
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 use fractions to represent even smaller quantities. In systems such as the degrees minutes and seconds[?] system, it is possible to represent one second of arc, equal to
of a circle.
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 realworld objects are:
Other small numbers are found in particle physics and quantum physics:
Although all these numbers above are very small, they are all still real numbers greater than zero. Some fields of mathematics define infinitesimal 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 numbers can be generated. Some mathematical systems such as surreal numbers and hyperreal numbers generate elaborate systems of infinitesimals with amazing properties.
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