AMWA Journal / V36 N3 / 2021 / amwa.org 121 To be a good medical writer, you need to know something about mathematics. Mathematics is the art of number, and numbers originated from words that were coined for the pur- pose of counting. However, some things can be counted, and some things cannot. Count and Noncount Nouns Discrete (as opposed to discreet!) means separate and distinct from other things. Objects that are discrete can be counted. For example, you might count the number of apples in a basket, but you can never count the number of gasoline in a tank. For this reason, apple is a count noun, but gasoline is a noncount noun. So you can ask, “How many apples are in the basket?” but it would be ungrammatical to ask, “How many gasolines are left in the tank?” Instead, you might ask, “How much gaso- line is left in the tank?” Many is used with count nouns; much is used with non- count nouns. Some quantifiers (eg, all, any, enough, most, plenty of, some, and no) can be used with count or noncount nouns. However, there are some quantifiers that are used only with count nouns (eg, every, many, a few) and others that are used only with noncount nouns (eg, much, less, a little). If we want to talk about more than 1 of something in English, we use the plural form of the noun. The plural form is usually made by adding s or es to the end of the noun. However, there are many exceptions (see Table on next page). These include some words of Anglo-Saxon origin, such as child/children, or woman/women, ox/oxen, goose/geese. Note that many of the animal nouns that came from Anglo-Saxon are the same in singular and plural: fish, sheep, moose. Many words of Greek or Latin origin that are important in medicine have irregular plural forms: bacterium/bacteria, corpus/ corpora, genus/genera, medium/media, species/species, stigma/stigmata. For some nouns with irregular plurals and some non- count nouns, a regular plural form has become commonplace or is used in specific circumstances. For example, water is a noncount noun. However, the word waters is used to refer to a watery geographical area (eg, the navigable waters of the United States) or in some poetic contexts. Amniotic fluid, which surrounds the fetus in the womb, is also sometimes called waters. Collective Nouns A collective noun is a noun that refers to a group (set) of per- sons or things. For example, a swarm refers to a group of insects, and a choir refers to a group of singers. This raises problems of agreement with pronouns and verbs. Should you refer to the collective as “it” or “them”? Should you use the singular or plural form of the verb to refer to the collective’s actions? The answer depends on whether the individual mem- bers or the group is being emphasized: • The emergency department staff are trained in the latest resuscitation techniques (emphasizing individuals). • The hospital’s emergency department staff is the best in the city (emphasizing the group). Note that British people are more likely than Americans to use plural pronouns and verbs for collectives, such as businesses: • Bloomingdale’s is having a sale on swimsuits (United States). • Fenwick are having a sale on swimming costumes (Britain). Units of Measure Many things that cannot be counted can nevertheless be mea- sured. To measure them, we need to find some unit of mea- sure. For example, we can say “1 liter of water” or “2 bushels of wheat.” Please notice the grammatical structure: the number Counting and Measuring Laurie Endicott Thomas, MA, ELS / Madison, NJ 122 AMWA Journal / V36 N3 / 2021 / amwa.org Singular Plural addendum addenda, also addendums aircraft aircraft alumna alumnae alumnus alumni analysis analyses antenna antennae, also antennas antithesis antitheses apex apices, also apexes appendix appendices, also appendixes axis axes bacillus bacilli bacterium bacteria basis bases beau beaux, also beaus bison bison bureau bureaus, also bureau cactus cacti, also cactus or cactuses château châteaux, also châteaus child children codex codices, also codexes concerto concerti, also concertos corpus corpora crisis crises criterion criteria, also criterions curriculum curricula, also curriculums datum data deer deer diagnosis diagnoses die dice, also dies dwarf dwarves, also dwarfs ellipsis ellipses erratum errata faux pas faux pas fez fezzes, also fezes fish fish, also fishesa focus foci, also focuses foot feet, sometimes foot formula formulae, also formulas fungus fungi, also funguses genus genera, also genuses goose geese graffito graffiti grouse grouse, also grouses half halves hoof hooves, also hoofs hypothesis hypotheses index indices, also indexes lacuna lacunae larva larvae leaf leaves libretto libretti, also librettos loaf loaves locus loci louse lice man men matrix matrices, also matrixes medium media, also mediums memorandum memoranda, also memorandums minutia minutiae moose moose mouse mice nebula nebulae, also nebulas nucleus nuclei oasis oases octopus octopuses or octopodes offspring offspring opus opera ovum ova ox oxen, also ox parenthesis parentheses phenomenon phenomena phylum phyla quiz quizzes radius radii referendum referenda, also referendums salmon salmon scarf scarves schema schemata, also schemas self selves series series sheep sheep shrimp shrimp, also shrimpsa species species stigma stigmata stimulus stimuli stratum strata swine swine syllabus syllabi, also syllabuses symposium symposia, also symposiums synopsis synopses tableau tableaux, also tableaus thesis theses thief thieves tooth teeth trout trout, also troutsa tuna tuna, also tunasa vertebra vertebrae, also vertebras vertex vertices, also vertexes vita vitae vortex vortices, also vortexes wharf wharves, also wharfs wife wives wolf wolves woman women Singular Plural Table. Irregular English Plurals aThe former is typically used to refer to more than 1 individual of the same species, and the latter is typically used to refer to more than 1 species. AMWA Journal / V36 N3 / 2021 / amwa.org 123 is an adjective modifying the unit of measure. The unit of mea- sure is a noun. The material being measured is now the object of a prepositional phrase (“of water”). If you are talking about some noncount noun that is being measured in this way, it will be treated as if it were a singular: 20 kilometers is (not are) a long walk. A unit of measure is arbitrarily defined; thus, the number associated with a measurement is meaningless unless the unit of measure has been defined, so you must carefully specify units of measure. Units of measure typically relate to some nat- ural phenomenon. For example, the inch was originally based on the width of a man’s thumb, and the foot was based on the length of a man’s foot. The main basis of the International System of Units (SI, for Système International [d’unités]) is the meter. The SI grew out of the metric system that was developed in Revolutionary France. The meter was supposedly based on 1/10,000 of the distance from the North Pole to the Equator. Other units of measure in the SI were derived from the meter. A centimeter is 1/100th of a meter. A liter is 1,000 cubic cen- timeters, and a kilogram originally represented the mass of a liter of water. A newton is the amount of force to make a 1-kg object accelerate 1 meter per second per second. Thus, mea- surements of force involve units of time as well as units of mass and distance. The SI also includes many units that are important in phys- ics and chemistry. A coulomb (C) is a measure of electrical charge, and an ampere (A) is a measure of electrical current. A candela (cd) is a measure of luminous intensity. A mole (mol) is a measure of the amount of a substance. A mole is defined as 6.02214076 × 1023 particles (eg, atoms or molecules). The number of particles in a mole is called Avogadro’s number. Our measurements of time were originally derived from the duration of a day. Each day is divided into 24 hours, and each hour into 60 minutes, each minute into 60 seconds. The 60-minute hour and 60-second minute are legacies of the ancient Mesopotamians, who used the number 60 as the basis of their number system. (The number 60 is the small- est number that can be divided evenly by every whole number from 1 to 6.) The ancients also divided a circle into 360°—partly because 360 can be evenly divided by so many different num- bers and partly because 360 is close to the number of days in the year. (The lunar calendar has 355 days, and the solar cal- endar has 365 days). Thus, the sun would advance roughly 1° along the ecliptic (its circular path relative to the background of stars) every day. Whole and Real Numbers When we count objects, the result will be an integer. But when we measure the amount of something, as opposed to count- ing the number of items, the result would theoretically be a real number, along with a unit of measure. A real number is a number that can be expressed as some point along a number line. To express a measurement, however, we will end up using a rational number, as we will report only a limited number of digits after the decimal point. A rational number is one that can be expressed as a quotient or fraction of 2 integers. Its decimal expansion, if it has one, will either terminate or end up repeating itself endlessly. For example, ¾ = 0.75 and ¹/³ = 0.33333…. (sometimes written 0.3, with the overbar repre- senting the repeating decimal expansion). In contrast, irratio- nal numbers have a decimal expansion that continues forever without repeating. Examples include p (the ratio of the circum- ference of a circle to its diameter), e (Euler’s number, which is useful for calculating compound interest), and the square root of 2. Using different units (eg, miles vs kilometers) will yield a different number, so you have to include the units with the number. Which unit of measure should you use? In scientific writing, you should use the SI (meters, kilometers, kilograms, etc). But if you are writing for consumers in the United States, you should probably use the units that are familiar to consum- ers (feet and inches, miles, pounds and ounces, etc). Accuracy, Precision, and Uncertainty Accuracy refers to how well a measurement agrees with the truth. In contrast, precision refers to the agreement among repeated measurements (made under the same conditions). Thus, a measurement that is accurate may be imprecise, and a measurement that is precise may be inaccurate. When choos- ing between methods of measurement, you often have to make a tradeoff between precision and accuracy. For example, a digi- tal clock displays a precise, rational number, but that reading does not represent the true time. In contrast, an analog clock expresses time as a real number that cannot be read precisely. Both inaccuracy and imprecision contribute to uncer- tainty. All measurements, and all quantities calculated from measurements, will have some degree of uncertainty. The degree of uncertainty can be expressed in various ways. One is by showing only a limited number of significant digits. For example, a reported value of 3.5 implies that the actual value is probably somewhere between 3.45 and 3.55. In contrast, a reported value of 3.50 implies that the actual value is prob- ably somewhere between 3.495 and 3.505—a much narrower range. You can also express the uncertainty in units of measure or as a percentage of the total value: 25.2 mL ± 0.05 mL can be expressed as 2.52 mL ± 0.2%. 124 AMWA Journal / V36 N3 / 2021 / amwa.org Even when we are dealing with counts, such as the number of people who live in a city, we sometimes have to deal with uncertainty. As a result, we may have to settle for an approxi- mate number, such as when we say that the population of New York City was 8.40 million in 2018. Nor should we report too many digits after a decimal point: we shouldn’t report a value as 5.38761 when the precision of the value really only lets us say 5.4. Fractions and Percentages A fraction is made by division. The top number (numerator) is divided by the bottom number (denominator). A percentage is a fraction whose denominator is 100. Whenever you encoun- ter a percentage or any other fraction, try to figure out what the numerator and denominator represent. For example, the forced expiratory volume in 1 second (FEV1) is the amount of air that a patient can exhale in 1 second and is measured in liters. This value can then be divided by the full forced vital capacity (FVC), which is the total amount of air that the person can exhale after taking the biggest possible breath, to yield the Tiffeneau-Pinelli index (FEV1/FVC), which is a unitless rational number. The FEV1 and FVC can also be expressed as a percent- age of the values that are predicted, given the patient’s sex, age, height, and race. When talking about values that are already expressed in percentages, be cautious about using percentages to express changes. For example, if a value increased from 10% to 20%, that’s an increase of 10 percentage points, not 10% (it’s a 100% increase; see Percentage Increase and Decrease). Negative Numbers and Vectors Addition is the arithmetic operation that originally represented adding objects to a collection. Its opposite is subtraction, which originally represented the removal of objects from a col- lection. If you have 5 apples in a basket, you cannot remove more than 5 apples from that basket. But if you have $100 in your checking account and write a check for $200, you will end up with a balance of −$100 in your account. You would have to deposit $100 in the account to bring the balance up to 0. Accountants sometimes use parentheses instead of a minus sign to indicate negative numbers. Addition and subtraction are often represented by right- ward or leftward movement, respectively, on a number line. Thus, addition and subtraction involve not just quantity but direction. In mathematics, a geometrical object that has a direction as well as a magnitude is called a vector. A line is one-dimensional, so there are only 2 directions. In contrast, a map is two-dimensional, which allows for an infinite number of directions. If I walk 1 block north, then 1 block west, then 1 block south, then 1 block east, I will have walked a distance of 4 blocks, but I will end up back where I started. Human beings can easily think in terms of 4 dimensions: the 3 dimensions of Euclidean geometry plus time. However, mathematicians often deal with problems that involve more than 4 dimensions. This allows them to develop a mathematical model of relationships among many variables at the same time. Exponents and Logarithms Exponentiation is when you multiply a base number (b) by itself n number of times (b n). For example, 23 = 2 × 2 × 2 = 8. The n is called an exponent, and we often say that b has been raised to the nth power. If n = 2, we say that the base is squared. If n = 3, we say that the base is cubed. Medical communica- tors often deal with powers of 10: 10 = 101, 100 = 102, 1,000= 103, 10,000 = 104, and so on. However, any real number could serve as the base or the exponent. The use of exponents can turn a multiplication problem into an addition problem: if 2 exponential expressions have the same base, you can multiply the 2 by adding their exponents: 102 × 103 = 105. To divide, you subtract the exponents: 105 ÷ 103 = 102. Any nonzero number divided by itself is 1; therefore, b 0 = 1. You can also have negative exponents: b -n = 1/b n. You can also raise neg- ative numbers to any power. Note, however, that if you raise a negative number to an even power (eg, −1 × −1), the product will be a positive number. If you raise it to an odd power, the result will be a negative number (eg, −1 × −1 × −1 = −1). In medical writing, you will often see powers of 10, espe- cially in scientific notation: 5.23 × 105 = 523,000, but 5.23 × 10−5 = 0.0000523. Sometimes E notation is used to express powers of 10. 5.23E5 means 5.23 × 105, and 5.23E−5 means 5.23 × 10−5. A logarithm is the inverse function of exponentiation. If b n = x, then n = logb(x). Because 23 = 8, log2(8) = 3. Because exponents can be negative, you can have negative loga- rithms, which represent the inverse of a number. Because ½ is the inverse of 2, the log2(½) = −1. Likewise, the log10(¹⁄10) = −1 (Figure on next page). Some units of measure are based on a logarithmic scale. For example, pH is based on the negative of the base-10 logarithm of the activity of the H+ ion (as measured in moles per liter). pH= –log10 (aH +) = log10 A solution of pure water has hydrogen activity of 1 × 10−7 mol/L. The reciprocal of that is 1 ×107, or 107; log10(107) = 7. So the pH of pure water is 7. Water with a pH of 6 would have a hydrogen activity of 1 ×10−6 mol/L, which is 10 times as many hydrogen ions as in pure water! 1 aH +( ) AMWA Journal / V36 N3 / 2021 / amwa.org 125 Base-10 logarithms are used so often that they are often just written as log(x). The natural logarithm, abbreviated ln(x), has Euler’s number (e) as its base. Euler’s number is an irrational number that is useful in many different areas of mathematics. Medical writers should be aware that viral load is often expressed in base-10 logarithms. I once edited a news article that described a patient as having a viral load of 5 copies/mL. That value was dubious: a value that low had to be below the limit of detection of any available assay. When I looked at the source material, I found out that the patient’s reported viral load was actually 5 log10 copies/mL, which meant 100,000 copies/mL. Big difference! Stevens’ Taxonomy of Measurement Medical writers must be aware that numbers do not always represent counts, or some point along a number line, or a vector. To explain this problem, Stanley Smith Stevens explained that there are 4 types of measurement scale.1 • Nominal— A nominal scale is used when items or individu- als are being assigned to groups that do not overlap. Such groups may be labeled with numbers: group 1, group 2, group 3. However, these numbers are simply being used as labels and do not express any sort of quantity. • Ordinal— An ordinal scale is used when items or individuals are being ranked according to how they compare with each other in terms of some property. For example, the runners in a race will be ranked first, second, third, and so on, accord- ing to how fast they ran. However, these ordinal numbers merely show rank. They do not use units of measure, and they do not show absolute quantities or ratios. For exam- ple, the second-place finisher in a race was faster than the fourth-place finisher—but not necessarily twice as fast. The Wong-Baker FACES pain rating scale2 is an ordinal scale. So are the Likert scales that are used in opinion research (eg, 1 = strongly disagree, 2 = disagree, 3 = neither agree nor disagree, 4 = agree, 5 = strongly agree). A logarithm is the inverse function of exponentiation. If bn = x, then n = logb(x). Since 23 = 8, log2(8) = 3. Since exponents can be negative, you can have negative logarithms, which represent the inverse of a number. Since ½ is the inverse of 2, the log2(½) = −1. Likewise, the log10(¹⁄₁₀) = −1 (Figure 1). Some units of measure are based on a logarithmic scale. For example, pH is based on the negative of the base-10 logarithm of the activity of the H+ ion (as measured in moles per liter). 𝑝𝑝𝑝𝑝 = −𝑙𝑙𝑙𝑙𝑙𝑙!"(𝑎𝑎#!) = 𝑙𝑙𝑙𝑙𝑙𝑙!" , 1 𝑎𝑎#! . A solution of pure water has hydrogen activity of 1 × 10−7. The reciprocal of that is 1 ×107, or 107; log10(107) = 7. So the pH of pure water is 7. Water with a pH of 6 would have a hydrogen activity of 1 ×10−6, which is 10 times as many hydrogen ions as in pure water! Figure 1. Logarithms are the inverse function of exponentiation: If bn = x, then n = logb(x). Exponentiation allows you to raise any real number (any point along a number line) to any real power. Note that x0 always equals 1, and x1 always equals x. For this reason, logb(0) is always 1 and logb(1) is always b, regardless of the value of b (i.e., regardless of the base of the logarithm). A negative exponent represents the inverse of a number (e.g., 2−1 = ½). Thus, a negative logarithm equals the logarithm of the inverse of the value: log2(½) = −1 and −log2(½) = 1. Courtesy of Richard F. Lyon via Wikimedia Commons. Base-10 logarithms are used so often that they are often just written as log (x). The natural logarithm, abbreviated ln (x), has Euler’s number (e) as its base. Euler’s number is an irrational number that is useful in many different areas of mathematics. Medical writers should be aware that viral load is often expressed in base-10 logarithms. I once edited a news article that described a patient as having a viral load of 5 copies/mL. That value was dubious: a value that low had to be below the limit of detection of any available assay. When I looked at the source material, I found out that the patient’s reported viral load was actually 5 log10 copies/mL, which meant 100,000 copies/mL. Big difference! Figure. You can raise any real number (b) to any real power (y), even negative and fractional powers. The logarithmic function is the inverse function of exponentiation: If by = x, then y = logb(x). The graph shows the value of logb(x) for x>0 and some nonzero values of b. The value of log0(x) is undefined; but for b≠0, logb(1)=0 and logb(b)=1. Also, logb(x) = −logb(1/x). Thus, the value of logb(0) is undefined but approaches −∞ as x approaches 0. The logarithm of a negative number is not a real number but involves a complex expression. Graph courtesy of Richard F. Lyon via Wikimedia Commons. Percentage Increase and Decrease The formula for calculating percentage increase and decrease is: Percentage Increase = [(Final Value − Starting Value)/ |Starting Value|] × 100 If you weigh 50 kg and gain 100 kg, that’s a 200% increase in weight: [(150 – 50)/50] × 100 = 200% But if you then lose that 100 kg, that’s only a 67% decrease in weight: [(50-150)/150] × 100 = −67% Fold Increase and Decrease A fold is a ratio between 2 values. The formula for calculating fold increase is: Fold change (for increases) = Final value/Starting value. If you weigh 50 kg and gain 100 kg, then that’s a 3-fold increase in weight: (150/50) = 3 A fold decrease is calculated as follows: Fold change (for decreases) = −(Starting value/ Final value). If you weigh 150 kg and lose 100 kg, then that’s a −3-fold change (3-fold decrease) in weight: − (150/50) = −3 However, some people use the fold increase formula for calculating fold decreases. As a result, they would describe a change from 150 kg to 50 kg as a 0.33-fold decrease in weight. So if you see someone express a fold decrease, make sure you know what they really meant! 126 AMWA Journal / V36 N3 / 2021 / amwa.org • Interval— An interval scale not only orders items or indi- viduals according to some characteristic but also establishes equal intervals between the units of measurement. This allows you to do some mathematical operations, such as calculating averages. However, the 0 point may be meaning- less. For this reason, the measurements cannot be expressed in ratios. For example, in the Celsius or centigrade scale, 0° was set to represent the freezing point of water, whereas 100° was set to represent the boiling point of water. However, the Fahrenheit scale sets 0° at a different point and uses different intervals. Water that is at 40 °C (104 °F) is warmer than water at 20 °C (68 °F), but it is not twice as warm! So don’t express such a change in temperature as a multiple or a percentage. • Ratio scale— A ratio scale has a meaningful 0 point as well as equal intervals. This allows you to calculate ratios. For example, you can say that 1 thing weighs twice as much as another, or that something costs twice as much as some- thing else. Conclusion So if you are being asked to report on the meaning of the out- come measures used in a study, you must present the numbers accurately and specify the units (if any). You must also think about what those numbers really mean! For example, it is an established convention that the 0 point on a psychological or educational measurement is arbitrary and may be meaningless. Author declaration and disclosures: The author notes no commercial associations that may pose a conflict of interest in relation to this article. Author contact: www.nottrivialbook.com; lthomas521@verizon.net References 1. Stevens SS. Measurement, statistics, and the schemapiric view. Like the faces of Janus, science looks two ways—toward schematics and empirics. Science. 1968;161(3844):849-856. 2. Wong DL, Baker CM. Pain in children: comparison of assessment scales. Pediatr Nurs. 1988;14(1):9-17. A Career in Medical Communication: Steps to Success Learn about the skills and attributes needed to be a successful medical communicator and discover opportunities in the field. www.amwa.org/career_steps http://www.nottrivialbook.com mailto:lthomas521@verizon.net http://www.amwa.org/career_steps