Integer Number Lines in U.S. School Mathematics - Number Line Descriptions

In some texts, part of the descriptions for positive and negative integers in algebra textbooks mirrored properties of a number line, even though actual illustrations of number lines were not present.  These texts were considered as “supporting” number line descriptions. Such descriptions contrasted with the previously-mentioned accounts in the sense that the text that contained context-free number line descriptions supported theoretical accounts of number line properties.

Descriptions of number lines, or of properties of number lines, in arithmetic and algebra texts in the 1800s varied in their complexity, as is evident in the following examples from texts written by Jeremiah Day and John Farrar.

Jeremiah Day

Jeremiah Day (1773-1867) was Head of Mathematics at Yale College before he became President of that institution. In 1814 he became the first North American author to write a textbook entirely devoted to algebra and aimed at North American students (Ellerton & Clements, 2017; Karpinski, 1940). Day's An introduction to algebra, being the first part of a course of mathematics, adapted to the method of instruction in American colleges (1814) provided an extensive number line description. Although he began his treatment of the positive and negative integers by referring to contexts, he eventually transitioned to more theoretical and context-free discussion. The following excerpt is Day’s contextualized account:

If, for instance, the profits of trade are the subject of calculus, the gain is considered positive; the loss will be negative; because the latter must be subtracted from the former, to determine clear profit. If the sums of a book account, are brought into an algebraic process, the debt and credit are distinguished by opposite signs. If a man on a journey, is by any accident necessitated to return several miles, this backward motion is considered to be negative, because that, in determining his real progress, it must be subtracted from the distance which he has travelled in the opposite direction. If the ascent of a body from the earth be called positive, its descent will be negative. These are only different examples of the same general principle. In each of the instances, one of the quantities is to be subtracted from the other. When a boat, in attempting to ascend a river, is occasionally driven back by the current; if the progress up the stream, to any particular point, is considered positive, every succeeding instance of forward motion will be positive, while the backward motion will be negative. (pp. 21-23)

Within his discussion of various contexts for negative integers, Day discussed the “relative” nature of the negative integers. He began with: “The terms positive and negative, as used in the mathematics, are merely relative” (p. 22), and concluded his explanations of positive and negative integers with a number line description drawn from Euler and Newton:

When a quantity continually decreasing is reduced to nothing, it is sometimes said to become afterwards less than nothing. But this is an exceptionable manner of speaking.* [References were made to Newton & Euler in a footnote.] No quantity can be really less than nothing. It may be diminished, till it vanishes, and gives place to an opposite quantity. The latitude of a ship crossing the equator is first made less than nothing, and afterwards contrary to what it was before. The north and south latitudes may therefore be properly distinguished, by the signs + and –; all the positive degrees being on one side of 0, and all the negative on the other; thus, +6, +5, +4, +3, +2, +1, 0, –1, –2, –3, –4, –5, &c. The numbers belonging to any other series of opposite quantities may be arranged in a similar manner. So that 0 may be conceived to be a kind of dividing point between positive and negative numbers. On a thermometer, the degrees above 0 may be considered positive, and those below 0, negative. (p. 24)

The above passage provides an example of a text used commonly in school mathematics that listed numbers as if they were on a number line, in order, described 0 as a dividing point, described two sides of the number line as positive and negative, and referred to the context of a thermometer (see, e.g., Figure 1). Although an extensive description, it is not a completely developed explanation. There is no evidence, for example, that Day recognized that negative integers could be subsumed into rational numbers or real numbers.

The above excerpt from Day can be interpreted in terms of the lag-time theoretical perspective. As stated previously, Jeremiah Day was a major early nineteenth century American educator and the first North American scholar to publish a text specifically devoted to algebra. His book would appear in many editions (Karpinski, 1940). It is interesting to note that his text, which was essentially written as a text for students, and not for mathematicians, offered a number line description which lagged in both time and development from texts offered by mathematicians such as Euler, Newton, and Wallis.

Figure 1. A passage from Jeremiah Day's Introduction to Algebra (1814, p. 24) included an ordering of integers from +6 to –5. Although Day’s Algebra was widely utilized, this 1814 edition has been rarely examined (Swetz, 2014).

John Farrar

John Farrar (1779-1853) was Hollis Professor of Mathematics at Harvard University between 1807 and 1836 (Ellerton & Clements, 2017; Karpinski, 1940). In his book, An introduction to the elements of algebra, designed for the use of those acquainted only with the first principles of arithmetic. Selected from the algebra of Euler (1821), Farrar provided an extensive “number line description” which attempted to differentiate between natural numbers and integers, as well as touching on the density of the number line. Farrar (1821) first introduced negative integers using the contexts of debts and temperature. After descriptions in these contexts, he then stated:

In the same manner therefore as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now we obtain positive numbers by adding 1 to 0, that is to say, to nothing; and by continuing always to increase thus from unity. This is the origin of the series of numbers called natural numbers; the following are the leading terms of this series: 0, +1, +2, +3, +4, +5, +6, +7, +8, +9, +10. But if instead of continuing this series by successive additions, we continued it in the opposite direction, by perceptually subtracting unity, we should have the series of negative numbers: 0, –1, –2, –3, –4, –5, –6, –7, –8, –9, –10, and so on to infinity. (p. 5)

In this excerpt, Farrar (1821) not only offered a definition of the natural numbers, but also appealed to the perceptual or visual nature of ordering the negative integers. A “less than nothing” description of negative was provided with a footnote that expounded on this with contextual explanations. Farrar introduced the following “definition” of the integers:

All of these numbers, whether positive or negative, have the known appellation of whole numbers, or integers, which consequently are either greater or less than nothing. We call them integers to distinguish them from fractions, from several other kinds of numbers of which we shall hereafter speak (p. 6).

After defining natural numbers and integers and referring to the perceptual nature of ordering the integers, Farrar presented a description that describes the density of the real numbers or number line. Farrar stated:

All these numbers, whether positive or negative, have the appellation of whole numbers, or integers, which consequently are either great[er] or less than nothing. We call them integers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak. For instance, 50 being greater by an entire unit than 49, it is easy to comprehend that there may be between 49 and 50 an infinity of intermediate numbers, all greater than 49, and yet all less than 50. We need only imagine two lines, one 50 feet the other 49 feet long, and it is evident that there may be drawn an infinite number of lines all longer than 49 feet, and yet shorter than 50. (p. 6)

When Farrar developed the operations with the integers, he did not refer to a number line. The treatment and description of a number line remained solely the description of the identity of negative numbers themselves.