# 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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### References

#### Bibliography

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