Number and operations in society

SUMMARY: Number, arithmetic, and estimation are parts of daily life. 

In early human societies, wealth was often measured in terms of physical possessions. Commerce depended on finding an “even” or fair exchange of goods. Counting and arithmetic were fundamental skills for enumerating goods and engaging in trade. Later agrarian societies also needed these skills to plan for activities like planting crops and storing or dispersing the harvest in an equitable way. When humans began to travel farther from home, they needed to be able to measure and calculate distances and directions. 

The introduction of money and more advanced tools and technology did not change the need to count and calculate; in most cases, they merely altered what was counted and the way in which the counting and arithmetic were done. In modern society, numbers and their basic operations are pervasive. Numeracy (or quantitative literacy) is a primary concern of government and educators in the twenty-first century. The increasingly quantitative nature of society requires some level of basic proficiency in all its citizens, not merely from mathematicians, engineers, scientists, and others in traditionally quantitative professions. In modern society, studies have shown that lack of basic number and operations skills can be associated with negative outcomes, such as financial mismanagement, consumer debt, poor risk assessment, and limited job prospects. Some individuals who experience difficulty with arithmetic operations have a condition called “dyscalculia,” which may be caused by neurological lesions. 

Early Number History

Rows of tally marks have been found in many archaeological sites, indicating that people not only counted, but also recorded their counts. However, it is difficult to quickly know a total merely by looking at a long row of tally marks. Recognizing a quantity without counting one by one is called “subitizing.” Psychologists note that humans usually can subitize accurately only up to quantities of about 5, 6, or 7 without making some combinations, so a line of 23 tally marks would allow only a guess of its number. Because of this limitation of subitizing ability, ancient humans arranged the marks into groups—usually equal groups—and counted the groups to find the total (many people also commonly do this in the twenty-first century by drawing every fifth tally mark over the first 4 to make groups of 5 for easier counting). 

This system appears to exist in many parts of the world and led to both the idea of place value and the operation of multiplication. The counting system of nearly every language uses terms of grouping; many, including English, group by 10s and then 10s of 10s (100s) and continue with higher powers of 10. Probably the group size of 10 was chosen for physiological reasons—humans have 10 fingers—since other choices, such as dozens, might have made for more convenience (especially for fractions) and mathematical efficiency. Some languages do use other bases, including 4, 5, and 12, and the English words “dozen” and “score” indicate an earlier use of groupings of 12s and 20s in old English. Similarly, multiplication in objective terms amounts to finding a total by counting groups of equal quantities (even when used for area or combination calculations). Learning the “times tables” is simply learning how those groupings come together and grow into such totals. 

The groupings were originally oral linguistic terms, but the idea also translated into written numerals—the ancient Egyptian, Babylonian, Mayan, and Chinese symbols (and others) were fitted into various types of place-value frameworks, some more structured than others. It is thought that the current widespread system of written numerals, the Hindu–Arabic system, originally grew out of earlier place-value grouping systems. It developed in India, early in the common era, where repeated marks were replaced by a single cipher (for example, “7” instead of “///////”), allowing more efficient writing and calculation. The Islamic mathematicians added more convenient algorithms—smoother techniques for doing arithmetic calculations and handling rational numbers. 

Operations

Although people in all parts of the world needed to do arithmetic and developed their own methods, the standard algorithms most widely used in the twenty-first century developed from the Islamic algorithms (the word “algorithm” even comes from the name of Al-Khowarizmi, a mathematician who worked in Baghdad around the year 800). These algorithms were modified and refined over the centuries as the techniques were carried into Europe. Usually, the adding and multiplying methods were straightforward, mostly collecting and regrouping symbols, but subtracting and dividing were more complicated and led to a greater variety of algorithms—especially different ways and sequences to regroup numbers, both in conceptual terms and in the written expositions. An important difference involved either starting with the units and moving to the higher groupings, often called a “right-to-left method,” or the reverse of working first with the larger groups and then taking care of the smaller unit details. 

The standard symbols of the numerals stabilized in medieval Europe (c. sixteenth century), as did most of the algorithmic methods. Along with this standardization came the symbols for the operations (+, -, ×, ÷, and =) and later other notations such as exponents for powers, the square root symbol, various kinds of brackets for groupings, and symbols for fractions and decimals. These symbols are not completely standardized; for example, Americans use a dot (.) between the whole number and a decimal fraction, while many Europeans use a comma (,). Also, there are some remaining discrepancies in the terminology of large numbers, as Americans say a thousand millions is a “billion,” but British usage is that a “billion” is a million millions. 

Types of Numbers

Languages usually differentiate number usage according to the purpose of the number. If the number is an adjective that tells the numbers of members in a set or collection, is it called a “cardinal number.” Thus, “three houses” describes the quantity of houses being discussed. In higher mathematics, especially in number theory, the quantity of the members of a set is called the set’s “cardinality,” which, significantly, can be infinite or even different infinities. In many languages, especially in eastern Asia, an extra word is inserted after the number to describe the category of the item being counted. For example, a certain word for the category of flat items might be used to describe quantities of paper, boards, or leaves, but a different category word would be used for quantities of round objects. 

Another everyday use of numbers is to describe the place of something in a sequence. If there is a line of houses along a street, the third house may be noted—counting from the beginning of the line to position number 3. Since the houses are considered in order, this is called the “ordinal number”: first, second, third, fourth, and so on, with most of the higher values using a “-th” at the end of the number word. 

A third kind of number is as a name, so it can be called a “nominal number.” These are used when counting in general or referring to the number itself, as in, “I am writing a five.” In English, this is usually the same word as the cardinal form, though in other languages, this is not always the case. Also, the nominal form is often used in nonmathematical language, where a number term is used to name a person or something else. Examples include numbers of Social Security accounts, house street addresses, bank accounts, telephones, routes of highways, buses, or planes, and car license plates. Sometimes they are arranged in a numerical order for convenience, but usually do not represent true ordinal usage. These are only a convenience (for example, they work well in computers) and usually have no mathematical meaning—one would not think of adding two phone numbers! Distinctions of cardinal, ordinal, and nominal usage are (1) taking three buses with the necessary changes from one to another (cardinal), (2) waiting as two buses pass and then taking the next one—the third bus (ordinal), and (3) looking for a sign on an approaching bus that says it is bus #3 (nominal) and then getting aboard. Often, when people complain of modern society reducing everyone to a number, they are in fact referring to the nominal usage. Since nominal numbers are so pervasive, it is important for children to learn the distinctions, so they will understand that these nominal usages are not mathematical. 

Economics and Demographics

Beyond the nominal names, actual quantities are used in nearly all aspects of society. At the heart of economic activity is the need to quantify money and compare this quantity with measurements of value, which may also be quantified. Accountants and bankers may not be mathematicians, but they constantly use numbers and carry out operations that may be based on simple arithmetic but used in very complex applications. These users may range from high-level financial managers to retailers to children selling lemonade. Some have suggested that, especially in the modern world, economic uses of numbers may be the biggest application of mathematics in society. Another important subject of counting is people—for records of population, attendance at schools and events, families, public health, television viewing, and many others. 

Measurement

When numbers are applied to comparisons, measurement is happening. Measurements of length, weight, volume, and many of the technical quantities—such as electrical conductivity, strength of magnetism, blood pressure, engine power, and acoustic properties—are used by scientists, engineers, architects, medical workers, mechanics, and even artists and musicians to deal with properties essential to their work. Numbers are not needed for sophisticated technicalities but may help with shopping for shoes, getting a first down in football, and giving directions to the library. Operations include totaling a shopping bill, converting currency, checking the movement of a comet, and building an oil rig. Few people may calculate comet orbits, but nearly everyone needs to check their shopping and bank account calculations. Over time, many systems of units of measurement developed, showing the importance of this use of numbers. Many measurements, especially linear measures, were comparisons with human body parts, such as the length of a handspan, the distance from the elbow to the fingertips (called a “cubit”), or the distance of a walking pace. It became clear that standardized measurements were needed for fair comparisons, especially in trade, so governments as early as the ancient Egyptians and Romans developed standardized systems. Many traditional measures were converted to standard systems, but often the units did not fit well into an organized system. In the late 1700s in France, the metric system was devised to serve as a well-organized standard system for world use. In the two centuries since then, that goal has almost been achieved. 

Statistics

The tools of statistics are used to analyze and report results of counting and measuring. Tables arrange data in columns and rows for easier comparisons as well as summations, averaging, and other calculations. In the twenty-first century, computerized spreadsheets have given new power to the calculation and manipulation of data in tabular form. Graphical displays make the information visible for quicker comprehension. Bar graphs and histograms sort data into comparative categories, while line graphs are especially useful to show changes over time. Circle graphs show comparative portions of a total. Newer displays include bar-and-whisker charts, which show the distribution of a collection of data, and stem-and-leaf charts, which are used to assemble data for bar charts. Statistics educators often warn that the ease of display of statistical graphs can also be misused to offer misleading implications, so a familiarity with statistics is considered important in evaluating displays in advertising and reports. 

Arithmetic is considered part of the basic foundation of the school curriculum because the need to deal with numbers and arithmetic is central to so many aspects of daily life and is the starting point of all higher mathematics and applications of mathematics in science, engineering, and technology. Usually instruction in counting begins even before formal schooling, the basic arithmetic operations are taught in the early grades, and work with fractions, percentages, and ratios in the upper grades of elementary school. Even in areas where few children may have the opportunity to attend higher levels of school, it is considered essential that they learn this foundational material—in school or perhaps on the job—because of the central role of number and operations in so much of life activity. 

Mental Arithmetic

Mental arithmetic is the operational counterpart to estimation, in which calculations are done without writing or using other calculation tools. A variety of techniques for mental arithmetic have been developed. Sometimes, it simply means using rounded off estimates to make the calculation easier. In addition and subtraction, the technique might mean ignoring the ones column or even more. Also, using factors can often simplify multiplication and division. Sometimes there are special “tricks” to using specific numbers in calculations, such as adjusting numbers to fit together to make 10s, adding a reciprocal to carry out subtraction, or applying algebraic techniques to simplify the numerical work. 

Calculation Tools

Even though mental arithmetic is fast and convenient, many mathematical calculations are too complex for such methods. Very early in history, people realized that they needed various tools to assist their computational work. It might even be argued that the process of writing numerals and using written algorithms is the most fundamental tool—though perhaps counting on one’s fingers is an even earlier tool. More than 2000 years ago, tools were developed to handle basic arithmetic. Romans made shallow grooves in the ground to represent the place-value positions and moved stones within the grooves to represent the value of each position. Adding and removing stones from the grooves carried out addition and subtraction operations, often requiring regrouping or exchanging 10 of a smaller position for 1 of the next larger position in order to have enough stones for the results or to reduce an overloaded position. The abacus uses the same principles of mechanizing arithmetic, but does it with beads strung onto wires in a frame instead of with stones in grooves. Since the beads cannot be physically added or removed from the wires, various new techniques were developed to handle the regrouping, often involving reciprocal adding or mental regrouping. 

As early as the 1600s, more sophisticated mechanical devices were being developed to make arithmetic even more automated. Two famous mathematicians, Blaise Pascal (1623–1662) and Gottfried Leibniz (1646–1716), both made mechanical devices with gear wheels and a ratchet mechanism to handle regrouping. John Napier (1550–1617) invented two very different tools, one for ordinary people and one for scientists. For ordinary people, Napier took the idea of lattice multiplication, which had come from Islamic mathematicians, and used small four-sided rods of multiplication tables to arrange like lattices to ease the multiplication of multidigit numbers. They were called “Napier’s Rods” or “Napier’s bones,” since he sometimes made the rods from bones. More significantly, he (and others) introduced the concept of logarithms, which are actually representations of powers of a common base (usually 10 or e). Since multiplication of two numbers written as exponents of the same base can be done by addition (and handling powers can be done with multiplication), logarithms simplified multiplication to addition and exponentiation to multiplication and thus allowed scientists to deal with much more complicated powers and roots than other techniques allowed, greatly speeding their calculations. 

Charles Babbage (1791–1871) is sometimes called the “father of computers,” but also he was a very frustrated man, since he was trying to invent devices one century too early—in the first half of the nineteenth century. Noting that calculations by hand often had errors (even errors in transcription), he wanted to avoid errors by substituting the handwork with complicated machinery. His inventions—the “difference engine,” followed by plans (which he could never completely carry out) for the “analytical engine”—had the same basic parts as modern computers: input/output, a storage memory, and a central processor. He used a system of programming to input data and to instruct the machine on what to do and then had the results printed out—all aiming to keep the work away from human error. Unfortunately, his plans were beyond the technical capability of his day. His support from the British government was used up as he struggled unsuccessfully to overcome technical problems. 

Later in the nineteenth century, Herman Hollerith (1860–1929) also worked to mechanize data handling. He noticed the “programming” of Jacquard textile weaving looms was implemented by wooden plates with holes arranged in particular patterns to control the movement and alignment of the threads. He realized that paper cards similarly punched with holes could be used to direct the movement and combination of data. He convinced the U.S. Census Bureau to use the idea in tabulating its data, and later he joined Thomas Watson in starting the company that became IBM. His punch cards were a staple of data processing and, later, computing for many decades. 

Computers

Computers finally came on the scene from considerable theoretical work in the 1930s, the pressures of war needs in the 1940s, and the growth of technology in general in the 1950s and 1960s. Technical developments, such as transistors, integrated circuits, and interactive interfaces, moved the development toward enabling the common person to compute. Interactivity opened the door for word processing and publishing, e-mail and other communications, and, eventually, the Internet. Meanwhile, fitting greater power into smaller and smaller devices allowed cell phones, thin television sets, laptops, and the explosion of handheld devices with thousands of applications. 

Computers have become such a central part of modern life that some concern has been raised about their role and their power. Even as computers may seem cold and inhuman, programming and merging of data files allow many more individualized responses than humans would be able to handle efficiently. Mathematicians and mathematics teachers sometimes debate the use of calculators and computers in both school mathematics and mathematical research. In both cases, the main argument is the efficiency and accuracy of using electronic tools against the sense that doing mathematics should be a human, mental activity. 

In a broader sense, this same question comes to the role of numbers and operations in society: quantitative versus qualitative. Certainly, numbers and operations are essential to science, business, and in fact all of modern life (and were quite essential even in ancient times). Some would argue, however, that the essence of humanity is found in the arts, philosophy, and religion. The division of the two worlds has long been debated. However, a convergence may have been found as quantitative measures are increasingly applied to the humanities and the sciences have researched the mysteries of the brain and cognition, quantum mechanics and cosmology, and multiple dimensions and infinities. 

This tension between proponents of the sciences and the humanities has existed for centuries. Some have argued that the gulf between the two human dispositions is such that the two are mutually exclusive; that is, both communities are unable to combine their gifts to solve problems facing humankind. Others maintain that humanities can indeed make a contribution but only in a supporting role to the sciences. The leading advocate of this philosophy of thought is generally recognized to be British chemist C.P. Snow in a 1959 work titled The Two Cultures and the Scientific Revolution. More recent thought suggests this tension is both unnecessary and inaccurate. In their 2024 article “Integrating the Humanities and the Social Sciences: Six Approaches and Case Studies,” authors Brendan Case and Tyler VanderWeele suggest a new framework for this discussion where the humanities inform the study of social sciences, which, in turn, shape the interpretation of empirical data.

Bibliography

Bulaitis, Zoe Hope. "Controversy and Conversation: The Relationship between the Humanities and the Sciences." Value and the Humanities, Palgrave Macmillan, 30 July 2020, pp. 81-111. Springer Nature Link, link.springer.com/chapter/10.1007/978-3-030-37892-9‗3. Accessed 28 Oct. 2024.

Case, Brendan, and Tyler VanderWeele. "Integrating the Humanities and the Social Sciences: Six Approaches and Case Studies." Humanities and Social Science Communications, vol. 11, no. 231, 2024. Nature, doi.org/10.1057/s41599-024-02684-4. Accessed 10 Nov. 2024.

Ifrah, Georges. The Universal History of Numbers. Hoboken, Wiley, 1994.

Katz, Victor. A History of Mathematics: An Introduction. New York, HarperCollins, 1998.

Lombrozo, Tania. "Embracing the Humanities: A Perspective from Physics." NPR, 25 July 2016, www.npr.org/sections/13.7/2016/07/25/487327634/embracing-the-humanities-a-perspective-from-physics. Accessed 28 Oct. 2024.

Paulos, J. A. Innumeracy: Mathematical Illiteracy and its Consequences. New York, Vintage Books. 1988.

Willis, Martin, et al. "Humanities and Science Collaboration Isn’t Well Understood, but Letting Off STEAM Is Not the Answer." The Conversation, 25 Mar. 2018, theconversation.com/humanities-and-science-collaboration-isnt-well-understood-but-letting-off-steam-is-not-the-answer-92146. Accessed 28 Oct. 2024.