# Definition: Subtraction 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 $$0$$. The subtraction of real numbers, written $$x-y$$, is defined as the addition of the first real number $$x$$ with the inverse of the second real number with respect to addition $$(-y)$$, formally $\begin{array}{rcl}x-y&=&((x_n)_{n\in\mathbb N} + I)-((y_n)_{n\in\mathbb N} + I)\\ &=&((x_n)_{n\in\mathbb N} + I)+((-y_n)_{n\in\mathbb N} + I)\\ &=&(x_n-y_n)_{n\in\mathbb N} + I\\ &=&x+(-y).\end{array}$

The result of the subtraction is called the difference of the real numbers $x$ and $y$.

Definitions: 1 2
Parts: 3
Proofs: 4 5 6 7 8
Propositions: 9 10 11

Github: ### References

#### Bibliography

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