
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Consider two numbers, x<sub>0</sub> and x<sub>1</sub>, in scientific notation. They may be written as follows.
  
 
:<math>x_0=a_0\times10^{b_0}</math>
 
:<math>x_0=a_0\times10^{b_0}</math>
  
 
:<math>x_1=a_1\times10^{b_1}</math>
 
:<math>x_1=a_1\times10^{b_1}</math>
 +
 +
'''Multiplication and division:'''
  
 
[[Multiplication]] and [[division (mathematics)|division]] are performed using the rules for operation with exponential functions:
 
[[Multiplication]] and [[division (mathematics)|division]] are performed using the rules for operation with exponential functions:
Line 66: Line 79:
 
:<math>\frac{x_0}{x_1}=\frac{a_0}{a_1}\times10^{b_0-b_1}</math>
 
:<math>\frac{x_0}{x_1}=\frac{a_0}{a_1}\times10^{b_0-b_1}</math>
  
some examples are:
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Examples:
 +
 
 
:<math>5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2}  </math>
 
:<math>5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2}  </math>
  
:<math>\frac{2.34\times10^2}{5.67\times10^{-5}}  \approx 0.413\times10^{7} = 4.13\times10^6  </math>
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:<math>\frac{2.34\times10^2}{5.67\times10^{-5}}  \approx 0.413\times10^{7} = 4.13\times10^6  </math>  
  
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'''Addition and subtraction:'''
  
[[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.
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To add or subtract two (or more) numbers in scientific notation, they must first be represented such that they have the same exponential part. After that, the mantissas can be simply added or subtracted.
  
 
:<math>x_1^\star = a_1^\star \times10^{b_0}</math>
 
:<math>x_1^\star = a_1^\star \times10^{b_0}</math>
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:<math>x_0 \pm x_1=x_0 \pm x_1^\star=(a_0\pm a_1^\star)\times10^{b_0}</math>
 
:<math>x_0 \pm x_1=x_0 \pm x_1^\star=(a_0\pm a_1^\star)\times10^{b_0}</math>
  
an example:
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Example:
  
 
:<math>2.34\times10^{-5} + 5.67\times10^{-6} = 2.34\times10^{-5} + 0.567\times10^{-5} \approx 2.91\times10^{-5}</math>
 
:<math>2.34\times10^{-5} + 5.67\times10^{-6} = 2.34\times10^{-5} + 0.567\times10^{-5} \approx 2.91\times10^{-5}</math>
 
==See also==
 
*[[SI prefixes]]
 
*International standard [[ISO 31-0]]
 
 
== Footnotes ==
 
<references />
 
  
 
== References ==
 
== References ==
  
*Fearon, Globe. ''Access to Math: Exponents and Scientific Notation'' 1996.
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* Fearon, Globe. ''Access to Math: Exponents and Scientific Notation.'' 1996. ISBN 0835915751
 +
* Dept. of Chemistry, Texas A&M University. [http://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html Math Skills Review: Scientific Notation.] Retrieved December 2, 2006.
 +
* Division of Chemical Education, Purdue University. [http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch1/scinot.html Scientific Notation.] Retrieved December 3, 2006.
  
 
==External links==
 
==External links==
 +
All links retrieved January 25, 2023.
  
*[http://members.aol.com/profchm/sci_not.html What Is Scientific Notation And How Is It Used?]
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*[http://www.math.toronto.edu/mathnet/plain/questionCorner/scinot.html Scientific Notation in Everyday Life].
*[http://www.math.toronto.edu/mathnet/plain/questionCorner/scinot.html Scientific Notation in Everyday Life]
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*[http://science.widener.edu/svb/tutorial/scinot.html An exercise in converting to and from scientific notation].
*[http://science.widener.edu/svb/tutorial/scinot.html An exercise in converting to and from scientific notation]
 
  
 
[[Category:Physical sciences]]
 
[[Category:Physical sciences]]

Latest revision as of 17:25, 25 January 2023

Scientific notation is a compact form of writing numbers that are very large or very small. It is especially useful when expressing measurements such as distances to stars, eons of time, or the sizes of objects that are extremely large or small. This type of notation helps simplify calculations and comparisons of these numbers, and it helps express the precision with which quantities have been measured.

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 (using "mantissa" may cause confusion because it can also refer to the fractional part of the common logarithm).

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

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

Notation using E

For devices such as calculators, typewriters, and some computers, it may be difficult to show a number with an exponent, such as 107. In such cases, an alternate notation is often used: "×10" is replaced by the letter E or e (for exponent), followed by the exponent. (Note that this e is not related to the mathematical constant e.) The sign of the exponent, whether positive or negative, is often given. For example, 1.56234 E+29 is a representation of the number 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. By including only the significant figures (the digits that are known to be reliable, plus one uncertain digit) in the coefficient, one can convey the precision 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 digit potentially in doubt by half a unit.

Consider, for example, the Earth's mass given above in conventional notation (5,973,600,000,000,000,000,000,000 kg). That representation gives no indication of the accuracy of the reported value, so a reader could incorrectly assume from the twenty-five digits shown that the mass is known right down to the last kilogram. By writing the number in scientific notation, one indicates that the Earth's mass is known with a precision of ± 0.00005×1024 kg, or ± 5×1019 kg.

In cases 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" by the number 4, and therefore the proton is roughly four orders of magnitude (about 10,000 times) more massive than the electron.

Converting numbers to scientific notation

A number can be readily converted to scientific notation in a few simple steps. Consider the following examples.

Example A

To convert the number 123.4 to scientific notation, one can use the steps given below.

1. Write 123.4 in a form that shows it multiplied by 100. (Note that 100 = 1, so this step is the same as multiplying the number by 1.)

2. Divide the mantissa (123.4) by 100, by shifting the decimal point two places to the left. Also multiply 100 by 100, by adding 2 to the exponent.

Thus the scientific notation for 123.4 is 1.234×102.

Example B

To convert the number 0.001234 to scientific notation, one can use the following steps.

1. Write 0.001234 in a form that shows it multiplied by 100.

2. Multiply the mantissa (0.001234) by 1,000, by shifting the decimal point three places to the right. Also divide 100 by 1,000, by adding −3 to the exponent.

Thus the scientific notation for 0.001234 is 1.234×10−3.

Basic operations with numbers in scientific notation

Consider two numbers, x0 and x1, in scientific notation. They may be written as follows.

Multiplication and division:

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

Examples:

Addition and subtraction:

To add or subtract two (or more) numbers in scientific notation, they must first be represented such that they have the same exponential part. After that, the mantissas can be simply added or subtracted.

Example:

References
ISBN links support NWE through referral fees

External links

All links retrieved January 25, 2023.

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