Textbook accounts of the rules of indices with rational exponents

ABSTRACT

The rules of indices, e.g. $a^n{b}^n=(ab{)}^n$, are a particularly important part of elementary algebra. This paper reports results from a textbook analysis which examined how the shift from integer to rational exponents in the rules of indices is discussed in school textbooks. The analysis also considered related issues, such as notation and the introduction of complex numbers. A selection of popular textbooks from the period 1800–2000 was examined and the nature of the justification given for the extension of meaning to rational indices considered. In both the definition and computational rules, when extending the domain of n in $a^n$ to rational numbers the (potential) contraction of the domain of a to positive numbers was often quietly ignored. A wide variety of approaches are used in choosing what is to be a definition, what is to follow, and how this is justified. The difference between computational rules for practical algebraic manipulation and a formal definition was often blurred.

KEYWORDS:

• Modes of reasoning
• algebra
• textbooks
• secondary mathematics

MATHS SUBJECT CLASSIFICATION:

• 97H20

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Christopher James Sangwin  http://orcid.org/0000-0002-3725-8625

Notes

1. The notation $\sqrt{x+a}$ is a combination of the radical sign $\sqrt{}$ and the vinculum. For example, [51] uses both parentheses and the vinculum for grouping, sometimes in the same expression. [52] writes, e.g. $\sqrt[3]{}(2x+3)$. Historic typography separated these $\sqrt[3]{}\overline{x+a}$ which is now written as $\sqrt[3]{x+a}$.