**7.1 Value and numerical value of a quantity **

The *value* of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the *numerical value* of the quantity expressed in that unit.

More formally, the value of quantity *A* can be written as *A* = {*A*}[*A*], where {*A*} is the numerical value of *A* when the value of *A* is expressed in the unit [*A*]. The numerical value can therefore be written as {*A*} = *A* / [*A*], which is a convenient form for use in figures and tables. Thus, to eliminate the possibility of misunderstanding, an axis of a graph or the heading of a column of a table can be labeled "*t*/°C" instead of "*t* (°C)" or "Temperature (°C)." Similarly, an axis or column heading can be labeled "*E*/(V/m)" instead of "*E* (V/m)" or "Electric field strength (V/m)."

*Examples:*

- In the SI, the value of the velocity of light in vacuum is c = 299 792 458 m/s exactly. The number 299 792 458 is the numerical value of c when c is expressed in the unit m/s, and equals c/(m/s).
- The ordinate of a graph is labeled
*T*/(10^{3}K), where*T*is thermodynamic temperature and K is the unit symbol for kelvin, and has scale marks at 0, 1, 2, 3, 4, and 5. If the ordinate value of a point on a curve in the graph is estimated to be 3.2, the corresponding temperature is*T*/ (10^{3}K) = 3.2 or*T*= 3200 K. Notice the lack of ambiguity in this form of labeling compared with "Temperature (10^{3}K)." - An expression such as ln(p/MPa), where p is the quantity symbol for pressure and MPa is the unit symbol for megapascal, is perfectly acceptable, because p/MPa is the numerical value of p when p is expressed in the unit MPa and is simply a number.

*Notes:*

- For the conventions concerning the grouping of digits, see Sec. 10.5.3.
- An alternative way of writing
*c*/(m/s) is {*c*}_{m/s}, meaning the numerical value of*c*when*c*is expressed in the unit m/s.

**7.2 Space between numerical value and unit symbol**

In the expression for the value of a quantity, the unit symbol is placed after the numerical value and a space is left between the numerical value and the unit symbol.

The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle: °, ′, and ″, respectively (see Table 6), in which case no space is left between the numerical value and the unit symbol.

*Example:* α = 30°22′8″

*Note:* α is a quantity symbol for plane angle.

This rule means that:

(a) The symbol °C for the degree Celsius is preceded by a space when one expresses the values of Celsius temperatures.

*Example:* *t* = 30.2 °C *but not:* * t* = 30.2°C *or* * t* = 30.2° C

(b) Even when the value of a quantity is used as an adjective, a space is left between the numerical value and the unit symbol. (This rule recognizes that unit symbols are not like ordinary words or abbreviations but are mathematical entities, and that the value of a quantity should be expressed in a way that is as independent of language as possible—sees Secs. 7.6 and 7.10.3.)

*Examples:*

a 1 m end gauge *but not:* a 1-m end gauge

a 10 kΩ resistor *but not:* a 10-kΩ resistor

However, if there is any ambiguity, the words should be rearranged accordingly. For example, the statement "the samples were placed in 22 mL vials" should be replaced with the statement "the samples were placed in vials of volume 22 mL."

*Note:* When unit names are spelled out, the normal rules of English apply. Thus, for example, "a roll of 35-millimeter film" is acceptable (see Sec. 7.6, note 3).

**7.3 Number of units per value of a quantity **

The value of a quantity is expressed using no more than one unit.

*Example:*: *l* = 10.234 m *but not*: *l* = 10 m 23 cm 4 mm

*Notes:* Expressing the values of time intervals and of plane angles are exceptions to this rule. However, it is preferable to divide the degree decimally. Thus one should write 22.20° rather than 22°12′, except in fields such as cartography and astronomy.

**7.4 Unacceptability of attaching information to units **

When one gives the value of a quantity, it is incorrect to attach letters or other symbols to the unit in order to provide information about the quantity or its conditions of measurement. Instead, the letters or other symbols should be attached to the quantity.

*Example:* *V*_{max} = 1000 V *but not:* *V* = 1000 V_{max}

*Note: V* is a quantity symbol for potential difference.

**7.5 Unacceptability of mixing information with units **

When one gives the value of a quantity, any information concerning the quantity or its conditions of measurement must be presented in such a way as not to be associated with the unit. This means that quantities must be defined so that they can be expressed solely in acceptable units (including the unit one — see Sec. 7.10).

*Examples:*

the Pb content is 5 ng/L* but not: *5 ng Pb/L or 5 ng of lead/L

the sensitivity for NO_{3} molecules is 5 × 10^{10}/cm^{3}* but not: *the sensitivity is 5 × 10^{10} NO^{3} molecules/cm^{3}

the neutron emission rate is 5 × 10^{10}/s* but not: *the emission rate is 5 × 10^{10} n/s

the number density of O_{2} atoms is 3 × 10^{18}/cm^{3}* but not: *the density is 3 × 10^{18} O_{2} atoms/cm^{3}

the resistance per square is 100 Ω* but not: *the resistance is 100 Ω/square

**7.6 Symbols for numbers and units versus spelled-out names of numbers and units **

This *Guide* takes the position that the key elements of a scientific or technical paper, particularly the results of measurements and the values of quantities that influence the measurements, should be presented in a way that is as independent of language as possible. This will allow the paper to be understood by as broad an audience as possible, including readers with limited knowledge of English. Thus, to promote the comprehension of quantitative information in general and its broad understandability in particular, values of quantities should be expressed in acceptable units using

— the Arabic symbols for numbers, that is, the Arabic numerals, not the spelled-out names of the Arabic numerals; and

— the symbols for the units, *not* the spelled-out names of the units.

*Examples:*

the length of the laser is 5 m* but not: *the length of the laser is five meters

the sample was annealed at a temperature of 955 K for 12 h* but not: *the sample was annealed at a temperature of 955 kelvins for 12 hours

*Notes:*

- If the intended audience for a publication is unlikely to be familiar with a particular unit symbol, it should be defined when first used.
- Because the use of the spelled-out name of an Arabic numeral with a unit symbol can cause confusion, such combinations must strictly be avoided. For example, one should never write "the length of the laser is five m."
- Occasionally, a value is used in a descriptive or literary manner and it is fitting to use the spelledout name of the unit rather than its symbol. Thus, this Guide considers acceptable statements such as "the reading lamp was designed to take two 60-watt light bulbs," or "the rocket journeyed uneventfully across 380 000 kilometers of space," or "they bought a roll of 35-millimeter film for their camera."
- The United States Government Printing Office Style Manual (Ref. [3], pp. 181-189) gives the rule that symbols for numbers are always to be used when one expresses (a) the value of a quantity in terms of a unit of measurement, (b) time (including dates), and (c) an amount of money. This publication should be consulted for the rules governing the choice between the use of symbols for numbers and the spelled-out names of numbers when numbers are dealt with in general.

**7.7 Clarity in writing values of quantities **

The value of a quantity is expressed as the product of a number and a unit (see Sec. 7.1). Thus, to avoid possible confusion, this *Guide* takes the position that values of quantities must be written so that it is completely clear to which unit symbols the numerical values of the quantities belong. Also to avoid possible confusion, this *Guide* strongly recommends that the word "to" be used to indicate a range of values for a quantity instead of a range dash (that is, a long hyphen) because the dash could be misinterpreted as a minus sign. (The first of these recommendations once again recognizes that unit symbols are not like ordinary words or abbreviations but are mathematical entities—see Sec. 7.2.)

51 mm × 51 mm × 25 mm | but not: |
51 × 51 × 25 mm |

225 nm to 2400 nm or (225 to 2400) nm | but not: |
225 to 2400 nm |

0 ºC to 100 ºC or (0 to 100) ºC | but not: |
0 ºC – 100 ºC |

0 V to 5 V or (0 to 5) V | but not: |
0 – 5 V |

(8.2, 9.0, 9.5, 9.8, 10.0) GHz | but not: |
8.2, 9.0, 9.5, 9.8, 10.0 GHz |

63.2 m ± 0.1 m or (63.2 ± 0.1) m | but not: |
63.2 ± 0.1 m or 63.2 m ± 0.1 |

129 s – 3 s = 126 s or (129 – 3) s = 126 s | but not: |
129 – 3 s = 126 s |

*Note:* For the conventions concerning the use of the multiplication sign, see Sec. 10.5.4.

**7.8 Unacceptability of stand-alone unit symbols **

Symbols for units are never used without numerical values or quantity symbols (they are not abbreviations).

*Examples:*

there are 10^{6} mm in 1 km *but not:* there are many mm in a km

it is sold by the cubic meter *but not:* it is sold by the m^{3}

t/°C, E/(V/m), p/MPa, and the like are perfectly acceptable (see Sec. 7.1).

The selection of the appropriate decimal multiple or submultiple of a unit for expressing the value of a quantity, and thus the choice of SI prefix, is governed by several factors.

These include:

— the need to indicate which digits of a numerical value are significant,

— the need to have numerical values that are easily understood, and

— the practice in a particular field of science or technology.

A digit is significant if it is required to express the numerical value of a quantity. In the expression *l* = 1200 m, it is not possible to tell whether the last two zeroes are significant or only indicate the magnitude of the numerical value of *l*. However, in the expression l = 1.200 km, which uses the SI prefix symbol for 10^{3} (kilo, symbol k), the two zeroes are assumed to be significant because if they were not, the value of *l* would have been written *l* = 1.2 km.

It is often recommended that, for ease of understanding, prefix symbols should be chosen in such a way that numerical values are between 0.1 and 1000, and that only prefix symbols that represent the number 10 raised to a power that is a multiple of 3 should be used.

*Examples:*

3.3 × 10^{7} Hz may be written as 33 × 10^{6} Hz = 33 MHz

0.009 52 g may be written as 9.52 × 10^{-3} g = 9.52 mg

2703 W may be written as 2.703 × 10^{3} W = 2.703 kW

5.8 × 10^{-8} m may be written as 58 × 10^{-9} m = 58 nm

However, the values of quantities do not always allow this recommendation to be followed, nor is it mandatory to try to do so.

In a table of values of the same kind of quantities or in a discussion of such values, it is usually recommended that only one prefix symbol should be used even if some of the numerical values are not between 0.1 and 1000. For example, it is often considered preferable to write "the size of the sample is 10 mm × 3 mm × 0.02 mm" rather than "the size of the sample is 1 cm × 3 mm × 20 μm."

In certain kinds of engineering drawings it is customary to express all dimensions in millimeters. This is an example of selecting a prefix based on the practice in a particular field of science or technology.

**7.10 Values of quantities expressed simply as numbers: the unit one, symbol 1 **

Certain quantities, such as refractive index, relative permeability, and mass fraction, are defined as the ratio of two mutually comparable quantities and thus are of dimension one (see Sec. 7.14). The coherent SI unit for such a quantity is the ratio of two identical SI units and may be expressed by the number 1. However, the number 1 generally does not appear in the expression for the value of a quantity of dimension one. For example, the value of the refractive index of a given medium is expressed as *n* = 1.51 × 1 = 1.51.

On the other hand, certain quantities of dimension one have units with special names and symbols which can be used or not depending on the circumstances. Plane angle and solid angle, for which the SI units are the radian (rad) and steradian (sr), respectively, are examples of such quantities (see Sec. 4.2.1).

**7.10.1 Decimal multiples and submultiples of the unit one **

Because SI prefix symbols cannot be attached to the unit one (see Sec. 6.2.6), powers of 10 are used to express decimal multiples and submultiples of the unit one.

*Example:* μr = 1.2 × 10^{-6} *but not:* μ_{r} = 1.2 μ

*Note:* μr is the quantity symbol for relative permeability.

**7.10.2 %, percentage by, fraction**

In keeping with Ref. [4: ISO 31-0], this *Guide* takes the position that it is acceptable to use the internationally recognized symbol % (percent) for the number 0.01 with the SI and thus to express the values of quantities of dimension one (see Sec. 7.14) with its aid. When it is used, a space is left between the symbol % and the number by which it is multiplied [4: ISO 31-0]. Further, in keeping with Sec. 7.6, the symbol % should be used, not the name "percent."

*Example:* x_{B} = 0.0025 = 0.25 % *but not:* x_{B} = 0.0025 = 0.25% or x_{B} = 0.25 percent

*Note:* x_{B} is the quantity symbol for amount-of-substance fraction of B (see Sec. 8.6.2).

Because the symbol % represents simply a number, it is not meaningful to attach information to it (see Sec. 7.4). One must therefore avoid using phrases such as "percentage by weight," "percentage by mass," "percentage by volume," or "percentage by amount of substance." Similarly, one must avoid writing, for example, "% (*m/m*)," "% (by weight)," "% (*V/V*)," "% (by volume)," or "% (mol/mol)." The preferred forms are "the mass fraction is 0.10," or "the mass fraction is 10 %," or "*w*_{B} = 0.10," or "*w*_{B} =10 %" (*w*_{B} is the quantity symbol for mass fraction of B—see Sec. 8.6.10); "the volume fraction is 0.35," or "the volume fraction is 35 %," or " φB = 0.35," or "φB = 35 %" (φB is the quantity symbol for volume fraction of B—see Sec. 8.6.6); and "the amount-of-substance fraction is 0.15," or "the amount-of-substance fraction is 15 %," or "*x*_{B} = 0.15," or "*x*_{B} = 15 %." Mass fraction, volume fraction, and amount-of-substance fraction of B may also be expressed as in the following examples: *w*_{B} = 3 g/kg; φB = 6.7 mL/L; *x*_{B} = 185 mmol/mol. Such forms are highly recommended (see also Sec. 7.10.3).

In the same vein, because the symbol % represents simply the number 0.01, it is incorrect to write, for example, "where the resistances *R*^{1} and *R*^{2} differ by 0.05 %," or "where the resistance R1 exceeds the resistance *R*2 by 0.05 %." Instead, one should write, for example, "where *R*^{1} = *R*^{2} (1 + 0.05 %)," or define a quantity Δ via the relation Δ = (*R*^{1} - *R*^{2}) / *R*2 and write "where Δ = 0.05 %." Alternatively, in certain cases,the word "fractional" or "relative" can be used. For example, it would be acceptable to write "the fractional increase in the resistance of the 10 kΩ reference standard in 2006 was 0.002 %."

In keeping with Ref. [4: ISO 31-0], this *Guide* takes the position that the language-dependent terms part per million, part per billion, and part per trillion, and their respective abbreviations "ppm," "ppb," and "ppt" (and similar terms and abbreviations), are not acceptable for use with the SI to express the values of quantities. Forms such as those given in the following examples should be used instead.

*Examples:*

a stability of 0.5 (μA/A)/min *but not:* a stability of 0.5 ppm/min

a shift of 1.1 nm/m *but not:* a shift of 1.1 ppb

a frequency change of 0.35 × 10^{-9} f *but not:* a frequency change of 0.35 ppb

a sensitivity of 2 ng/kg *but not:* a sensitivity of 2 ppt

the relative expanded uncertainty of the resistance R is Ur = 3 μΩ/Ω

or

the expanded uncertainty of the resistance *R* is *U* = 3 × 10^{-6}*R*

or

the relative expanded uncertainty of the resistance R is U_{r} = 3 × 10^{-6}*but not:*

the relative expanded uncertainty of the resistance R is U_{r} = 3 ppm

Because the names of numbers 10^{9} and larger are not uniform worldwide, it is best that they be avoided entirely (in many countries, 1 billion = 1 × 10^{12}, not 1 × 10^{9} as in the United States); the preferred way of expressing large numbers is to use powers of 10. This ambiguity in the names of numbers is one of the reasons why the use of ppm, ppb, ppt, and the like is deprecated. Another, and a more important one, is that it is inappropriate to use abbreviations that are language dependent together with internationally recognized signs and symbols, such as MPa, ln, 10^{13}, and %, to express the values of quantities and in equations or other mathematical expressions (see also Sec. 7.6).

*Note:* This *Guide* recognizes that in certain cases the use of ppm, ppb, and the like may be required by a law or a regulation. Under these circumstances, Secs. 2.1 and 2.1.1 apply.

It is unacceptable to use Roman numerals to express the values of quantities. In particular, one should not use C, M, and MM as substitutes for 10^{2}, 10^{3}, and 10^{6}, respectively.

**7.11 Quantity equations and numerical-value equations **

A quantity equation expresses a relation among quantities. An example is *l* = *ν*t, where *l* is the distance a particle in uniform motion with velocity *ν* travels in the time *t*.

Because a quantity equation such as l = *νt* is independent of the units used to express the values of the quantities that compose the equation, and because *l, ν*, and *t* represent quantities and not numerical values of quantities, it is incorrect to associate the equation with a statement such as "where *l* is in meters, *ν* is in meters per second, and *t* is in seconds."

On the other hand, a numerical value equation expresses a relation among numerical values of quantities and therefore does depend on the units used to express the values of the quantities. For example, {*l*}m = 3.6^{-1} {*ν*}_{km/h} {*t*}_{s} expresses the relation among the numerical values of l, ν, and t only when the values of *l, ν*, and *t* are expressed in the units meter, kilometer per hour, and second, respectively. (Here {*A*}_{X} is the numerical value of quantity *A* when its value is expressed in the unit X—see Sec. 7.1, note 2.)

An alternative way of writing the above numerical value equation, and one that is preferred because of its simplicity and generality, is *l*/m = 3.6^{-1} [*ν*/(km/h)](t / s). NIST authors should consider using this preferred form instead of the more traditional form "l = 3.6^{-1} νt, where l is in meters, ν is in kilometers per hour, and t is in seconds." In fact, this form is still ambiguous because no clear distinction is made between a quantity and its numerical value. The correct statement is, for example, "*l** = 3.6^{-1} ν* *t* *, where *l** is the numerical value of the distance l traveled by a particle in uniform motion when *l* is expressed in meters, ν* is the numerical value of the velocity ν of the particle when ν is expressed in kilometers per hour, and *t** is the numerical value of the time of travel t of the particle when t is expressed in seconds." Clearly, as is done here, it is important to use different symbols for quantities and their numerical values to avoid confusion.

It is the strong recommendation of this Guide that because of their universality, quantity equations should be used in preference to numerical-value equations. Further, if a numerical-value equation is used, it should be written in the preferred form given in the above paragraph, and if at all feasible the quantity equation from which it was obtained should be given.

*Notes:*

- Two other examples of numerical-value equations written in the preferred form are as follows, where E
_{g}is the gap energy of a compound semiconductor and*k*is the conductivity of an electrolytic solution:

E_{g}/eV = 1.425 - 1.337*x*+ 0.270*x*^{2}, 0 ≤*x*≤ 0.15, where*x*is an appropriately defined amount-of-substance fraction (see Sec. 8.6.2).*k*/(S / cm) = 0.065 135 + 1.7140 × 10^{-3}(t / °C) + 6.4141 × 10^{-6}(t / °C)^{2}- 4.5028 × 10^{-8}(t / °C)^{3}, 0 °C ≤ t ≤ 50 °C, where*t*is Celsius temperature. - Writing numerical-value equations for quantities expressed in inch-pound units in the preferred form will simplify their conversion to numerical-value equations for the quantities expressed in SI units.

**7.12 Proper names of quotient quantities **

Derived quantities formed from other quantities by division are written using the words "divided by" or per rather than the words "per unit" in order to avoid the appearance of associating a particular unit with the derived quantity.

*Example:* pressure is force divided by area or pressure is force per area *but not:* pressure is force per unit area

**7.13 Distinction between an object and its attribute **

To avoid confusion, when discussing quantities or reporting their values, one should distinguish between a phenomenon, body, or substance, and an attribute ascribed to it. For example, one should recognize the difference between a body and its mass, a surface and its area, a capacitor and its capacitance, and a coil and its inductance. This means that although it is acceptable to say "an object of mass 1 kg was attached to a string to form a pendulum," it is not acceptable to say "a mass of 1 kg was attached to a string to form a pendulum."

Any SI derived quantity *Q* can be expressed in terms of the SI base quantities length (*l* ) , mass (*m*), time (*t*), electric current (*l* ) , thermodynamic temperature (*T* ) , amount of substance (*n*), and luminous intensity (*I*_{v} ) by an equation of the form

where the exponents α, β, γ, . . . are numbers and the factors a_{k} are also numbers. The dimension of *Q* is defined to be

dim *Q* = L^{α} M^{β} T^{γ} I^{δ} Ө^{ε} N^{ζ} J^{η},

where L, M, T, I, θ, N, and J are the *dimensions* of the SI base quantities length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The exponents α, β, γ, . . . are called "dimensional exponents." The SI derived unit of Q is m^{α}·kg^{β} s^{γ} ·A^{δ}·K^{ε} mol^{ζ} cd^{η}, which is obtained by replacing the dimensions of the SI base quantities in the dimension of Q with the symbols for the corresponding base units.

*Example:* Consider a nonrelativistic particle of mass *m* in uniform motion which travels a distance *l* in a time *t* . Its velocity is *ν* = *l* / *t* and its kinetic energy is E_{k} = *mν*^{2} / 2 = *l* ^{2}*mt*.^{-2} / 2. The dimension of *E*_{k} is dim *E*_{k} = L^{2}MT^{-2} and the dimensional exponents are 2, 1, and -2. The SI derived unit of *E*_{k} is then m^{2}·kg·s^{-2}, which is given the special name "joule" and special symbol J.

A derived quantity of dimension one, which is sometimes called a "dimensionless quantity," is one for which all of the dimensional exponents are zero: dim *Q* = 1. It therefore follows that the derived unit for such a quantity is also the number one, symbol 1, which is sometimes called a "dimensionless derived unit."

*Example:* The mass fraction *w*_{B} of a substance B in a mixture is given by *w*_{B} = *m*_{B} / m, where *m*_{B} is the mass of B and *m* is the mass of the mixture (see Sec. 8.6.10). The dimension of *w*_{B} is dim *w*_{B} = M^{1}M^{-1} = 1; all of the dimensional exponents of wB are zero, and its derived unit is kg^{1}·kg^{-1} = 1 also.

Created January 28, 2016, Updated June 2, 2021