# Scientific and Mathematical Notation

Mathematical Material

In a series, units of measurement are usually omitted for all but the last number, except for percent and degree signs: 22, 30 and 35 cm; 20%, 30% and 42%; 68° to 70°.

Italic and Roman

Mathematical symbols (variable quantities) are set in italic characters (A, B, C, a, b, c, x, y, z). Chemical symbols and letters indicating shape are set in roman characters (O2, CaCl2, T network, S-shaped). Roman is also used for abbreviations (lb, pH, KE, Re, sin, cos) and Greek letters.

Examples:

 log k = log kH++ log [H2O] (1) KE = 1/2 mV2 (2) Wrev= h1 - h2= 100 Btu/lb (3) β= µ G2 = µ(dU/dy)2,/sup> (4) sin2 β + cos2 β = 1 (5)

Use Roman for Constant Values

Constants in a mathematical equation are set in roman.

Example:
y
= mx + c
m and c are constants.

Subscript Notation

Words, chemical formulas, numerals, and abbreviations in subcripts are set in roman characters. Italics are reserved for mathematical symbols (variable quantities) even though they are, in fact, a kind of abbreviation.

Examples:
Vflow, Emax, Foct, Na, Sco, Savg, Pref, H1, H23, Vbc

When a symbol carries two subscripts, a comma without a space is generally used between them (R1,max). Follow author’s text on use or omission of a comma in subscripts such as c33, n12, n1,2, nij.

Negative Values

Use en dashes to represent minus signs.

Subscript and Superscript

In keyboarding, the subscript usually precedes the superscript (D min2) as this arrangement is relatively simple to set. The alignment of exponents and subscripts may involve a separate operation.

Numbering Equations

For reference purposes some or all equations may be numbered. The numbers appear in parentheses at the right margin.

Examples:
xsin x = 1 (1)

If an equation occupies more than one line, the equation number is aligned with the last line.
0 a)2x-1 = pβa[(µ’ - µ’’ - 1 + δ2)tan2 π+ (δ’ - δ’’)]
x (µ’ -µ’’ - 1 + t2)1/2 (2)

When two or more equations are designated by the same number the equation number can be placed on the right hand margin below the whole group or, if the equations are short, the number may be placed midway in the margin opposite the equations.

x + y = a
(3)
x & y = a

Distinguishing between Zero (0) and Oh (o)

Subscript letter o (as in po) and subscript number zero (p0) are differentiated typographically. Ask author if in doubt on whether the character is an o or a 0. In chemical formulas, the letter is generally an o, O (as in H2O).

The Exponential

The term exp indicates "exponent." It is a symbol or number placed above and after another symbol or number to denote the power to which the latter is to be raised. It is always in Roman type.

Example:
5n+1=5 exp(n+1)

The exponential e (a constant equal to 2.7182818...) can be replaced by exp when the expression of the power is long or complicated, and the exponent set on the line if it includes radical signs (e.g., square root √) or summations signs (Ε t0) with limits, or other special symbols not readily available.

Examples:
u
= πe - δ dx can be replaced with π exp(- δdx)
y = c0e√ (a + b)/kt can be replaced with y = c0exp[(a + b)/kt]1/2

Breaking Equations

An expression short enough to be embedded in the text, e.g., sin (x + y) or a should not be broken at the end of a line. If the expression is so long that placing it on the second line makes the preceding line too short, the expression should be centered on a line by itself.

If an expression is so long that it must be broken, the break may come at an equal sign, at an operation sign
(+, -,×÷), or after a parenthesis, bracket, or brace, i.e.,),],}.

Defining Lists Following Equations

Lists in which the symbols and units of measure used in an equation are defined are set up as shown below whenever the material is adaptable to such an arrangement.

Example:
NRT = 60 W/S (1)

where
NRT = nip width residence time, s
W = nip width, mm
S = machine speed, m/min

Words in Equations

The first word of the left hand member of an equation is capitalized in the numerator and denominator of a fraction; other words are set in lower case.

Example:
Downward force/Upward force = net effect
Abbreviations and units of measure that are normally lower case remain lower case.
rpm = 120 H cps/P

Referring to Equations

Equations in the text are referred to by number and the word Equation, Eqs., or Eq. Thus:

Examples:
Beginning of a sentence - Equation (9) shows ...
Within a sentence - ....as shown by Eq. (9)...

CHEMICAL FORMULAS AND EQUATIONS

Chemical symbols are always roman, except in the column headings of tables

Treatment with SO2Cl, Treatment with SO2Cl2

For reference purposes some or all chemical equations appearing in a text may be numbered. The numbers appear in parentheses at the right margin.

Arrows are commonly used in chemical equations:
CaCl2 + H2CO3 ↔ CaCO3 + 2HCl (1)
NaOCl + H2O + 2e ↔ NaCl + 2OH­ (2)
Mg(OH)2 + 2H2SO3 - Mg(HSO32 + 2H2O (3)

If equal signs are used in chemical equations, these should not be changed to arrows without the author’s approval.

Arrows with single barbs are used for reversible or equilibrium reactions:
2CO + 2H2 CO2CH4 (1)
Double headed arrows ( ) signify resonance.

Valences are shown as follows:
Cl­ Cu++ SO4­­ N Sn4+

Chemical Kinetics

Variable mathematical symbols are set in italics while chemical formulas and symbols are in roman.
Khydr + {[H+][Cl­] [HOCl]}/[Cl2] (1)
[] denote concentration.

METRIC AND SI UNITS

The United States and other nations have agreed to use the metric system of measurement, also known as SI units (for Systçme International d’Unités). Scientific work published in TAPPI JOURNALshould use SI units. Because other units of measurement are still commonly used, those units may be used for clarity and accuracy, but metric equivalents should be included in parentheses.

There are several Internet sites that provide helpful information on conversions and proper use of SI units. Those include:

The U.S. Metric Association -http://lamar.colostate.edu/~hillger

The National Institute of Standards and Technology (NIST), Metric Program --
http://ts.nist.gov/ts/200/202/mp_home.htm
http://ts.nist.gov/ts/htdocs/200/202/mpo_pubs.htm

The NIST has also published a comprehensive "Guide for the Use of the International System of Units (SI)," NIST Special Publication 811, 1995 Edition, by Barry N. Taylor. It is available from the U.S. Government Printing Office or online: http://physics.nist.gov/Pubs/SP811/sp811sl.pdf