Magnitude

In number systems and scientific arithmetic, the order of magnitude is the factor that is necessary to increase or decrease a value by one digit in the respective number representation , while maintaining the individual digits and their order.

In particular, the order of magnitude is also the power with the base 10 ( decimal order of magnitude ) or 2 ( binary order of magnitude ).
The order of magnitude of a physical quantity is expressly referred to as the powers of ten with respect to its base unit .
In addition, “order of magnitude” generally describes value ranges or scales that are plotted against these powers of a base.

It is shown in the exponential notation (floating point number).

Decimal order of magnitude

Mostly a decimal system is assumed, which is why an order of magnitude usually denotes a factor (or divisor ) of 10 . For example, the sizes "2 meters" and "200 meters" differ by two orders of magnitude , i.e. by a factor of 10 ² = 100. In general, an additive change in the order of magnitude indicates an exponential change in the actual size, or that you get from the actual size to the order of magnitude (multiplied by a constant factor) by taking the logarithm .

The orders of magnitude that appear in the respective context differ drastically. A scientific pocket calculator calculates up to 10 ⁹⁹ , but the order of magnitude of the number of elementary particles in the universe is estimated to be “only” 10 ⁸⁷ , and the universe is about 10 ¹⁸ seconds old. In contrast, the order of magnitude of the number of different possible routes between 100 cities for the traveling ^salesmanproblem is already 10 ¹⁵⁸ .

Binary order of magnitude

A binary order of magnitude corresponds to doubling or halving. In computer technology in particular, it depends on the data type .

Order of magnitude and unit of measure

In scientific practice, however, an order of magnitude is often used as a rather imprecise description of proportions and generally refers to the power of the floating point number. The sense of this application arises from the context and is mostly in the designation of large or very large differences in numbers. For example, the next one is star to five orders of magnitude farther from the earth away than the sun . What is meant here are decimal orders of magnitude, rounded to an integer . The order of magnitude in this sense is millimeter (one thousandth of a meter ) → centimeter (one hundredth) → decimeter (one tenth of a meter) → meter. For example, it is said that a size is “in the centimeter range”.

In the SI system of units , the prefixes for units of measurement , which determine the decimal order of magnitude of the base unit , are precisely regulated. In engineering, the technical notation is used with a factor of 1000 as the order of magnitude, i.e. restricted to nanometers → micrometers → millimeters → meters → kilometers, and so on.

Examples of physical quantities with units in different orders of magnitude

Mass: gram (g), kilogram (kg), ton (t)
Energy: Electron volt (eV), megaelectron volt (MeV), gigaelectron volt (GeV), joule (J), kilowatt hour (kWh), terawatt hour (TWh)
Power: Milliwatt (mW), watt (W), kilowatt (kW), megawatt (MW)
Time: femtosecond (fs), picosecond (ps), nanosecond (ns), microsecond (μs), millisecond (ms), second (s), minute (min), hour (h), day (d), year (a )
Frequency: Hertz (Hz), Kilohertz (kHz), Megahertz (MHz)
Length: Nanometer (nm), micrometer (µm), millimeter (mm), centimeter (cm), decimeter (dm), meter (m), kilometer (km), astronomical unit (AU), light year (Lj), parsec ( pc)
Area: square meters (m²), ar (a), hectares (ha), square kilometers (km²)
Volume: milliliter (ml), centiliter (cl), deciliter (dl), liter (l), cubic meter (m³)
Temperature: nanokelvin (nK), microkelvin (µK), millikelvin (mK), Kelvin (K)
Pressure: Kilopascal (kPa), hectopascal (hPa), Pascal (Pa)

Scales of the order of magnitude of various elementary sizes

The relevant value range of physical quantities in nature and technology often covers many orders of magnitude. Therefore, logarithmic scales - which arrange the powers linearly - are particularly suitable for representing such scalings .

The following articles provide an overview of the magnitudes of the most important sizes, using exemplary phenomena:

Basic sizes

Derived quantities

Thematic compilations

Orders of magnitude of the data rates
Geographic scale (macro, meso, micro)

and display information about the image

Scenes from every known natural scale. 10 ⁻³² m quantum foam . 10 ⁻²⁴ m cross-sectional ^radius of 1 MeV neutrinos . 10 ⁻²² m top quark , the smallest quark. 10 ⁻²⁰ m bottom and charm quarks. 10 ⁻¹⁸ m up and down quarks. 10 ⁻¹⁶ m protons and neutrons. 10 ⁻¹⁴ m electrons and nuclei. 10 ⁻¹² m Longest wavelength of gamma radiation . 10 ⁻¹¹ m hydrogen and helium atom . 10 ⁻¹⁰ m carbon atoms . 10 ⁻⁹ m diameter of the deoxyribonucleic acid helix. 10 ⁻⁸ m Smallest virus ( Porcines_Circovirus-1 ). 10 ⁻⁷ m Largest virus ( megavirus ). 10 ⁻⁶ m X chromosome . 10 ⁻⁵ m Typical size of a red blood cell. 0.1mm width of human hair. 10 mm fingertip. 1 m size of an infant. 10m tallest dinosaur. 1 km Barringer Crater. 100 km Jamaica Island (235 km long). 10000 km diameter of the planet Earth (12,742 km). 10 ⁸ m moon orbit (770,000 km). 10 ⁹ m diameter of the sun (1391400km). 10 ¹¹ m diameter of the inner solar system . (600,000,000 km) 10 ¹³ m diameter of the solar system. 10 ¹⁵ m outer limit of the Oort cloud . 10 ¹⁶ m from Alpha Centauri . 10 ¹⁸ m Messier 13 globular clusters. 10 ²⁰ m diameter of the Milky Way. 10 ²² m Local group of galaxies . 10 ²³ m Typical cluster of galaxies (2 to 10 Mpc). 10 ²⁴ m Laniakea Supercluster . (160 Mpc) 10 ²⁵ m honeycomb structure of the universe . 10 ²⁶ m diameter of the observable universe . The universe is larger than 10 ²⁷ m and possibly infinite.