In scientific notation, numbers are written in the form:
|Ordinary decimal notation||Scientific notation (non-normalized)|
Any given number can be written in the form in many ways; for example 350 can be written as , or , or .
In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). For example, 350 is written as . This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g. minus one half is ). The 10 and exponent are usually omitted when the exponent is 0.
In many fields, scientific notation is normalized in this way, except during intermediate calculations or when an unnormalized form, such as engineering notation, is desired. (Normalized) scientific notation is often called exponential notation — although the latter term is more general and also applies when a is not restricted to the range 1 to 10 (as in engineering notation for instance), and to bases other than 10 (as in ).
Examples from computing:
6.0221415E23is equivalent to .
Numbers in this form are easily read out using magnitude prefixes like mega- (b = 6), kilo- (b = 3), milli- (b = −3), micro- (b = −6) or nano- (b = −9). For example, can be read as "twelve point five nanometers" or written as .
As with ordinary decimal notation, the number of digits in scientific notation may or may not indicate significant figures. For example, using scientific notation, the speed of light in SI units is and the inch is ; both numbers are exact.
It is possible to use scientific notation in conjunction with significant figures, but this is not mandatory and should never be assumed. It is always better to state the uncertainty explicitly. For instance, the accepted value of the unit of elementary charge can properly be expressed as (Coulomb), where the (40) indicates 40 counts of uncertainty in the last decimal place. If a number has been rounded off, it can be written in the form to explicitly indicate that there is a half-count of uncertainty in the last digit.
Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as 'billion', which might indicate either 10 or 10.
To convert from ordinary decimal notation to scientific notation, move the decimal separator the desired number of places to the left or right, so that the mantissa will be in the desired range (between 1 and 10 for the normalized form). If you moved the decimal point n places to the left then multiply by 10n; if you moved the decimal point n places to the right then multiply by 10−n. For example, starting with 1,230,000, move the decimal point six places to the left yielding 1.23, and multiply by 106, to give the result . Similarly, starting with 0.000000456, move the decimal point seven places to the right yielding 4.56, and multiply by 10−7, to give the result .
If the decimal separator did not move then the exponent multiplier is logically 100, which is correct since 100 = 1. However, the exponent part "× 100" is normally omitted, so, for example, 1.234 is just written as 1.234 rather than .
To convert from scientific notation to ordinary decimal notation, take the mantissa and move the decimal separator by the number of places indicated by the exponent — left if the exponent is negative, or right if the exponent is positive. Add leading or trailing zeroes as necessary. For example, given 9.5 × 1010, move the decimal point ten places to the right to yield 95,000,000,000.
Conversion between different scientific notation representations of the same number is achieved by performing opposite operations of multiplication or division by a power of ten on the mantissa and the exponent parts. The decimal separator in the mantissa is shifted n places to the left (or right), corresponding to division (multiplication) by 10n, and n is added to (subtracted from) the exponent, corresponding to a cancelling multiplication (division) by 10n. For example:
some examples are:
Addition and subtraction require the numbers to be represented using the same exponential part, so that the mantissas can be simply added or subtracted. These operations may therefore 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. In this example, x1 is rewritten as:
Next, add or subtract the mantissas: