Scientific notation is a scheme for writing numbers that is often used by scientists and mathematicians to easily write large and small numbers. A number that is written in scientific notation has several properties that make it very useful to scientists.

In scientific notation, numbers are written using powers of ten in the form a×10^b where b is an integer exponent and the coefficient a is any real number, called the significand or mantissa (using "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm).

In normalized form, a is chosen such that 1 ≤ a < 10. Following this rule allows easy comparison of two positive numbers as the one with the larger power of ten must be larger. In normalized form, the exponent b gives the number's order of magnitude. It is implicitly assumed that scientific notation should always be normalized except during calculations or when an unnormalized form is desired (e.g. engineering notation).

Variations

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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 is read as "twelve point five nanometers".

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 e—which is 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 (positive or negative) is also generally given (e.g. E+11 not E11). For example, 1.56234 E+29 is the same as 1.56234×10²⁹. This representation is used only because exponents like 10⁷ are difficult to typeset; computers, typewriters and calculators aren't always able to handle superscripts. This representation is commonly called exponential notation.

Motivation

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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—significant figures and order of magnitude.

Examples

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

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) in the coefficient conveys the precison of the value. A physical quantity in scientific notation is assumed to have been measured to at least the quoted number of digits of precision.

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 that it is known down to the last digit shown. The scientific notation indicates that it is known with a precision of ± 0.00005×10²⁴ kg, or ± 5×10¹⁹ kg.

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

Order of magnitude

Scientific notation also enables simple order of magnitude comparisons. A proton's mass is 0.0000000000000000000000000016726 kg. If this is written as 1.6726×10⁻²⁷ 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, '−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 avoids misunderstandings due to regional differences in certain quantifiers, such as 'billion', which might indicate either 10⁹ or 10¹².

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:

$123.4\; =\; 123.4\backslash times10^0\; =\; (123.4/10^2)\; \backslash times\; (10^0\backslash times10^2)\; =\; 1.234\backslash times10^2$

$.001234\; =\; .001234\backslash times10^0\; =\; (.001234\backslash times\; 10^3)\; \backslash times\; (10^0\; /\; 10^3)\; =\; 1.234\backslash times10^\{-3\}$

Basic operations

Given two numbers in scientific notation,

$x\_0=a\_0\backslash times10^\{b\_0\}$

$x\_1=a\_1\backslash times10^\{b\_1\}$

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

$x\_0\; x\_1=a\_0\; a\_1\backslash times10^\{b\_0+b\_1\}$

$\backslash frac\{x\_0\}\{x\_1\}=\backslash frac\{a\_0\}\{a\_1\}\backslash times10^\{b\_0-b\_1\}$

some examples are:

$5.67\backslash times10^\{-5\}\; \backslash times\; 2.34\backslash times10^2\; \backslash approx\; 13.3\backslash times10^\{-3\}\; =\; 1.33\backslash times10^\{-2\}$

$\backslash frac\{2.34\backslash times10^2\}\{5.67\backslash times10^\{-5\}\}\; \backslash approx\; 0.413\backslash times10^\{7\}\; =\; 4.13\backslash times10^6$

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.

$x\_1^\backslash star\; =\; a\_1^\backslash star\; \backslash times10^\{b\_0\}$

$x\_0\; \backslash pm\; x\_1=x\_0\; \backslash pm\; x\_1^\backslash star=(a\_0\backslash pm\; a\_1^\backslash star)\backslash times10^\{b\_0\}$

an example:

$2.34\backslash times10^\{-5\}\; +\; 5.67\backslash times10^\{-6\}\; =\; 2.34\backslash times10^\{-5\}\; +\; 0.567\backslash times10^\{-5\}\; \backslash approx\; 2.91\backslash times10^\{-5\}$

See also

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• SI prefixes
• International standard ISO 31-0

External links

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• What Is Scientific Notation And How Is It Used?
• Scientific Notation in Everyday Life
• An exercise in converting to and from scientific notation

Science

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