# Proof

The set of real numbers without $$0$$ (i.e. the number zero), denoted by $$\mathbb R^*$$, together with the specific multiplication operation "$$\cdot$$" is a commutative group. This is because:

1. The multiplication operation is associative, i.e. $$(x\cdot y)\cdot z=x\cdot (y\cdot z)$$ is valid for all $$x,y,z\in\mathbb R^*$$.
2. We have shown the existence of a neutral element of multiplication - the number $$1\in\mathbb R^*$$, i.e. such that $$1\cdot x=x$$ for all $$x\in\mathbb R^*$$.
3. For every $$x\in\mathbb R^*$$, there there exists an inverse real number $$x^{-1}\in\mathbb R^*$$, such that $$x\cdot x^{-1}=1$$.
4. For every $$x\in\mathbb R^*$$, there there exists an inverse real number $$x^{-1}\in\mathbb R^*$$, such that $$x\cdot x^{-1}=1$$.

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013