Parts-per notation

In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are ppm (parts-per-million, 10−6), ppb (parts-per-billion, 10−9), ppt (parts-per-trillion, 10−12) and ppq (parts-per-quadrillion, 10−15). This notation is not part of the SI system and its meaning is ambiguous.

Fluorescein (2)
Fluorescein aqueous solutions, diluted from 1–10,000 parts-per-million

Overview

Parts-per notation is often used describing dilute solutions in chemistry, for instance, the relative abundance of dissolved minerals or pollutants in water. The quantity “1 ppm” can be used for a mass fraction if a water-borne pollutant is present at one-millionth of a gram per gram of sample solution. When working with aqueous solutions, it is common to assume that the density of water is 1.00 g/mL. Therefore, it is common to equate 1 kilogram of water with 1 L of water. Consequently, 1 ppm corresponds to 1 mg/L and 1 ppb corresponds to 1 μg/L.

Similarly, parts-per notation is used also in physics and engineering to express the value of various proportional phenomena. For instance, a special metal alloy might expand 1.2 micrometers per meter of length for every degree Celsius and this would be expressed as “α = 1.2 ppm/°C.” Parts-per notation is also employed to denote the change, stability, or uncertainty in measurements. For instance, the accuracy of land-survey distance measurements when using a laser rangefinder might be 1 millimeter per kilometer of distance; this could be expressed as “Accuracy = 1 ppm.”[1]

Parts-per notations are all dimensionless quantities: in mathematical expressions, the units of measurement always cancel. In fractions like “2 nanometers per meter” (2 nm/m = 2 nano = 2 × 10−9 = 2 ppb = 2 × 0.000000001) so the quotients are pure-number coefficients with positive values less than 1. When parts-per notations, including the percent symbol (%), are used in regular prose (as opposed to mathematical expressions), they are still pure-number dimensionless quantities. However, they generally take the literal “parts per” meaning of a comparative ratio (e.g., “2 ppb” would generally be interpreted as “two parts in a billion parts”).[2]

Parts-per notations may be expressed in terms of any unit of the same measure. For instance, the coefficient of thermal expansion of a certain brass alloy, α = 18.7 ppm/°C, may be expressed as 18.7 (µm/m)/°C, or as 18.7 (µin/in)/°C; the numeric value representing a relative proportion does not change with the adoption of a different unit of measure.[3] Similarly, a metering pump that injects a trace chemical into the main process line at the proportional flow rate Qp = 125 ppm, is doing so at a rate that may be expressed in a variety of volumetric units, including 125 µL/L, 125 µgal/gal, 125 cm3/m3, etc.

In nuclear magnetic resonance (NMR) spectroscopy

In nuclear magnetic resonance spectroscopy (NMR), chemical shift is usually expressed in ppm. It represents the difference of a measured frequency in parts per million from the reference frequency. The reference frequency depends on the instrument's magnetic field and the element being measured. It is usually expressed in MHz. Typical chemical shifts are rarely more than a few hundred Hz from the reference frequency, so chemical shifts are conveniently expressed in ppm (Hz/MHz). Parts-per notation gives a dimensionless quantity that does not depend on the instrument's field strength.

Parts-per expressions

Visualisation parts per
Visualisation of 1%, 1‰, 1‱ and 1 ppm as the fraction of the red cube to its respective block (larger version)
  • One part per hundred is generally represented by the percent sign (%) and denotes one part per 100 parts, one part in 102, and a value of 1 × 10−2. This is equivalent to about fifteen minutes out of one day.
  • One part per thousand should generally be spelled out in full and not as "ppt" (which is usually understood to represent "parts per trillion"). It may also be denoted by the permille sign (‰). Note however, that specific disciplines such as oceanography, as well as educational exercises, do use the "ppt" abbreviation. "One part per thousand" denotes one part per 1000 parts, one part in 103, and a value of 1 × 10−3. This is equivalent to about one and a half minutes out of one day.
  • One part per ten thousand is denoted by the permyriad sign (‱). Although rarely used in science (ppm is typically used instead), one permyriad has an unambiguous value of one part per 10000 parts, one part in 104, and a value of 1 × 10−4. This is equivalent to about nine seconds out of one day.
    In contrast, in finance, the basis point is typically used to denote changes in or differences between percentage interest rates (although it can also be used in other cases where it is desirable to express quantities in hundredths of a percent). For instance, a change in an interest rate from 5.15% per annum to 5.35% per annum could be denoted as a change of 20 basis points (per annum). As with interest rates, the words "per annum" (or "per year") are often omitted. In that case, the basis point is a quantity with a dimension of (time−1).[4]
  • One part per million (ppm) denotes one part per 1000000 parts, one part in 106, 1/1000000 × 100% = 0.0001% (or 1% = 10000 ppm), and a value of 1 × 10−6. This is equivalent to about 32 seconds out of a year; in distance measurement, it is equivalent to 1 mm of error per km of distance traversed.
  • One part per billion (ppb) denotes one part per 1000000000 parts, one part in 109, 1/1000000000 × 100% = 0.0000001% (or 1% = 10000000 ppb) and a value of 1 × 10−9. This is equivalent to about three seconds out of a century.
  • One part per trillion (ppt) denotes one part per 1000000000000 parts, one part in 1012, and a value of 1 × 10−12. This is equivalent to about thirty seconds out of every million years.
  • One part per quadrillion (ppq) denotes one part per 1000000000000000 parts, one part in 1015, and a value of 1 × 10−15. This is equivalent to about two and a half minutes out of the age of the Earth (4.5 billion years). Although relatively uncommon in analytical chemistry, measurements at the ppq level are sometimes performed.[5]

Criticism

Although the International Bureau of Weights and Measures (an international standards organization known also by its French-language initials BIPM) recognizes the use of parts-per notation, it is not formally part of the International System of Units (SI).[2] Note that although “percent” (%) is not formally part of the SI, both the BIPM and the ISO take the position that “in mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01” for dimensionless quantities.[2][6] According to IUPAP, “a continued source of annoyance to unit purists has been the continued use of percent, ppm, ppb, and ppt.”[7] Although SI-compliant expressions should be used as an alternative, the parts-per notation remains nevertheless widely used in technical disciplines. The main problems with the parts-per notation are set out below.

Long and short scales

Because the named numbers starting with a “billion” have different values in different countries, the BIPM suggests avoiding the use of “ppb” and “ppt” to prevent misunderstanding. The U.S. National Institute of Standards and Technology (NIST) takes the stringent position, stating that “the language-dependent terms [ . . . ] are not acceptable for use with the SI to express the values of quantities.”[8]

Thousand vs. trillion

Although "ppt" usually means "parts per trillion", it occasionally means "parts per thousand". Unless the meaning of "ppt" is defined explicitly, it has to be determined from the context.

Mass fraction vs. mole fraction vs. volume fraction

Another problem of the parts-per notation is that it may refer to mass fraction, mole fraction or volume fraction. Since it is usually not stated which quantity is used, it is better to write the unit as kg/kg, mol/mol or m3/m3 (even though they are all dimensionless).[9] The difference is quite significant when dealing with gases and it is very important to specify which quantity is being used. For example, the conversion factor between a mass fraction of 1 ppb and a mole fraction of 1 ppb is about 4.7 for the greenhouse gas CFC-11 in air. For volume fraction, the suffix "V" or "v" is sometimes appended to the parts-per notation (e.g., ppmV, ppbv, pptv).[10][11] Unfortunately, ppbv and pptv are also often used for mole fractions (which is identical to volume fraction only for ideal gases).

To distinguish the mass fraction from volume fraction or mole fraction, the letter w (standing for weight) is sometimes added to the abbreviation (e.g., ppmw, ppbw).[12]

The usage of the parts-per notation is generally quite fixed inside most specific branches of science, leading some researchers to draw the conclusion that their own usage (mass/mass, mol/mol, volume/volume, or others) is the only correct one. This, in turn, leads them to not specify their usage in their publications, and others may therefore misinterpret their results. For example, electrochemists often use volume/volume, while chemical engineers may use mass/mass as well as volume/volume. Many academic papers of otherwise excellent level fail to specify their usage of the parts-per notation.

SI-compliant expressions

SI-compliant units that can be used as alternatives are shown in the chart below. Expressions that the BIPM explicitly does not recognize as being suitable for denoting dimensionless quantities with the SI are shown in red text.

Notations for dimensionless quantities
Measure SI
units
Named
parts-per ratio
(short scale)
Parts-per
abbreviation
or symbol
Value in
scientific
notation
A strain of… 2 cm/m 2 parts per hundred     2%[13] 2 × 10−2
A sensitivity of… 2 mV/V 2 parts per thousand 2 ‰ 2 × 10−3
A sensitivity of… 0.2 mV/V 2 parts per ten thousand 2 ‱ 2 × 10−4
A sensitivity of… 2 µV/V 2 parts per million 2 ppm 2 × 10−6
A sensitivity of… 2 nV/V 2 parts per billion 2 ppb 2 × 10−9
A sensitivity of… 2 pV/V 2 parts per trillion 2 ppt 2 × 10−12
A mass fraction of… 2 mg/kg 2 parts per million 2 ppm 2 × 10−6
A mass fraction of… 2 µg/kg 2 parts per billion 2 ppb 2 × 10−9
A mass fraction of… 2 ng/kg 2 parts per trillion 2 ppt 2 × 10−12
A mass fraction of… 2 pg/kg 2 parts per quadrillion 2 ppq 2 × 10−15
A volume fraction of… 5.2 µL/L 5.2 parts per million 5.2 ppm 5.2 × 10−6
A mole fraction of… 5.24 µmol/mol 5.24 parts per million 5.24 ppm 5.24 × 10−6
A mole fraction of… 5.24 nmol/mol 5.24 parts per billion 5.24 ppb 5.24 × 10−9
A mole fraction of… 5.24 pmol/mol 5.24 parts per trillion 5.24 ppt 5.24 × 10−12
A stability of… 1 (µA/A)/min. 1 part per million per min. 1 ppm/min. 1 × 10−6/min.
A change of… 5 nΩ/Ω 5 parts per billion 5 ppb 5 × 10−9
An uncertainty of… 9 µg/kg 9 parts per billion 9 ppb 9 × 10−9
A shift of… 1 nm/m 1 part per billion 1 ppb 1 × 10−9
A strain of… 1 µm/m 1 part per million 1 ppm 1 × 10−6
A temperature coefficient of… 0.3 (µHz/Hz)/°C 0.3 part per million per °C 0.3 ppm/°C 0.3 × 10−6/°C
A frequency change of… 0.35 × 10−9 ƒ 0.35 part per billion 0.35 ppb 0.35 × 10−9

Note that the notations in the “SI units” column above are all dimensionless quantities; that is, the units of measurement factor out in expressions like “1 nm/m” (1 nm/m = 1 nano = 1 × 10−9) so the quotients are pure-number coefficients with values less than 1.

Uno

Because of the cumbersome nature of expressing certain dimensionless quantities per SI guidelines, the International Union of Pure and Applied Physics (IUPAP) in 1999 proposed the adoption of the special name "uno" (symbol: U) to represent the number 1 in dimensionless quantities.[7] This symbol is not to be confused with the always-italicized symbol for the variable "uncertainty" (symbol: U). This unit name "uno" and its symbol could be used in combination with the SI prefixes to express the values of dimensionless quantities that are much less—or even greater—than one.[14]

Common parts-per notations in terms of the uno are given in the table below.

IUPAP's "uno" proposal
Coefficient Parts-per example Uno equiv. Symbol form Value of quantity
10−2 2% 2 centiuno 2 cU 2 × 10−2
10−3 2 ‰ 2 milliuno 2 mU 2 × 10−3
10−4 2 ‱ 0.2 milliuno 0.2 mU 2 × 10−4
10−6 2 ppm 2 microuno 2 µU 2 × 10−6
10−9 2 ppb 2 nanouno 2 nU 2 × 10−9
10−12 2 ppt 2 picouno 2 pU 2 × 10−12

In 2004, a report to the International Committee for Weights and Measures (known also by its French-language initials CIPM) stated that response to the proposal of the uno "had been almost entirely negative" and the principal proponent "recommended dropping the idea".[15] To date, the uno has not been adopted by any standards organization and it appears unlikely it will ever become an officially sanctioned way to express low-value (high-ratio) dimensionless quantities. The proposal was instructive, however, as to the perceived shortcomings of the current options for denoting dimensionless quantities.

Improper applications

Parts-per notation may properly be used only to express true dimensionless quantities; that is, the units of measurement must cancel in expressions like "1 mg/kg" so that the quotients are pure numbers with values less than 1. Mixed-unit quantities such as "a radon concentration of 15 pCi/L" are not dimensionless quantities and may not be expressed using any form of parts-per notation, such as "15 ppt". Other examples of measures that are not dimensionless quantities are as follows:

Note however, that it is not uncommon to express aqueous concentrations—particularly in drinking-water reports intended for the general public—using parts-per notation (2.1 ppm, 0.8 ppb, etc.) and further, for those reports to state that the notations denote milligrams per liter or micrograms per liter. Although "2.1 mg/L" is not a dimensionless quantity, it is assumed in scientific circles that "2.1 mg/kg" (2.1 ppm) is the true measure because one liter of water has a mass of about one kilogram. The goal in all technical writing (including drinking-water reports for the general public) is to clearly communicate to the intended audience with minimal confusion. Drinking water is intuitively a volumetric quantity in the public’s mind so measures of contamination expressed on a per-liter basis are considered to be easier to grasp. Still, it is technically possible, for example, to "dissolve" more than one liter of a very hydrophilic chemical in 1 liter of water; parts-per notation would be confusing when describing its solubility in water (greater than a million parts per million), so one would simply state the volume (or mass) that will dissolve into a liter, instead.

When reporting air-borne rather than water-borne densities, a slightly different convention is used since air is approximately 1000 times less dense than water. In water, 1 µg/m3 is roughly equivalent to parts-per-trillion whereas in air, it is roughly equivalent to parts-per-billion. Note also, that in the case of air, this convention is much less accurate. Whereas one liter of water is almost exactly 1 kg, one cubic meter of air is often taken as 1.143 kg—much less accurate, but still close enough for many practical uses.

See also

References

  1. ^ This is a simplified explanation. Laser rangefinders typically have a measurement granularity of one to ten millimeters; thus, the complete specification for distance measurement accuracy might read as follows:  Accuracy: ±(1 mm + 1 ppm). Consequently, a distance measurement of only a few meters would still have an accuracy of ±1 mm in this example. 
  2. ^ a b c BIPM: 5.3.7 Stating values of dimensionless quantities, or quantities of dimension one 
  3. ^ In the particular case of coefficient of thermal expansion, the change to inches (one of the U.S. customary units) is typically also accompanied by a change to degrees Fahrenheit. Since a Fahrenheit-sized interval of temperature is only ​59 that of a Celsius-sized interval, the value is typically expressed as 10.4 (µin/in)/°F rather than 18.7 (µin/in)/°C. 
  4. ^ "What are Basis Points (BPS)?".
  5. ^ Measurements of dioxin are routinely made at the sub-ppq level. The U.S. Environmental Protection Agency (EPA) currently sets a hard limit of 30 ppq for dioxin in drinking water but once recommended a voluntary limit of 0.013 ppq. Also, radioactive contaminants in drinking water, which are quantified by measuring their radiation, are often reported in terms of ppq; 0.013 ppq is equivalent to the thickness of a sheet of paper versus a journey of 146000 trips around the world. 
  6. ^ Quantities and units - Part 0: General principles, ISO 31-0:1992. 
  7. ^ a b Petley, Brian W. (September 1998). "Report on recent Committee activities on behalf of IUPAP to the 1999 IUPAP General Assembly". Archived from the original on 2017-08-15. Retrieved 2017-08-15.
  8. ^ NIST: Rules and Style Conventions for Expressing Values of Quantities: 7.10.3 ppm, ppb, and ppt 
  9. ^ Schwartz and Warneck (1995). "Units for use in atmospheric chemistry" (PDF). Pure Appl. Chem. 67: 1377–1406. doi:10.1351/pac199567081377. Retrieved March 9, 2011.
  10. ^ "EPA On-line Tools for Site Assessment Calculation: Indoor Air Unit Conversion". Environmental Protection Agency.
  11. ^ Milton R. Beychok (2005). "Air Dispersion Modeling Conversions and Formulas". Fundamentals of Stack Gas Dispersion (4th ed.). Milton R. Beychok. ISBN 0964458802.
  12. ^ "Introduction to Green Engineering".
  13. ^ Compliance with the SI regarding the percent symbol (%) is limited in this chart. According to the BIPM’s SI brochure: Subsection 5.3.3, Formatting the value of a quantity, a space is always used to separate the unit symbol from the numeric value. Notable exceptions are the unit symbols for degree, minute, and second for plane angle, °, ′, and ″ (e.g., a latitude of 47° 38′ 8.8″). However, according to 5.3.7 Stating values of dimensionless quantities, or quantities of dimension one, the exception does not apply to the “%” symbol; it states as follows: “When it [the percent symbol] is used, a space separates the number and the symbol %.” This practice has not been well adopted with regard to the % symbol, is contrary to Wikipedia’s Manual of Style, and is not observed here. 
  14. ^ Certain mathematical functions can produce proportional quantities with values greater than 1. 
  15. ^ Consultative Committee for Units (13–14 May 2004). "Report of the 16th meeting (13–14 May 2004) to the International Committee for Weights and Measures, of the International Bureau of Weights and Measures" (PDF). Archived from the original (PDF) on 2014-03-10.

External links

  • National Institute of Standards and Technology (NIST): Home page
  • International Bureau of Weights and Measures (BIPM): Home page
Basis point

A per ten thousand sign or basis point (often denoted as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one ten thousandth. The related concept of a permyriad is literally one part per ten thousand. Figures are commonly quoted in basis points in finance, especially in fixed income markets.

Concentration

In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration. A concentration can be any kind of chemical mixture, but most frequently solutes and solvents in solutions. The molar (amount) concentration has variants such as normal concentration and osmotic concentration.

Deformation (mechanics)

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body.

A deformation may be caused by external loads, body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

Strain is a description of deformation in terms of relative displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called elastic deformations. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is viscous deformation, which is the irreversible part of viscoelastic deformation.

In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one and the corresponding unit of measurement in the SI is one (or 1) unit and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: and 17/3) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

We begin with positive common fractions, where the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero because zero parts can never make up a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates or ​34 of a cake.

A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10−2 all equal the fraction 1/100. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and division. Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four). The non-zero denominator in the case using a fraction to represent division is an example of the rule that division by zero is undefined.

We can also write negative fractions, which represent the opposite of a positive fraction. For example if 1/2 represents a half dollar profit, then −1/2 represents a half dollar loss. Because of the rules of division of signed numbers, which require that, for example, negative divided by positive is negative, −1/2, -1/2 and 1/-2, all represent the same fraction, negative one-half. Because a negative divided by a negative produces a positive, -1/-2 represents positive one-half.

In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as 2/2 (see square root of 2) and π/4 (see proof that π is irrational).

Glossary of fuel cell terms

The Glossary of fuel cell terms lists the definitions of many terms used within the fuel cell industry. The terms in this fuel cell glossary may be used by fuel cell industry associations, in education material and fuel cell codes and standards to name but a few.

History of mathematical notation

The history of mathematical notation includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

The development of mathematical notation can be divided in stages. The "rhetorical" stage is where calculations are performed by words and no symbols are used. The "syncopated" stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. From ancient times through the post-classical age, bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light. The "symbolic" stage is where comprehensive systems of notation supersede rhetoric. Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This symbolic system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century, and has continued to develop in the contemporary era.

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, the focus here, the investigation into the mathematical methods and notation of the past.

Index of meteorology articles

This is a list of meteorology topics. The terms relate to meteorology, the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. (see also: List of meteorological phenomena)

Millionth

One millionth is equal to 0.000 001, or 1 x 10−6 in scientific notation. It is the reciprocal of a million, and can be also written as 1/1 000 000. Units using this fraction can be indicated using the prefix "micro-" from Greek, meaning "small". Numbers of this quantity are expressed in terms of µ (the Greek letter mu)."Millionth" can also mean the ordinal number that comes after the nine hundred, ninety-nine thousand, nine hundred, ninety-ninth and before the million and first.

Nine (purity)

Nines are an informal, yet common method of grading the purity of materials.

Per mille

A per mille (from Latin per mīlle, "in each thousand"), also spelled per mil, per mill, permil, permill, or permille is a sign indicating parts per thousand. Per mil should not be confused with parts per million (ppm).

The sign is written ‰, which looks like a percent sign (%) with an extra zero in the divisor. It is included in the General Punctuation block of Unicode characters: U+2030 ‰ PER MILLE SIGN (HTML ‰ · ‰). It is accessible in Windows using ALT+0137, and is accessible via a Compose Key using %o.

The term occurs so rarely in English that major dictionaries do not agree on the spelling or pronunciation even within a single dialect of English and some major dictionaries such as Macmillan and Longman do not even contain an entry. The term is more common in other European languages where it is used in contexts, such as blood alcohol content, that are usually expressed as a percentage in English-speaking countries.

Percent sign

The percent (per cent) sign (%) is the symbol used to indicate a percentage, a number or ratio as a fraction of 100. Related signs include the permille (per thousand) sign ‰ and the permyriad (per ten thousand) sign ‱ (also known as a basis point), which indicate that a number is divided by one thousand or ten thousand respectively. Higher proportions use parts-per notation.

Percentage

In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number (pure number).

Percentage point

A percentage point or percent point is the unit for the arithmetic difference of two percentages. For example, moving up from 40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured. In the literature, the percentage point unit is usually either written out, or abbreviated as pp or p.p. to avoid ambiguity. After the first occurrence, some writers abbreviate by using just "point" or "points".

Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent smoked. One can thus say that from 1980 to 1990, the prevalence of smoking decreased by 10 percentage points although smoking did not decrease by 10 percent (it decreased by 20 percent) – percentages indicate ratios, not differences.

Percentage-point differences are one way to express a risk or probability. Consider a drug that cures a given disease in 70 percent of all cases, while without the drug, the disease heals spontaneously in only 50 percent of cases. The drug reduces absolute risk by 20 percentage points. Alternatives may be more meaningful to consumers of statistics, such as the reciprocal, also known as the number needed to treat (NNT). In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. Thus if 5 patients are treated with the drug, one could expect to heal one more case of the disease than would have occurred in the absence of the drug.

For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation).

Ratio

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers in a ratio may be quantities of any kind, such as counts of persons or objects, or such as measurements of lengths, weights, time, etc. In most contexts both numbers are restricted to be positive.

A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient a/b, since the product of the quotient and the second number yields the first, as required by the above definition.

Consequently, a ratio may be considered as an ordered pair of numbers, as a fraction with the first number in the numerator and the second as denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.

Snake Projection

The Snake Projection is a coordinate system which projects geographical coordinates onto an easting and northing grid. The parameters defining the Snake Projection must be tailored for specific projects; the most typical use is with large-scale linear engineering projects such as rail infrastructure, however the projection is equally applicable to any application requiring a low distortion grid along a linear route (e.g. pipelines and roads). The name of the projection is derived from the sinuous nature of the projects it may be designed for. Typical map projection distance distortion characteristics of a Snake projection are minimal over the whole route within approximately 20 kilometres of the centre line. The principal advantage of the projection is that, for the corridor defining the design space, distances measured on the ground have a one to one relationship with distances in coordinate space (i.e. no scale factor need be applied to convert between distances in grid and distances on the ground). The main disadvantage is that away from the design corridor the distortion of the projection is not controlled.

The Snake Projection is the engineering coordinate system used for a significant proportion of primary rail routes in the UK, including that of the HS2 London to Birmingham line. For the London to Glasgow West Coast Main Line the distortion in the Snake Projection used is no greater than 20 parts per notation within 5 kilometres of either side of the track.

Ultrapure water

Ultrapure water (also UPW or high-purity water) is water that has been purified to uncommonly stringent specifications. Ultrapure water is a commonly used term in the semiconductor industry to emphasize the fact that the water is treated to the highest levels of purity for all contaminant types, including: organic and inorganic compounds; dissolved and particulate matter; volatile and non-volatile, reactive and inert; hydrophilic and hydrophobic; and dissolved gases.

UPW and commonly used term deionized (DI) water are not the same. In addition to the fact that UPW has organic particles and dissolved gases removed, a typical UPW system has three stages: a pretreatment stage to produce purified water, a primary stage to further purify the water, and a polishing stage, the most expensive part of the treatment process.A number of organizations and groups develop and publish standards associated with the production of UPW. For microelectronics and power, they include Semiconductor Equipment and Materials International (SEMI) (microelectronics and photovoltaic), American Society for Testing and Materials International (ASTM International) (semiconductor, power), Electric Power Research Institute (EPRI) (power), American Society of Mechanical Engineers (ASME) (power), and International Association for the Properties of Water and Steam (IAPWS) (power). Pharmaceutical plants follow water quality standards as developed by pharmacopeias, of which three examples are the United States Pharmacopeia, European Pharmacopeia, and Japanese Pharmacopeia.

The most widely used requirements for UPW quality are documented by ASTM D5127 "Standard Guide for Ultra-Pure Water Used in the Electronics and Semiconductor Industries" and SEMI F63 "Guide for ultrapure water used in semiconductor processing".Ultra pure water is also used as boiler feed water in the UK AGR fleet.

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