Scientific notation

As scientific notation (English: scientific notation ) refers to two variants of modern numerical representation : the exponential , even traditional scientific notation or standard presentation called, and the engineering notation (English: engineering notation ). In both, the numerical value to be displayed is divided into mantissa and exponent (to base 10):

{\ displaystyle a \ cdot 10 ^ {b}}

The mantissa is a point number (with additional conditions), the exponent is an integer. ${\ displaystyle a}$ ${\ displaystyle b}$

The traditional scientific notation in the article exponential discussed in detail. Here, only one , zero, left-decimal, so . The advantage in science is the quick overview of the order of magnitude and the possible comparison of several numerical values. Usually a number is given in the format . The disadvantage of this notation format is that the results must be "reformatted" if they are to be expressed with the prefixes of the SI symbol units. ${\ displaystyle a}$ ${\ displaystyle 1 \ leq a <10}$ ${\ displaystyle a \ cdot 10 ^ {b}}$

In technical notation , only whole-number multiples of 3 are used as exponents , i.e. whole-number powers of a thousand. (Then it is usually in the range .) This notation therefore deals with the use of units of measure - prefixes , because the standardized orders of magnitude ( micro, milli, kilo, mega, ... ) correspond to powers of 10 ³ . ${\ displaystyle a}$ ${\ displaystyle 1 \ leq a <1000}$

Scientific calculator

Most modern pocket calculators can automatically display numbers in scientific notation (display shows for example: SCI ). In the case of very large numbers or very small decimal fractions, this is usually not possible any other way.

The term scientific notation is, however, not used in a completely uniform way, but very often simply - especially in English - used synonymously with traditional scientific notation - i.e. for exponential representation. On pocket calculators, the technical notation is usually referred to as ENG ( engineering notation ).

If no superscript digits are available, the following notation is used: 1 · 10 becomes ¹⁸1 E18 . The number 3200 z. B. can thus also be 3,2 E3noted. (See also exponential representation )

Precision in SI and ENG format

Sometimes both the SI magnitudes and the engineering format were accused of raising doubts about the precision of the values determined.

In fact, the exponential represents the precision of the results in a very simple and clear way, namely by the number of digits after the figure. For example, the results 5 E-4 m, 5.0 E-4 m and 5.00 E-4 m just don't mean the same thing. These three different results would have to be reduced indiscriminately to 500 µm and 500 E-6 m in both the SI and the ENG format.

This apparent shortcoming of the SI and ENG formats can be overcome by specifying the results as decimal fractions of the higher order of magnitude, in the above example as 0.5 mm, 0.50 mm and 0.500 mm or as 0.5 E. -3 m, 0.50 E-3 m and 0.500 E-3 m. The specification of the precision is restored. In any case, this procedure is only necessary for results that can be determined to no more than two decimal places, which is a rare case in science.

uncertainty

If a variable is subject to a random error (to be distinguished from a systematic error ), a standard uncertainty is given; so is z. B. the gravitational constant

${\ displaystyle 6 {,} 674 \, 30 (15) \ cdot 10 ^ {- 11} \, \ mathrm {\ frac {m ^ {3}} {kg \ cdot s ^ {2}}}}$ , short for

${\ displaystyle (6 {,} 674 \, 30 \ pm 0 {,} 000 \, 15) \ cdot 10 ^ {- 11} \, \ mathrm {\ frac {m ^ {3}} {kg \ cdot s ^ {2}}}}$ .

Orders of magnitude of technical notation

10 ^N.	symbol	Surname	decimal number	1000 ^N	Numeral
10 ²⁴	Y	Yotta	1 000 000 000 000 000 000 000 000	1000 ⁸	Quadrillion
10 ²¹	Z	Zetta	1 000 000 000 000 000 000 000	1000 ⁷	Trillion
10 ¹⁸	E.	Exa	1,000,000,000,000,000,000	1000 ⁶	Trillion
10 ¹⁵	P	Peta	1 000 000 000 000 000	1000 ⁵	Billiards
10 ¹²	T	Tera	1,000,000,000,000	1000 ⁴	trillion
10 ⁹	G	Giga	1,000,000,000	1000 ³	billion
10 ⁶	M.	Mega	1,000,000	1000 ²	million
10 ³	k	kilo	1,000	1000 ¹	thousand
10 ²	H	Hecto	1 00		Hundred
10 ¹	there	Deka	1 0		ten
10 ⁰		unit	1	1000 ⁰	one
10 ⁻¹	d	Deci	0.1		tenth
10 ⁻²	c	Centi	0.01		Hundredths
10 ⁻³	m	Milli	0.001	1000 ⁻¹	Thousandths
10 ⁻⁶	µ	Micro	0.000 001	1000 ⁻²	Millionth
10 ⁻⁹	n	Nano	0.000 000 001	1000 ⁻³	billionth
10 ⁻¹²	p	Pico	0.000 000 000 001	1000 ⁻⁴	Trillionth
10 ^-15	f	Femto	0.000 000 000 000 001	1000 ^-5	Billiardstel
10 ⁻¹⁸	a	Atto	0.000 000 000 000 000 001	1000 ⁻⁶	Trillionth
10 ⁻²¹	z	Zepto	0.000 000 000 000 000 000 001	1000 ⁻⁷	Trillionths
10 ⁻²⁴	y	Yocto	0.000 000 000 000 000 000 000 001	1000 ⁻⁸	Quadrillionth