Magnitude (mathematics) in the context of Coordinate space

Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment.

If ${\displaystyle {\textbf {F}}}$ is the total of the forces acting on the system, ${\displaystyle m}$ is the mass of the system and ${\displaystyle {\textbf {a}}}$ is the acceleration of the system, Newton's second law states that ${\displaystyle {\textbf {F}}=m{\textbf {a}}\,}$ (the bold font indicates a vector quantity, i.e. one with both magnitude and direction). If ${\displaystyle {\textbf {a}}=0}$, then ${\displaystyle {\textbf {F}}=0}$. As for a system in static equilibrium, the acceleration equals zero, the system is either at rest, or its center of mass moves at constant velocity.

In physics, a force is an action (usually a push or a pull) that can cause an object to change its velocity or its shape, or to resist other forces, or to cause changes of pressure in a fluid. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity (force vector). The SI unit of force is the newton (N), and force is often represented by the symbol F.

Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include elastic, frictional, contact or "normal" forces, and gravitational. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In the case of multiple forces, if the net force on an extended body is zero the body is in equilibrium.

A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose.

The arithmetic mean, also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted using an overhead bar, ${\displaystyle {\bar {x}}}$. If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean (${\displaystyle {\bar {x}}}$) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted ${\displaystyle \mu }$ or ${\displaystyle \mu _{x}}$.

Size in general is the magnitude or dimensions of a thing. More specifically, geometrical size (or spatial size) can refer to three geometrical measures: length, area, or volume. Length can be generalized to other linear dimensions (width, height, diameter, perimeter). Size can also be measured in terms of mass, especially when assuming a density range.

In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter". Size is determined by the process of comparing or measuring objects, which results in the determination of the magnitude of a quantity, such as length or mass, relative to a unit of measurement. Such a magnitude is usually expressed as a numerical value of units on a previously established spatial scale, such as meters or inches.

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.

A unit of measurement, or unit of measure, is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement.

For example, a length is a physical quantity. The metre (symbol m) is a unit of length that represents a definite predetermined length. For instance, when referencing "10 metres" (or 10 m), what is meant is 10 times the definite predetermined length called "metre".

In geometry, an angle is formed by two lines that meet at a point. Each line is called a side of the angle, and the point they share is called the vertex of the angle. The term angle is used to denote both geometric figures and their size or magnitude as associated quantity. Angular measure or measure of angle are sometimes used to distinguish between the measure of the quantity and figure itself. The measurement of angles is intrinsically linked with circles and rotation, and this is often visualized or defined using the arc of a circle centered at the vertex and lying between the sides.

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.

The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others. According to Thomas W. Hawkins Jr., "It was primarily through the theory of multiple integrals and, in particular the work of Camille Jordan that the importance of the notion of measurability was first recognized."

In ecology, resilience is the capacity of an ecosystem to respond to a perturbation or disturbance by resisting damage and subsequently recovering. Such perturbations and disturbances can include stochastic events such as fires, flooding, windstorms, insect population explosions, and human activities such as deforestation, fracking of the ground for oil extraction, pesticide sprayed in soil, and the introduction of exotic plant or animal species. Disturbances of sufficient magnitude or duration can profoundly affect an ecosystem and may force an ecosystem to reach a threshold beyond which a different regime of processes and structures predominates. When such thresholds are associated with a critical or bifurcation point, these regime shifts may also be referred to as critical transitions.

Human activities that adversely affect ecological resilience such as reduction of biodiversity, exploitation of natural resources, pollution, land use, and anthropogenic climate change are increasingly causing regime shifts in ecosystems, often to less desirable and degraded conditions. Interdisciplinary discourse on resilience now includes consideration of the interactions of humans and ecosystems via socio-ecological systems, and the need for shift from the maximum sustainable yield paradigm to environmental resource management and ecosystem management, which aim to build ecological resilience through "resilience analysis, adaptive resource management, and adaptive governance". Ecological resilience has inspired other fields and continues to challenge the way they interpret resilience, e.g. supply chain resilience.

In kinematics, the speed (commonly referred to as v) of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero. Speed is the magnitude of velocity (a vector), which indicates additionally the direction of motion.

Speed has the dimensions of distance divided by time. The SI unit of speed is the metre per second (m/s), but the most common unit of speed in everyday usage is the kilometre per hour (km/h) or, in the US and the UK, miles per hour (mph). For air and marine travel, the knot is commonly used.

A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. Usually the term refers to the six classical simple machines that were defined by Renaissance scientists:

• Lever
• Wheel and axle
• Pulley
• Inclined plane
• Wedge
• Screw

A simple machine uses a single applied force to do work against a single load force. Ignoring friction losses, the work done on the load is equal to the work done by the applied force. The machine can increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the applied force is called the mechanical advantage.

In mathematics, the absolute value or modulus of a real number ${\displaystyle x}$, denoted ${\displaystyle |x|}$, is the non-negative value of ${\displaystyle x}$ without regard to its sign. Namely, ${\displaystyle |x|=x}$ if ${\displaystyle x}$ is a positive number, and ${\displaystyle |x|=-x}$ if ${\displaystyle x}$ is negative (in which case negating ${\displaystyle x}$ makes ${\displaystyle -x}$ positive), and ${\displaystyle |0|=0}$. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

In mathematics, a negative number is the opposite of a positive real number. Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −‍(−3) = 3 because the opposite of an opposite is the original value.

Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced and read as "minus three" or "negative three". Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign.

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a directed line segment. A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B, and denoted by ${\textstyle {\stackrel {\longrightarrow }{AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means 'carrier'. It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. In older texts, the phase of a periodic function is sometimes called the amplitude.

In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the orientation of the net force acting on that object. The magnitude of an object's acceleration, as described by Newton's second law, is the combined effect of two causes:

• the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force;
• that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object's mass.

The SI unit for acceleration is metre per second squared (m⋅s, ${\displaystyle \mathrm {\tfrac {m}{s^{2}}} }$).

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation respectively. The curl of a field is formally defined as the circulation density at each point of the field.

A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.