Scientific notation

Scientific notation, also sometimes known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favoured by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation, numbers are written in the form:
 * $$a\,\times\,10^b\!$$

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number, called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes the first of the decimal digits of a (as in ordinary decimal notation). For example:

Normalized notation
Any given number can be written in the form a × 10 $1$ in many ways; for example 350 can be written as $3$, or $5.72$, or $-6.1$.

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 $b$. 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 $3.5$). 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 315 × 220).

E notation
Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 107 cannot always be conveniently represented on computers, typewriters, and calculators, an alternative format is often used: the letter "E" or "e" represents "times ten raised to the power", thus replacing the "× 10 $35$", while the exponent is not superscripted but is left on the same level as the significand (e.g. a E−6 not a × 10 $350$). The sign is often given even if positive (e.g. a E+11 rather than a E11). For example, 6.0221415 E+23 or 6.0221415 e23 is the same as $3.5$ (Avogadro's number).

Note that this character "e" is not related to the mathematical constant e (a confusion not possible when using capital "E"); and though it is short for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation rather than (scientific) exponential notation (though the latter also occurs).

Examples from computing:
 * In the FORTRAN programming language 6.0221415E23 is equivalent to $-5$.
 * The ALGOL programming language also uses the E notation; alternatively&mdash;when available&mdash;either character '₁₀' or '\' can be used, for example: 6.0221415₁₀23 and 6.0221415\23.

Engineering notation
Engineering notation differs from normalized scientific notation in that the exponent b is restricted to multiples of 3. Consequently, the absolute value of a is in the range 1 ≤ |a| < 1000, rather than 1 ≤ |a| < 10. Though similar in concept, engineering notation is rarely called scientific notation.

Numbers in this form are easily read out using magnitude prefixes like mega- (b = 6), kilo- (b = 3) , milli- (b = &minus;3) , micro- (b = &minus;6) or nano- (b = &minus;9). For example, $n$ can be read as "twelve point five nanometers" or written as $-6$.

Use of spaces
In normalized scientific notation, in E notation, and in engineering notation, the space (which in typesetting may be represented by a normal width space or a thin space) that is allowed only before and after "×" or in front of "E" or "e" may be omitted, though it is less common to do so before the alphabetical character.

Motivation
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 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. In scientific notation, this is written $6.022$.
 * The Earth's mass is about 5,973,600,000,000,000,000,000,000 kg. In scientific notation, this is written $6.022$.
 * The Earth's circumference is approximately 40,000,000 m. In scientific notation, this is written $12.5 m$. In engineering notation, this is written $12.5 nm$. In SI writing style, this may be written "$9.109 kg$" ("40 megameters").

Significant figures
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 $5.974 kg$ and the inch is $4 m$; 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 $40 m$ (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 2.5 (½) × 10 $40 Mm$ to explicitly indicate that there is a half-count of uncertainty in the last digit.

Order of magnitude
Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is 0.000 000 000 000 000 000 000 000 001 672 6 kg. If this is written as $2.998 m/s$, 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 comparing the exponents instead of the more error-prone task of counting the leading zeros. In this case, '−27' is larger than '−31' and therefore the proton is roughly 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 10$2.54 m$ or 10$1.602 C$.

Converting
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&minus;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 $-2$. Similarly, starting with 0.000000456, move the decimal point seven places to the right yielding 4.56, and multiply by 10&minus;7, to give the result $1.673 kg$.

If the decimal separator did not move then the exponent multiplier is logically 100, which is correct since 100 = 1. However, the exponent part "&times; 100" is normally omitted, so, for example, 1.234 is just written as 1.234 rather than $9$.

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 &mdash; 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 &times; 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:


 * $$1.234 \times10^3 = 12.34 \times10^2 = 123.4 \times10^1 = 1234 \times10^0 = 1234.$$

Basic operations
Given two numbers in scientific notation,


 * $$x_0=a_0\times10^{b_0}$$


 * $$x_1=a_1\times10^{b_1}$$

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


 * $$x_0 x_1=a_0 a_1\times10^{b_0+b_1}$$


 * $$\frac{x_0}{x_1}=\frac{a_0}{a_1}\times10^{b_0-b_1}$$

some examples are:
 * $$5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2} $$


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

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:


 * $$x_1 = c \times10^{b_0}$$

Next, add or subtract the mantissas:


 * $$x_0 \pm x_1=(a_0\pm c)\times10^{b_0}$$

An example:


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