This paper intends to make an in-depth analysis of the structure of two pieces of knowledge that have provoked interesting discussions in relation to school mathematics. The first is recurring decimals, in which the digit 9 is repeated infinitely, and the second is rational exponents of negative bases. Understanding the structure of knowledge is seeing the underlying basic ideas of the knowledge. The algebraic permanence principle has been an important thinking pattern to extend numbers, thus it can be a powerful concept to see the underlying forms of numbers. From the perspective of the algebraic permanence principle, we reveal the improper structure of these pieces of mathematical knowledge in school mathematics, and suggest their proper didactical positions.