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Scientific notation (also known as Standard index notation) is a convenient way to write very small or large numbers. Formally, scientific notation is floating-point notation with base 10.

When using scientific notation numbers are written as $a\times10^b$

The exponent b is an integer and a is any real number called the significand, or also mantissa but this may give confusion with its alternative meaning of 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.

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

Most calculators and many computer programs present very large and very small results in scientific notation. usually the '10' is omitted and replaced the letter E, 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

1 Why scientific notation is useful
2 Using scientific notation
- 2.1 Converting
- 2.2 Basic operations
3 See also
4 External link

[edit] Why scientific notation is useful

Scientific notation is a very convenient way to write large or small numbers.

The Earth mass can be expressed as 5,973,600,000,000,000,000,000,000 Kg. That becomes much shorter and readable using scientific notation: 5.9736×10²⁴ Kg.

An electron mass is 0.00000000000000000000000000000091093826 Kg. That is quite harder to read than 9.1093826×10^-31 Kg.

It also enables easier comparisons. A proton mass is 0.0000000000000000000000000016726 Kg. If it is written as 1.6726×10^-27 Kg, comparing this to the electron mass above becomes easier – '-27' is larger than '-31' – and we also get a faster hint on the order of magnitude of the difference, simply comparing the exponents rather than counting all those zeroes.

Scientific notation is useful too for describing quantities which are only known within certain error limits. Giving just the significant figures, the digits that are known to be reliable, conveys an implicit indication of the precision.

When a physical quantity is quoted using scientific notation, it is usually assumed to be accurate to no less 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.

Looking at the Earth mass first expression above one would not know how accurate it is or, even worse, could wrongly assume 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.

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

[edit] Using scientific notation

[edit] Converting

Multiplication and division by 10 are easy to perform both with the mantissa and with the exponential part of a number represented 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

At the exponential part, multiplication by 10 results in adding 1 to the exponent: 10²×10=10³. Division by 10 results in adding -1 to 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 — done shifting the decimal point n times to the right) — requires the exponential part to be divided by 10 the same number of times — done by adding -n to the exponent. Some examples:

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

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

[edit] Basic operations

Given two numbers in scientific notation,

$x_0=a_0\times10^{b_0}$

$x_1=a_0\times10^{b_1}$

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 representing 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, usually done with the one with the smaller exponent. Second, add, or subtract, the mantissas.

$x_1^\star = a_1^\star \times10^{b_0}$