Definition of arithmetic operations for hyperreal numbers

Question

I read paper about hyperreal numbers (https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf)

I have few questions about definition of arithmetic operations.

What is idea of this definition? Why can we use componentwise operations? What is this used for?

Thanks.

Answer 1

Note that the author uses square brackets $[\;]$ for the equivalence class of the sequence. The point is to define an ordered field which properly extends $\mathbb R$. In order to do this, one uses sequences, and introduces a suitable equivalence relation among such sequences. At this stage, one needs to define operations of addition and multiplication among such equivalence classes. The way this is done is that one chooses a representative sequence, say $(r_n)$, and similarly for $(s_n)$, and defines the sum $[r] + [s]$ to be the equivalence class of a new sequence. The new sequence is the term-by-term sum of the representative sequences, namely the sequence $(r_n+s_n)$ (as $n$ runs over the natural numbers as usual). The equivalence class of this new sequence is denoted $[(r_n+s_n)]$, indicated by the square brackets as usual.

At this point, one's task is not finished. One needs to show that the operation is well-defined, namely independent of the choice of representatives. After that, one needs to show that the set of equivalence classes is indeed a field. This is all done rather well in Goldblatt's book:

Goldblatt, Robert. Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998. xiv+289 pp.