Definition of Integers

Question

I'm studying Mathematical Education about construction of number system. My teacher said that there was 2 ways to extend numbers sets: embedding and adding element.

As an example, to construct $\mathbb{Z}$, we can use equivalence classes of $\mathbb{N}\times \mathbb{N}$.

http://www.math.wustl.edu/~freiwald/310integers.pdf

I want to know that how people orginally constructed $\mathbb{Z}$ from $\mathbb{N}$? Did they use the method "adding element"? And was there a book about construction of numbers sets?

Thanks for your help.

You can see: Ethan Bloch, The real numbers and real analysis, Springer (2011), Ch.1Construction of the Real Numbers. — Mauro ALLEGRANZA, Apr 05 '17 at 11:36
See also http://math.stackexchange.com/questions/987564/books-that-follow-axiomatic-approach. — lhf, Apr 05 '17 at 11:53
The question is how did you define $\Bbb N$ in the first places. — Asaf Karagila, Apr 05 '17 at 13:09

score 2 · Answer 1 · answered Apr 05 '17 at 11:45

The formal construction of Z as equivalence classes in NxN is a modern approach. I think the historical route would have been more intuitive and more pragmatic. People noticed that they needed negative numbers to solve certain problems, either as intermediate results or to represent a full set of solutions. They found that they could perform arithmetic with negative numbers in a consistent way without producing contradictions like 1=0. And in the end they realised that using negative numbers in their calculations was much simpler than trying to avoid them.

Chinese, Indian and Islamic mathematicians were ahead of European mathematicians in this respect - the use of negative numbers was still being resisted by some European mathematicians as recently as the 18th century.

Definition of Integers

1 Answers1