15. Mathematical expressions

15.1 Letter symbols and units

Letter symbols defined in applicable IEEE standards (see Clause 2) should be used in preparing mathematical expressions. (See 14.4 for a discussion of letter symbols.)

All terms shall be defined, including both quantities and units, in a tabulation following the equation [see Equation (1)]. The list should be preceded by the word where, followed by the list of variables and corresponding definitions. See 4.5 in Annex B for an example.

15.2 Numbering of equations

If the standard contains more than one equation, then equations of key importance should be numbered consecutively in parentheses at the right margin. Derivations of equations or examples where values are substituted for variables need not be numbered.

An equation should be cited in the text by the word Equation and its number only [e.g., “see Equation (1)”]. If referring to two or more equations in the same sentence, each should be named separately. For example, use “see Equation (1), Equation (2), and Equation (3),” instead of “see Equations (1) through (3).”

Equations in annexes should be numbered beginning with the letter of the annex where they are found. For example, the first equation in Annex B would be numbered “(B.1)” and the reference to it would be to “see Equation (B.1).”

15.3 Presentation of equations

Certain types of material in displayed equations are automatically italic. Some simple general rules apply. All variables are italic. (e.g., x, y, n). Function names and abbreviations are Roman (sin, cos, sinc, sinh), as are units or unit abbreviations (e.g., deg, Hz), complete words (e.g., in, out), and abbreviations of words (e.g., max, min), or acronyms (e.g., SNR). Single letter superscripts and subscripts may be italic even if they are abbreviations, unless this leads to inconsistency between italic and roman characters for similar types of subscripts.

A multiplication sign (×), not the letter “x” or a multidot (∙), should be used to indicate multiplication of numbers and numerical values, including those values with units (e.g., 3 cm × 4 cm).

Although the stacked style of fractions is preferred, exceptions should be made in text to avoid printing more than two lines of type. For example, in text a/b is preferable to \(\frac{a}{b}\).

The general rules regarding the use of upright (Roman) and italic text in equations [see Equation (1)] are as follows:

• Quantity symbols (including the symbols for physical constants), subscripts or superscripts representing symbols for quantities, mathematical variables, and indexes are set in italic text.
• Unit symbols, mathematical constants, mathematical functions, abbreviations, and numerals are set in upright (Roman) text.

Example:

where

\(E_{0} \quad\) is the dc value of the waveform

\(N \quad\) is the number of sample data in one period

\(E_{j} \quad\) is the \(i^{\text{th}}\) sample data of the waveform

Table 1 lists a number of functions and operators that are set in upright (Roman) text.

Additional examples of the presentation of equations is provided in Annex B.

15.4 Quantity and numerical value equations

Equations shall be dimensionally correct. Equations may be in either quantity equation form or in numerical value equation form. Stipulation of units for substituted values in the variable list below the equation does not suffice to meet this requirement.

A quantity equation is valid regardless of the units used with the substituted values, once any unit conversions and prefix scaling factors have been taken into account. For example, F = ma is always correct.

A numerical value equation depends on the use of particular units and prefixes. Such equations may be presented in one of two forms. One form represents a numerical relationship among quantities whose dimensions have been reduced to 1 due to division by the appropriate (prefixed) units. For example,

$t/°C=T/\mathrm{K}-273.15$

The other form annotates the quantities with the units to be used. For example,

${\left\{t\right\}}_{°\mathrm{C}}={\left\{T\right\}}_{\mathrm{K}}-273.15$