Integer Number Lines in U.S. School Mathematics - Relative Number Lines

Descriptions of negative integers in early arithmetic and algebra texts that included illustrations of number lines were notably different from the aforementioned categories. However, not only did the accompanying descriptions vary, but also the illustrations of the number lines themselves included attributes of relativity. For example, some illustrations of number lines did not include zero. If the number line did not include zero and focused on the relativity aspects, then the text was classified as using a “relative number line.”

In their A School Algebra, Fletcher Durrell and Edward Robbins (1897) provided an example of a “relative number line” (see Figure 2). After presenting positive and negative integers in contexts (e.g., debts, temperature), Durrell and Robbins provided a “First Law for + and – taken together” with a “relative number line.” The number line provided in their text highlighted various points indicated by letters (K, B, O, A, and F). The point O was considered to be the initial position with positive and negative integers distinguished by walking to the right and left of this point. Rules were then presented for operating (e.g., “The sign – applied twice to a given positive quantity gives a + result.” (p. 22)), and these rules were then described in the context of this relative number line.

Figure 2. A relative number line in Fletcher Durrell and Edward Robbins' A School Algebra (1897, p. 20).