Difference between revisions of "Scientific notation" - New World Encyclopedia
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===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. | + | 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 digit potentially in doubt by half a unit. |
| − | + | Consider, for example, the Earth's mass given above in conventional notation. 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 ± 0.00005×10<sup>24</sup> kg, or ± 5×10<sup>19</sup> kg. | |
| − | + | In cases where precision in such measurements is crucial, more sophisticated expressions of [[measurement error]]* must be used. | |
===Order of magnitude=== | ===Order of magnitude=== | ||
| Line 34: | Line 34: | ||
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<sup>−27</sup> kg. As noted above, the electron's mass is 9.1093826×10<sup>−31</sup> 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. | 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<sup>−27</sup> kg. As noted above, the electron's mass is 9.1093826×10<sup>−31</sup> 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. | ||
| − | == | + | == Converting numbers to scientific notation == |
| − | |||
| − | A number can be readily converted to scientific notation in a few simple steps. | + | A number can be readily converted to scientific notation in a few simple steps. Consider the following examples. |
| − | 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 | + | '''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 10<sup>0</sup>. (Note that 10<sup>0</sup> = 1, so this step is the same as multiplying the number by 1.) | ||
:<math>123.4 = 123.4\times10^0</math> | :<math>123.4 = 123.4\times10^0</math> | ||
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. | 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. | ||
| + | :<math>123.4\times10^0 = (123.4/10^2) \times (10^0\times10^2) = 1.234\times10^2</math> | ||
| − | + | Thus the scientific notation for 123.4 is 1.234×10<sup>2</sup>. | |
| − | + | '''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 10<sup>0</sup>. | |
| + | :<math>0.001234 = 0.001234\times10^0</math> | ||
| − | Also | + | 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 −3 to the exponent. |
| + | :<math>.001234\times10^0 = (.001234\times 10^3) \times (10^0 / 10^3) = 1.234\times10^{-3}</math> | ||
| − | + | Thus the scientific notation for 0.001234 is 1.234×10<sup>−3</sup>. | |
| − | + | == Basic operations with numbers in scientific notation == | |
| − | < | + | 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: | ||
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:<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> | ||
| − | + | 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> | + | :<math>\frac{2.34\times10^2}{5.67\times10^{-5}} \approx 0.413\times10^{7} = 4.13\times10^6 </math> |
| + | '''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. | |
:<math>x_1^\star = a_1^\star \times10^{b_0}</math> | :<math>x_1^\star = a_1^\star \times10^{b_0}</math> | ||
| Line 85: | Line 92: | ||
:<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> | ||
| − | + | 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> | ||
Revision as of 18:49, 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 digit potentially in doubt by half a unit.
Consider, for example, the Earth's mass given above in conventional notation. 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", and therefore the proton is 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:
See also
- integer
- SI prefixes
- International standard ISO 31-0
Footnotes
ReferencesISBN links support NWE through referral fees
- Fearon, Globe. Access to Math: Exponents and Scientific Notation 1996.
External links
- What Is Scientific Notation And How Is It Used?
- Scientific Notation in Everyday Life
- An exercise in converting to and from scientific notation
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