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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, 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 contribution to an object's mass by electrons.
Any macroscopic system is always much more complicated than any idealized mathematical model 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), the reaction force of the road opposing the weight, the friction of the wheels on the road, the force of the engine[clarification needed], 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 have 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 electrical engineering
Electronic circuit designers make use of ideal circuit concepts in the design process. These include the perfect voltage source, the perfect current source, the perfect amplifier, a perfect ground 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, say 5 V dc, across any load, no matter what current may become necessary to do this. As the load impedance reduced toward zero ohms (a perfect short circuit - which also cannot truly exist) then the current flow and power 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 amps. 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.
In risk assessment
The continuing success of the global travel 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.
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 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.
A computer system's internal representation of floating point 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 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 where resolution and bit-rates are necessarily limited, financial 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 and mouse, a network 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, 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 of throwing and catching exceptions 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, 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 systems of the computer.
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