One may speak of infinite divisibility, or the lack thereof, of matter, space, time, money, or abstract mathematical objects such as the continuum.
The origin of the idea in the Western tradition can be traced to the 5th century BCE starting with the Ancient Greek pre-Socratic philosopher Democritus and his teacher Leucippus, who theorized matter's divisibility beyond what can be perceived by the senses until ultimately ending at an indivisible atom.
[3] Andrew Pyle gives a lucid account of infinite divisibility in the first few pages of his Atomism and its Critics.
As a result, the Greek word átomos (ἄτομος), which literally means "uncuttable", is usually translated as "indivisible".
In other words, the quantum-mechanical description of matter no longer conforms to the cookie cutter paradigm.
However, according to the best currently accepted theory in physics, the Standard Model, there is a distance (called the Planck length, 1.616229(38)×10−35 metres, named after one of the fathers of Quantum Theory, Max Planck) and therefore a time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.39116(13) × 10−44 seconds, known as the Planck time) at which the Standard Model is expected to break down – effectively making this the smallest physical scale about which meaningful statements can be currently made.
It is quite commonplace for prices of some commodities such as gasoline to be in increments of a tenth of a cent per gallon or per litre.
There is a point of precision in each transaction that is useless because such small amounts of money are insignificant to humans.
For example, when buying a million shares of stock, the buyer and seller might be interested in a tenth of a cent price difference, but it's only a choice.
Perhaps paradoxically, technical mathematics applied to financial markets is often simpler if infinitely divisible time is used as an approximation.
Infinite divisibility does not imply gaplessness: the rationals do not enjoy the least upper bound property.
Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic process { Xt : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability distribution of Xt − Xs depends only on t − s; independent increments means that that difference is independent of the corresponding difference on any interval not overlapping with [s, t], and similarly for any finite number of intervals).
This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.