Natural Numbers Read in the Integers and in the Rationals Agree with Their Embeddings

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $n\in\mathbb{N}_{0}$, read in a commutative ring as in Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals; let $\iota_{0}$ be as in The Natural Numbers with Zero and Their Embedding into the Integers §embedding and $j$ as in The Rational Numbers §embedding.

¶ In the commutative ring $\mathbb{Z}$ of The Integers §integers, $n=\iota_{0}(n)$.

¶ In the ordered field $\mathbb{Q}$ of The Rational Numbers §rationals, $n=j(\iota_{0}(n))$.