Multiple (mathematics) in the context of Fundamental quantity

In physics, a quantum (pl.: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This means that the magnitude of the physical property can take on only discrete values consisting of integer multiples of one quantum. For example, a photon is a single quantum of light of a specific frequency (or of any other form of electromagnetic radiation). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values. Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom. Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of energy and its influence on how energy and matter interact (quantum electrodynamics) is part of the fundamental framework for understanding and describing nature.

In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the codomain. Precisely, a binary relation over sets ${\displaystyle X}$ and ${\displaystyle Y}$ is a set of ordered pairs ${\displaystyle (x,y)}$, where ${\displaystyle x}$ is an element of ${\displaystyle X}$ and ${\displaystyle y}$ is an element of ${\displaystyle Y}$. It encodes the common concept of relation: an element ${\displaystyle x}$ is related to an element ${\displaystyle y}$, if and only if the pair ${\displaystyle (x,y)}$ belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "divides" relation over the set of prime numbers ${\displaystyle \mathbb {P} }$ and the set of integers ${\displaystyle \mathbb {Z} }$, in which each prime ${\displaystyle p}$ is related to each integer ${\displaystyle z}$ that is a multiple of ${\displaystyle p}$, but not to an integer that is not a multiple of ${\displaystyle p}$. In this relation, for instance, the prime number ${\displaystyle 2}$ is related to numbers such as ${\displaystyle -4}$, ${\displaystyle 0}$, ${\displaystyle 6}$, ${\displaystyle 10}$, but not to ${\displaystyle 1}$ or ${\displaystyle 9}$, just as the prime number ${\displaystyle 3}$ is related to ${\displaystyle 0}$, ${\displaystyle 6}$, and ${\displaystyle 9}$, but not to ${\displaystyle 4}$ or ${\displaystyle 13}$.

In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and ${\displaystyle x\cdot (2+x)}$ is the product of ${\displaystyle x}$ and ${\displaystyle (2+x)}$ (indicating that the two factors should be multiplied together).When one factor is an integer, the product is called a multiple.

The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well.

In mathematics, a divisor of an integer ${\displaystyle n,}$ also called a factor of ${\displaystyle n,}$ is an integer ${\displaystyle m}$ that may be multiplied by some integer to produce ${\displaystyle n.}$ In this case, one also says that ${\displaystyle n}$ is a multiple of ${\displaystyle m.}$ An integer ${\displaystyle n}$ is divisible or evenly divisible by another integer ${\displaystyle m}$ if ${\displaystyle m}$ is a divisor of ${\displaystyle n}$; this implies dividing ${\displaystyle n}$ by ${\displaystyle m}$ leaves no remainder.

The discovery of the neutron and its properties was central to the extraordinary developments in atomic physics in the first half of the 20th century. Early in the century, Ernest Rutherford used alpha particle scattering to discover that an atom has its mass and electric charge concentrated in a tiny nucleus. By 1920, isotopes of chemical elements had been discovered, the atomic masses had been determined to be approximately integer multiples of the mass of the hydrogen atom, and the atomic number had been identified as the charge on the nucleus. Throughout the 1920s, the nucleus was viewed as composed of combinations of protons and electrons, the two elementary particles known at the time, but that model presented several experimental and theoretical contradictions.

The essential nature of the atomic nucleus was established with the discovery of the neutron by James Chadwick in 1932 and the determination that it was a new elementary particle, distinct from the proton.

Decimal multiplicative prefixes have been a feature of all forms of the metric system, with six of these dating back to the system's introduction in the 1790s. Metric prefixes have also been used with some non-metric units. The SI prefixes are metric prefixes that were standardised for use in the International System of Units (SI) by the International Bureau of Weights and Measures (BIPM) in resolutions dating from 1960 to 2022. Since 2009, they have formed part of the ISO/IEC 80000 standard. They are also used in the Unified Code for Units of Measure (UCUM).

A base unit of measurement (also referred to as a base unit or fundamental unit) is a unit of measurement adopted for a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the subset can be expressed in terms of the others. The SI base units, or Systéme International d'unités, consists of the metre, kilogram, second, ampere, kelvin, mole and candela.

A unit multiple (or multiple of a unit) is an integer multiple of a given unit; likewise a unit submultiple (or submultiple of a unit) is a submultiple or a unit fraction of a given unit. Unit prefixes are common base-10 or base-2 powers multiples and submultiples of units.

Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience.

The term "harmonics" originated from the Ancient Greek word harmonikos, meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes. Still, the term has been generalized beyond its original meaning.