Scientific notation

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In scientific notation numbers are written as <math>a\times10^b</math> where the exponent b is an integer and a is any real number. The number represented by a is called the significand or the mantissa, but the latter may cause confusion as it can also refer to the fractional part of the common logarithm. Usually a is chosen in the range of 1 to 10, excluding 10. Such a fixed range allows easy comparison of two numbers since the one with the larger exponent is larger. In that case b is the number's order of magnitude. Scientific notation is a very beneficial tool for scientists.

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Engineering notation

Restricting the exponent b to multiples of 3 results in what is called engineering notation.

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Exponential notation

Most calculators and many computer programs present very large and very small results in scientific notation. Usually the '10' is omitted and replaced by the letter E or, confusingly, e—which is short for exponent. Note that this is not related to the mathematical constant e. For example 1.56234 E+29 is the same as 1.56234×1029. This is commonly called exponential notation.

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Motivation

Scientific notation is a very convenient way to write large or small numbers. It also quickly conveys two properties of a measurement that are useful to scientists—significant figures and order of magnitude.

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Examples

  • An electron's mass is 0.00000000000000000000000000000091093826 kg. In scientific notation, it is written 9.1093826×10−31 kg.
  • The Earth's mass is 5,973,600,000,000,000,000,000,000 kg. In scientific notation, it is written 5.9736×1024 kg.
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Significant digits

Scientific notation is useful for indicating the precision with which the quantity was measured. Including only the significant figures, the digits that are known to be reliable, in the mantissa implicitly conveys value's precision. Any physical quantity in scientific notation is assumed to be precise to no fewer than the quoted number of digits of precision. However, where precision in such measurements is crucial, more sophisticated expressions of measurement error must be used.

As an example, consider the Earth's mass as presented above in conventional notation. Since the representation gives no indication of the accuracy of the reported value, a reader could incorrectly assume that it is known down to the last digit displayed. The scientific notation implicitly shows it is known with a precision of 0.00005×1024 kg, or 5×1019 kg.

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Order of magnitude

Scientific notation also enables simple order of magnitude comparisons. A proton's mass is 0.0000000000000000000000000016726 kg. If it is written as 1.6726×10−27 kg, it is easy to compare this mass with that of the electron above. The difference in order of magnitude is obtained simply by comparing the exponents rather than counting all those zeroes. In this case, '−27' is larger than '−31' and therefore the proton is four orders of magnitude more massive than the electron.

Scientific notation also avoids regional differences in certain quantifiers, such as billion, which may be either 109 or 1012, thus avoiding misunderstanding.

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Using scientific notation

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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:

<math>123.4 = 123.4\times10^0 = (123.4/10^2) \times (10^0\times10^2) = 1.234\times10^2</math>

<math>.001234 = .001234\times10^0 = (.001234\times 10^3) \times (10^0 / 10^3) = 1.234\times10^{-3}</math>

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Basic operations

Given two numbers in scientific notation,

<math>x_0=a_0\times10^{b_0}</math>
<math>x_1=a_0\times10^{b_1}</math>

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

<math>x_0 x_1=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:

<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>


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.

<math>x_1^\star = 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:

<math>2.34\times10^{-5} + 5.67\times10^{-6} = 2.34\times10^{-5} + 0.567\times10^{-5} \approx 2.91\times10^{-5}</math>
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See also

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External links

ca:Notació científica de:Wissenschaftliche Notation es:Notación científica fr:Notation scientifique it:Notazione scientifica nl:Wetenschappelijke notatie pl:Notacja naukowa pt:Notação científica fi:Tieteellinen merkintätapa sv:Grundpotens zh:科学记数法

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