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CHAPTER 11
Numbers, Mathematics, and Units of Measure
Numbers Both numerals and words can be used to express numbers. The usage and style conventions for numerals and words are different for technical and nontechnical material.
Numeral and Word Usage ➤ Use numerals with units of time or measure, and use a space between the numeral and the unit, except %, $, ° (angular degrees), ′ (angular minutes), and ″ (angular seconds). 6 min 0.30 g 273 K 90 °F
25 mL 125 V/s 50% $250 47°8′23″ 180° (but 180 °C) 50 µg of compound/dL of water
exception Spell out numbers with units of measure used in a nontechnical sense. If you take five minutes to read this article, you’ll be surprised.
➤ With items other than units of time or measure, use words for cardinal
numbers less than 10; use numerals for 10 and above. Spell out ordinals “first” through “ninth”; use numerals for 10th or greater. three flasks
30 flasks
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third flask seven trees eighth example first century sixfold
12th flask 10 trees 33rd example 21st century 20-fold
exception 1 Use all numerals in a series or range containing numbers 10 or greater, even in nontechnical text. 5, 8, and 12 experiments 2nd and 20th samples 5–15 repetitions
exception 2 Use all numerals for numbers modifying nouns in parallel construction in the same sentence if one of the numbers is 10 or greater. Activity was reduced in 2 pairs, not significantly changed in 11 pairs, and increased in 6 pairs. We present new results pertaining to 12 phenanthrolines and 3 porphyrins.
exception 3 For very large numbers used in a nontechnical sense, use a combination of numerals and words. 1 billion tons 180 million people 2 million pounds (not lb) 4.5 billion years $15 million (not 15 million dollars)
➤ When a sentence starts with a specific quantity, spell out the number as well
as the unit of measure. Twelve species were evaluated in this study. Twenty slides of each blood sample were prepared. Fifteen milliliters of supernate was added to the reaction vessel. Twenty-five milliliters of acetone was added, and the mixture was centrifuged.
However, if possible, recast the sentence. Acetone (25 mL) was added, and the mixture was centrifuged. A 25 mL portion of acetone was added, and the mixture was centrifuged.
➤ Even when a sentence starts with a spelled-out quantity, use numerals when
appropriate in the rest of the sentence. Twenty-five milliliters of acetone and 5 mL of HCl were added. Three micrograms of sample was dissolved in 20 mL of acid. Fifty samples were collected, but only 22 were tested.
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➤ Use numerals for expressions used in a mathematical sense. The incidence of disease increased by a factor of 4. The yield of product was decreased by 6 orders of magnitude. The efficiency of the reaction was increased 2-fold. After 2 half-lives, the daughter product could be measured. The control group had 3 times the risk for colon cancer. The values are determined with 5 degrees of freedom.
➤ When the suffix “fold” is used in a nonmathematical sense, spell out the
accompanying number if it is less than 10. The purpose of this discussion is twofold.
➤ When the word “times” is used in a nonmathematical sense, spell out the
accompanying number if it is less than 10. The beaker was rinsed four times.
➤ Use numerals in ratios. a ratio of 1:10 a ratio of 1/10 a 1:1 (v/v) mixture a 1/1 (v/v) mixture
➤ In dates, use numerals without ordinal endings. January 3, Jan 3 (not January 3rd, Jan 3rd) September 5, Sept 5 (not September 5th, Sept 5th)
➤ Use numerals for decades, and form their plurals by adding an “s”. Do not use
apostrophes in any position. the 1960s (not the 1960’s, not the ’60s) values in the 90s (not the 90’s) She is in her 20s. (not her 20’s)
➤ Use numerals with a.m. and p.m. 12:15 a.m.
4:00 p.m.
➤ Spell out and hyphenate fractions whose terms are both less than 10. If one
of the terms is 10 or greater, use a piece fraction. one-quarter of the experiments two-thirds of the results
¹⁄₂₀ of the subjects ¹⁄₁₂ of the volume
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➤ Use numerals to label figures, tables, schemes, structures, charts, equations,
and references. Number sequentially; do not skip numbers or number out of sequence. Use arabic numerals for references, but for the other items, the use of arabic and roman numerals varies among ACS publications. Consult a recent issue or author instructions to determine what system is preferred. ➤ In journal articles and book chapters, instead of repeating chemical names
over and over, use numerals in boldface (not italic) type to identify chemical species. Use these identifiers only in text, not in article or chapter titles, and number consecutively. This paper describes the syntheses, structures, and stereodynamic behavior of the novel hexacoordinate silicon complexes 1–4. The cyclization of 1,3,5-hexatriene (6) to 1,3-cyclohexadiene (7) is predicted to proceed more rapidly in an electrostatic field. Complexes 8–12, in the presence of monoamine oxidase, produce active catalysts for propylene polymerization. Primary amines 2–5, 7, and 9 gave the same Cotton effect signs, depending on the configuration. Monomer III reacts with the initiator (I, Ar = 2,6-diisopropylphenyl) via a ringopening metathesis polymerization mechanism.
➤ Numerals may be used to name members of a series. Sample 1 contained a high level of contamination, but samples 2 and 3 were relatively pure. Methods 1 and 2 were used for water-soluble compounds, and methods 3 and 4 were used for oil-soluble compounds.
➤ When numerals are used as names and not enumerators, form their plurals
by adding an apostrophe and “s” to avoid confusion with mathematical expressions and to make it clear that the “s” is not part of the name. The athlete received five 9’s from the judges. Boeing 747’s are among the largest airplanes.
➤ Arabic numerals in parentheses may be used to enumerate a list of phrases or
sentences in text. Always use an opening and a closing parenthesis, not one alone. Some advantages of these materials are (1) their electrical properties after pyrolysis, (2) their ability to be modified chemically before pyrolysis, and (3) their abundance and low cost. The major conclusions are the following: (1) We have further validated the utility of molecular mechanical methods in simulating the kinetics of these reactions.
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(2) A comparison of the calculated structures with available X-ray structures revealed satisfactory agreement. (3) The combined use of different theoretical approaches permitted characterization of the properties of a new isomer.
➤ Arabic numerals followed by periods or enclosed in parentheses may be used
to enumerate a displayed list of sentences or to number paragraphs. Here are two acceptable ways to format a list. These results suggest the following: 1. Ketones are more acidic than esters. 2. Cyclic carboxylic acids are more acidic than their acyclic analogues. 3. Alkylation of the active methylene carbon reduces the acidity. These results suggest the following: (1) Ketones are more acidic than esters. (2) Cyclic carboxylic acids are more acidic than their acyclic analogues. (3) Alkylation of the active methylene carbon reduces the acidity.
Style for Numbers ➤ For very large numbers with units of measure, use scientific notation or
choose an appropriate multiplying prefix for the unit to avoid numbers of more than four digits. 1.2 × 106 s 3.0 × 104 kg 5.8 × 10–5 M or 58 µM 42.3 L (not 42,300 mL or 42 300 mL)
exception 1 In tables, use the same unit and multiplying prefix for all entries in a column, even if some entries therefore require four or more digits. exception 2 Use the preferred unit of a discipline, even when the numbers require four or more digits: g/L kg/m3 GPa kPa MPa
for mass density of fluids for mass density of solids for modulus of elasticity for fluid pressure for stress
➤ In four-digit numbers, use no commas or spaces.
exception Spaces or commas are inserted in four-digit numbers when alignment is needed in a column containing numbers of five or more digits. ➤ When a long number cannot be written in scientific notation, the digits must
be grouped. For grouping of digits in long numbers (five digits or greater), check the publication in which the manuscript will appear. Two styles are possible.
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style 1 In some publications, for numbers with five or more digits, the digits are grouped with commas placed between groups of three counting to the left of the decimal point. 4837 10,000 930,582 6,398,210 85,798.62578
style 2 In some publications (including ACS journals), for numbers with five or more digits, the digits are grouped with a thin space between groups of three, counting both to the left and to the right of the decimal point. 9319.4 74 183.0629 0.508 27 501 736.293 810 4
exceptions • U.S. monetary values are always written with commas: $5,000. • U.S. patent numbers are always written with commas: U.S. Patent 6,555,655. The patent numbers of other countries should be presented as on the original patent document. • Page numbers in reference citations are always printed solid: p 11597. ➤ Use the period as the decimal point, never a comma. ➤ Use numerals before and after a decimal point. 0.25 (not .25) 78.0 or 78 (not 78.)
➤ Use a decimal and a zero following a numeral only when such usage truly repre-
sents the precision of the measurement: 27.0 °C and 27 °C are not interchangeable. ➤ Use decimals rather than fractions with units of time or measure, except
when doing so would imply an unwarranted accuracy. 3.5 h (not 3½ h) 5.25 g (not 5¼ g)
➤ Standard deviation, standard error, or degree of accuracy can be given in two
ways: • with only the deviation in the least significant digit(s) placed in parentheses following the main number and closed up to it or • with all digits preceded by a ± and following the main number. Spaces are left on each side of the ±.
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2.0089(1) means 2.0089 ± 0.0001 1.4793(23) means 1.4793 ± 0.0023
The shorter version is better in tables. Always specify which measure (e.g., standard deviation or standard error) of uncertainty is being used. ➤ When two numbered items are cited in narrative, use “and”. Figures 1 and 2 refs 23 and 24 compounds I and II
➤ Use a comma between two reference callouts in parentheses or as superscripts. Lewis (12, 13) found Lewis12,13 found
When the reference numbers are on the line, the comma is followed by a space; when the numbers are superscripts, the comma is not followed by a space. ➤ Use an en dash in ranges or series of three or more numbered items, whether
on the line or in a superscript. 43–49 325–372 2005–2008 Tables 1–4 temperatures of 100–125 °C refs 3–5
aliquots of 50–100 mL eqs 6–9 samples 5–10 past results (27–31) past results27–31 pp 165–172
exception 1 Do not use an en dash in expressions with the words “from … to” or “between … and”. from 20 to 80 (not from 20–80) between 50 and 100 mL (not between 50–100 mL)
exception 2 When either one or both numbers are negative or include a symbol that modifies the number, use the word “to” or “through”, not the en dash. –20 to +120 K –145 to –30 °C ≈50 to 60 10 to >600 mL 353 M(x+y)+
exception 2 Leave no space around a slash (a/b), a ratio colon (1:10), or a centered dot (PM·V). ➤ Leave no space between simple variables being multiplied: xy. Do not use a
centered dot (·) or the times sign (×) with single-letter scalar variables. ➤ In multiplication involving the two-letter symbols for transport properties,
use a space, enclose them in parentheses, or use the times sign. When superscripts or subscripts are present, the symbols can be closed up. Re Nu (Re)(Nu) Re × Nu RexNuy
➤ Use a space for simple multiplication of functions of the type f(x) (one-dimen-
sional) or g(y, z) (multidimensional). Close up multipliers to such functions where applicable. You may also use additional enclosing marks instead of spaces. W = 2f(x) g(y, z) W = 2[f(x)][g(y, z)]
➤ When mathematical symbols are used as adjectives, that is, with one number
that is not part of a mathematical operation, do not leave a space between the symbol and the number. a conversion of >50% –12 °C a probability of ,
