- Small number
Within a set of positive numbers, a number is small if it is close to zero. A number is smaller if it is
less than another number.Within a set of positive and negative numbers there is ambiguity, because being closer to zero does not correspond to being less, but to being less in
absolute value . Depending on context a negative number may be called "smaller" if it is closer to zero, or if it is more negative.This article deals with positive numbers, and is also applicable to negative numbers by taking the absolute value.
Small numbers are
number s that are small compared with the numbers used in everyday life. Very small numbers often occur in fields such aschemistry ,electronics andquantum physics .Fractions
As soon as systems of
weights and measures were devised, units were subdivided into smaller units: pounds were divided into ounces, pounds into shillings and pence. Beyond the smallest units, there was a need to usevulgar fraction s to represent even smaller quantities. In systems such as thedegrees minutes and seconds system, it is possible to represent onesecond of arc , equal to:frac {1} {360 imes 60 imes 60}
of a circle.
Tiny numbers in science
Even smaller numbers are often found in science, which are so small that they are not easily dealt with using fractions.
Scientific notation was created to handle very small and very large numbers.Examples of small numbers describing everyday real-world objects are:
* size of a bit on a computer hard disk
*feature size of a structure on amicroprocessor chip
* wavelength ofgreen light: 5.5 × 10-7 m
* period of a 100 MHz FMradio wave: 1 × 10-8 s
* time taken by light to travel one meter: roughly 3 × 10-9 s
* radius of ahydrogen atom: 2.5 × 10-11 m
* the charge on anelectron : roughly 1.6 × 10-19 C (negative)
* "please add more to this list"Other small numbers are found in
particle physics andquantum physics :
* size of theatomic nucleus of a lead atom: 7.1 × 10-15 m
* thePlanck length : 1.6 × 10-35 mExtremely small numbers
Extremely small numbers can be described through their reciprocal, an extremely
large number . The notation is similar, with a minus sign at the first level of exponents, e.g.10^{,!-10^{10^{10^{10^{4.829*10^{183230
Infinitesimals
Although all these numbers above are very small, they are all still
real numbers greater than zero. Some fields of mathematics defineinfinitesimal numbers. An infinitesimal is a number greater than zero yet smaller than any positive real number.Infinitesimal numbers were originally developed to create the differential and integral calculus, but were replaced by systems using limits when they were shown to lack
theoretical rigor . More recent work has restored rigor to infinitesimals, making them once more a powerful mathematical tool.Systems of infinitesimals can be generated in the same way as systems of
transfinite number s can be generated.Some mathematical systems such assurreal number s andhyperreal number s generate elaborate systems of infinitesimals with amazing properties.ee also
*
Orders of magnitude
*floating-point number s
*large numbers
*human scale
* fractions
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