Negligible refers to the quantities so small that they can be ignored (neglected) when studying the larger effect. Although related to the more mathematical concepts of infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

, the idea of negligibility is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic engineering, computer programming and in everyday decision-making. A quantity can be said to be negligible when it is safe to ignore (neglect) it in the present case, within the margins for error that have been agreed to be acceptable in this case. Examples of quantities that are often considered negligible are the electrical resistance of a wire, and the mass of the electrons in an atom.

In physics

Any macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

 system is always much more complicated than any idealized mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

 describing it. In order to simplify real situations, some effects are generally regarded as insignificant because their magnitude is so small as to be negligible. Given a system described by a formula it is sometimes possible to make a Taylor expansion of the expression and then identify negligible terms.

An example would be a car moving at 10 km/h along a straight horizontal road. In total, there are five main forces acting on this car, gravity on the mass of the car (the weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...

), the reaction force of the road opposing the weight, the friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

 of the wheels on the road, the force of the engine, and air resistance against the car. The forces that have the most effect on the car will be the weight, the reaction opposing the weight and the friction. In order to describe the motion of the car mathematically, to a reasonable precision, only four of the forces has to be included, weight, engine, reaction and friction. Air resistance is "negligible" and can be disregarded because the car is moving at such a low speed. Even though air resistance has an effect, the effect is so minuscule that for most purposes it is safe to regard it as not being there at all, so to avoid any unnecessarily complicated calculations. At greater speeds, air resistance becomes significant and can no longer be neglected.

In electronic engineering

Electronic circuit designers make use of ideal
-In philosophy:* Ideal , values that one actively pursues as goals* Platonic ideal, a philosophical idea of trueness of form, associated with Plato-In mathematics:* Ideal , special subsets of a ring considered in abstract algebra...

 circuit concepts in the design process. These include the perfect voltage source
Voltage source
In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...

, the perfect current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...

, the perfect amplifier
Operational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

, a perfect ground
Ground (electricity)
In electrical engineering, ground or earth may be the reference point in an electrical circuit from which other voltages are measured, or a common return path for electric current, or a direct physical connection to the Earth....

 and many others.

In none of these cases can the perfect circuit element actually exist in practice. To take one example, consider the perfect voltage source. If a perfect voltage source existed, it would have no internal impedance and would continue to maintain its rated voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

, say 5 V dc, across any load, no matter what current may become necessary to do this. As the load impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

 reduced toward zero ohms (a perfect short circuit
Short circuit
A short circuit in an electrical circuit that allows a current to travel along an unintended path, often where essentially no electrical impedance is encountered....

 - which also cannot truly exist) then the current flow and power
Electric power
Electric power is the rate at which electric energy is transferred by an electric circuit. The SI unit of power is the watt.-Circuits:Electric power, like mechanical power, is represented by the letter P in electrical equations...

 delivery would approach infinity.
However, a real circuit is not made to function in every imaginable case or be "perfect".

To continue this example, we need to derive a specification for this practical voltage source. Perhaps the current draw will never exceed 2 amp
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

s. Perhaps the input voltages between 4.999 V and 5.001 V will produce errors that in themselves are negligible for the practical purposes of the remaining circuitry. If the output impedance of the voltage source can drop 0.002 V (5.001 - 4.999) at a current of 2 A, it must be no more than 0.001 Ω or one milliohm.

The voltage source with its negligible 1 mΩ output impedance will produce voltages that only deviate from 5.0 V by negligible amounts, provided the current requirements remain within spec.

In another case these discrepancies may be far too much as any voltage less than 4.999999 volts, or more than 5.000001 V, would be unacceptable.

The electronic engineer may continue to look upon the device, to a first approximation, as an ideal voltage source because as far as this requirement is concerned, that is what it is. Its practical discrepancies are negligible compared to the specification at this point. It is an important part of the engineer's skill, however, always to remember the assumptions and simplifications inherent in this thinking and to be able quickly to identify when cost savings can be made by reducing a specification requirement as well as when new requirements invalidate previously acceptable assumptions.

Similar examples could be created for any of the 'ideal' circuit elements listed above, and many more, from RF frequency mixer
Frequency mixer
In electronics a mixer or frequency mixer is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals at frequencies f1 and f2 are applied to a mixer, and it produces new signals at the sum f1 + f2 and difference f1 -...

s to the simplest switch
In electronics, a switch is an electrical component that can break an electrical circuit, interrupting the current or diverting it from one conductor to another....


In risk assessment

The continuing success of the global travel
Tourism is travel for recreational, leisure or business purposes. The World Tourism Organization defines tourists as people "traveling to and staying in places outside their usual environment for not more than one consecutive year for leisure, business and other purposes".Tourism has become a...

 industry depends upon the general public's perception that the personal risks involved in airline flight, as well as those involved in visiting foreign countries, are negligible compared to the pleasure to be gained by doing so.

Catastrophic failures or accidents however unlikely may, render the general public unable to neglect a certain risk however small, as seen in the by the economic effects arising from the September 11 attacks
Economic effects arising from the September 11 attacks
Major economic effects arose from the September 11 attacks, with initial shock causing global stock markets to drop sharply. The attacks themselves caused approximately $40 billion in insurance losses, making it one of the largest insured events ever....


Similarly in technical design, there are probabilities, in each case, that an electronic product may be used in the vicinity of a powerful radio transmitter, that mains-borne power surges may occur, that its batteries may go flat while in use etc. The designer has to consider each of these and write some off as outside of the specified requirements, while others clearly are not. There are clearly a very large number of uncontrollable possibilities of misuse and accidents and that any designed product may have to contend with.

By ignoring cases that that are unlikely, or that do not worry the general public when they occur, designers may be able to make significant cost savings on products and services.

In software engineering

One might expect that a deterministic
Deterministic system (mathematics)
In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...

 thing like a piece of software does not suffer from the vagaries of the negligible but this is not the case in at least two areas.

Representation errors

A computer system's internal representation of floating point
Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

 numbers may normally approximate so closely to real numbers as to produce only negligible errors under most circumstances. An exception is if two very similar values are subtracted, the result may be strongly affected by the floating point rounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...

 of the values.

There are many other ways that assumptions about the "negligible" errors involved in these digital representations may cause problems at run time or later including analog-to-digital conversion
Analog-to-digital converter
An analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...

 where resolution and bit-rates are necessarily limited, financial
FINANCIAL is the weekly English-language newspaper with offices in Tbilisi, Georgia and Kiev, Ukraine. Published by Intelligence Group LLC, FINANCIAL is focused on opinion leaders and top business decision-makers; It's about world’s largest companies, investing, careers, and small business. It is...

 calculations where floating points or other imprecise number systems do not 'take care of the pennies' etc.

Interaction with the outside world

Digital systems that interact with the outside world, whether though a keyboard
Computer keyboard
In computing, a keyboard is a typewriter-style keyboard, which uses an arrangement of buttons or keys, to act as mechanical levers or electronic switches...

 and mouse, a network
Computer network
A computer network, often simply referred to as a network, is a collection of hardware components and computers interconnected by communication channels that allow sharing of resources and information....

 or even via a disk drive gain an element of risk that also must be considered. There is a chance that the user may click another button before this calculation is complete, the network may be flooded with requests for this service quicker than the software can provide it or the disk may be full when we try to write to it, or the file we need to read may have been deleted or moved.

Modern computer programming languages provide the mechanism
Design pattern (computer science)
In software engineering, a design pattern is a general reusable solution to a commonly occurring problem within a given context in software design. A design pattern is not a finished design that can be transformed directly into code. It is a description or template for how to solve a problem that...

 of throwing and catching exceptions
Exception handling
Exception handling is a programming language construct or computer hardware mechanism designed to handle the occurrence of exceptions, special conditions that change the normal flow of program execution....

 so that the developer can handle these and many other possibilities without making the structure and logic of their code impenetrably complex to readers and future developers. Some languages, for example java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...

, are designed to remind the developer about the exceptions that may be thrown — and so that should be caught, handled or declared of negligible interest — at each point. Others, like C#, provide the mechanism but do not enforce the practice in this way.

These examples are related to probabilities introduced by the IO
In computing, input/output, or I/O, refers to the communication between an information processing system , and the outside world, possibly a human, or another information processing system. Inputs are the signals or data received by the system, and outputs are the signals or data sent from it...

systems of the computer.
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