nLab prealgebra real number

Idea

The real numbers as encountered in prealgebra and high school algebra.

Definition

Let $\mathbb{Q}$ be the set of rational numbers and let $[0, 9]$ denote the interval in the natural numbers of all natural numbers between $0$ and $9$ inclusive. Infinite decimals representations are elements of $\mathbb{Z} \times [0, 9]^\mathbb{N}$, with the idea that each pair $(i, d)$ consists of an integer $i$ and a sequence of digits $d(n)$ in the infinite decimal representation. The series

$$\sum_{n = 0}^\infty i + \frac{d(n)}{10^{n + 1}}$$

can be shown to be a Cauchy sequence.

The set of repeating infinite decimal representations is the subset of $\mathbb{Z} \times [0, 9]^\mathbb{N}$ such that for pairs $(i, d)$ in the subset, there exist natural number $k$ and positive natural number $n$ such that the sequence $\lambda m.d(m + k)$ factors through the cyclic group $\mathbb{Z}/n\mathbb{Z}$.

$$\lambda m.d(m + k):\mathbb{N} \to \mathbb{Z}/n\mathbb{Z} \to [0, 9]$$

Two infinite decimal representations $(i, d)$ and $(j, e)$ are said to be apart from each other if $i \neq j$ or there exists a natural number $n$ such that $d(n) \neq e(n)$. An infinite decimal representation $x$ is said to be strictly non-repeating if it is apart from every repeating base $b$ infinite radix expansion. Let $\mathbb{J}$ be the set of strictly non-repeating infinite decimal representations. These are referred to as the irrational numbers in the prealgebra and high school algebra literature.

Then the set of prealgebra real numbers or high school algebra real numbers is the (disjoint) union of $\mathbb{Q}$ and $\mathbb{J}$. The name “prealgebra real numbers” or “high school algebra real numbers” is because this is the definition of the real numbers which most commonly appears in prealgebra and high school algebra textbooks.

In constructive mathematics

In constructive mathematics, this definition only results in a subset of the real numbers, since not every real number can be shown to be either a rational number or an irrational number - this statement is equivalent to the analytic LPO when irrational is defined strictly in the sense of being apart from the rational numbers, and equivalent to the analytic WLPO when irrational is defined weakly in the sense of the negation of equality with any rational number.

In addition, not every strictly irrational number is the limit of the series

$$\sum_{n = 0}^\infty i + \frac{d(n)}{10^{n + 1}}$$

given by a strictly non-repeating infinite decimal representation. However, it is still the case that strictly irrational numbers which are also a Cauchy real number are interdefinable with strictly non-repeating infinite decimal representation.

See also

• real numbers, rational numbers, irrational numbers
• analytic LPO, analytic WLPO
• infinite decimal representation of a unit interval

References

• Nichols, Eugene D, et al. Holt Algebra with Trigonometry. Holt, Rinehart and Winston : Harcourt Brace Jovanovich, 1992.

• Marecek, Lynn, et al. Prealgebra 2e. OpenStax, Rice University, 2020.