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The neper (symbol: Np) is a logarithmic unit for
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of
John Napier John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioan ...
, the inventor of logarithms. As is the case for the
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
and bel, the neper is a unit defined in the
international standard international standard is a technical standard developed by one or more international standards organizations. International standards are available for consideration and use worldwide. The most prominent such organization is the International Org ...
ISO 80000. It is not part of the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
(SI), but is accepted for use alongside the SI.

# Definition

Like the
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
, the neper is a unit in a
logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
. While the bel uses the decadic (base-10)
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
to compute ratios, the neper uses the natural logarithm, based on
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an express ...
(). The level a ratio of two signal amplitudes or root-power quantities, with the unit neper, is given by :$L = \ln\frac\mathrm,$ where $x_1$ and $x_2$ are the signal amplitudes, and is the natural logarithm. The level of a ratio of two power quantities, with the unit neper, is given by :$L = \frac \ln\frac\mathrm,$ where $p_1$ and $p_2$ are the signal powers. In the
International System of Quantities The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlie ...
, the neper is defined as .

# Units

The neper is defined in terms of ratios of field quantities — also called ''root-power quantities'' — (for example, voltage or current amplitudes in electrical circuits, or pressure in
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
), whereas the decibel was originally defined in terms of power ratios. A power ratio 10 log ''r'' dB is equivalent to a field-quantity ratio 20 log ''r'' dB, since power in a linear system is proportional to the square ( Joule's laws) of the amplitude. Hence the decibel and the neper have a fixed ratio to each other: :$1\ \text = 20 \log_ e\ \text \approx \text$ and :$1\ \mathrm = \frac \ln\left(10\right)\ \mathrm \approx \text .$ The (voltage) level ratio is :$\begin L & = 10 \log_ \frac & \text \\ & = 10 \log_ ^2 & \text \\ & = 20 \log_ \frac & \text \\ & = \ln \frac & \text. \\ \end$ Like the decibel, the neper is a dimensionless unit. The
International Telecommunication Union The International Telecommunication Union is a specialized agency of the United Nations responsible for many matters related to information and communication technologies. It was established on 17 May 1865 as the International Telegraph Union ...
(ITU) recognizes both units. Only the neper is coherent with the SI.

# Applications

The neper is a natural linear unit of relative difference, meaning in nepers (logarithmic units) relative differences add rather than multiply. This property is shared with logarithmic units in other bases, such as the bel. The derived units decineper (1 dNp = 0.1 neper) and centineper (1 cNp = 0.01 neper) are also used. The centineper for root-power quantities corresponds to a
log point In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting'' va ...
or log percentage, see '.

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Nat (unit) The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information, based on natural logarithms and powers of ''e'', rather than the powers of 2 and base 2 logarithms, which define the shannon. This unit is ...

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