Proof

(related to Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers))

The (natural number) one \(1\) exists, since it is the first ordinal succeeding of the natural number \(0\), i.e.

\[1:=0^+.\] By the definition of the multiplication of natural numbers "\( \cdot \)", the commutativity of addition of natural numbers, and because \(0\) is neutral with respect to this addition operation, we have vor all \(n\in\mathbb N\)
\[\begin{array}{rcll} n \cdot 1&=& n\cdot 0^+&1\text{ is the successor of }0\\ &=&(n\cdot 0)+n&\text{by definition of multiplying natural numbers}\\ &=&0+n&\text{by definition of multiplying natural numbers}\\ &=&n+0&\text{by commutativity of addition of natural numbers}\\ &=&n&0\text{ is neutral with respect to addition of natural numbers} \end{array}\] In other words, \(1\) is neutral with respect to this operation (does not change the natural number \(n\), if multiplied by it).

It remains to be shown that also the equation \(1\cdot n=n\) holds for all \(n\in\mathbb N\). It follows immediately from the commutativity of multiplying natural numbers.


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References

Bibliography

  1. Kramer J├╝rg, von Pippich, Anna-Maria: "Von den nat├╝rlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013