TITLE: How addition will be if we don't define it only on pairs of numbers?
QUESTION [6 upvotes]: In calculus by Michael spivak  book
he say in page 4

It might seem reasonable to regard addition as an operation which can be performed on several numbers at once and consider the sum $a_1+...+a_n$ of n number $a_1,...,a_n$ as a basic concept . it is more convenient , however to consider addition of pairs of numbers only and to define other sums in terms of sums of this types

My questions :
1- How addition will be if we don't define it only on pairs of numbers ?
2- why it is more convenient , however to consider addition of pairs of numbers only and to define other sums in terms of sums of this types?

REPLY [3 votes]: How addition will be if we don't define it only on pairs of numbers ?
The short answer is, it will be summation, as in $\sum_{i = 1}^n a_i$.
The long answer is, Spivak was writing in the 1960s, for people fresh out of high school, who hadn't necessarily ever heard of associative or commutative laws, but were familiar with expressions like "$a + b + c$." He's about to show that that expression can be justified by use of the associative law of addition; he's anticipating objections from readers who don't see why it needs to be justified at all.
why it is more convenient , however to consider addition of pairs of numbers only and to define other sums in terms of sums of this types?
Because then you can define the algebraic properties of the real numbers as consequences of the field laws and order laws, Spivak's properties P1-P12, and these involve only the binary operation.