Definition: Division of Real Numbers

Let \(x\) and \(y\) be real numbers, which by definition means that they are the equivalence classes \[\begin{array}{rcl}x&:=&(x_n)_{n\in\mathbb N} + I,\\ y&:=&(y_n)_{n\in\mathbb N} + I.\end{array}\] In the above definition, \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\) denote elements of the set \(M\) of all rational Cauchy sequences, which represent the real numbers \(x\) and \(y\), while \(I\) denotes the set of all rational sequences, which converge to \(0\). In the following, let $(y_n)_{n\in\mathbb N}$ be not a zero sequence, i.e. $(y_n)_{n\in\mathbb N}\not\in I.$

The division of real numbers, written \(x\div y\), is defined as the product of the first real number \(x\) with the inverse of the second real number with respect to multiplication \((y^{-1})\), formally

\[x\div y:=((x_n)_{n\in\mathbb N} + I)\cdot ((y_n^{-1})_{n\in\mathbb N} + I):=(x_n\cdot y_n^{-1})_{n\in\mathbb N} + I=x+(y^{-1}),\]

where $(y_n^{-1})_{n\in\mathbb N}$ denotes a rational Cauchy sequence such that $|y_n|\neq 0$ for sufficiently large indices $n$, which is ensured by the assumptions that $(y_n)_{n\in\mathbb N}$ be not a zero sequence.

The result of the division is called quotient or ratio. The number $x$ is called the dividend (or the numerator) and the number $y$ is called the divisor (or the denominator).

Definitions: 1
Parts: 2
Proofs: 3
Propositions: 4

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013