what does it mean to say that rational numbers are obtained "from unity"?

Question

In Courant's Introduction to Calculus and Analysis (Introduction, page 2) he says that:

[The rational numbers] are all obtained from unity by using the "rational 
operations" of calculation, namely, addition, subtraction, multiplication, 
and division.

I can't quite figure out what he means by "obtained from unity." I've been googling around, and I see that unity is any element that behaves the same way as 1 under multiplication. Technically I guess you could obtain any rational number from 1 through a combination of the 4 rational operations, do you think that's what he means?

Maybe this is just a quirk of expression, but since I'm still in Chapter 1 I didn't want to ignore a point if it's potentially fundamental.

Answer 1

Yes. You can obtain any natural number $n$ by adding up $n\ 1$s. You can obtain any negative integer by subtraction. You can then obtain any rational by division. He may go on to say you can't get the irrationals by any similarly simple process. Even if you allow taking roots you only get algebraic numbers, not any transcendentals.

Answer 2

Every Rational are nothing but string of number '1'. You can right 10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. In Similar manner you can obtain any given number by adding up 1s. That is the most simplest example I can give to explain that rational numbers are obtained “from unity".

Answer 3

You might say that the rational numbers field is obtained by letting $1$ generate it ($0$ doesn't do much) in the most natural way - giving it 'free reign' to generate a minimal number system. As mentioned in another answer, using addition and subtraction, $1$ generates $\mathbb Z$.

If we look at dividing $1$ by $n \gt 1$ as cutting up a pie into $n$ equal sizes, or putting tick marks on a ruler, then we obtain from unity (dividing it up into $n$ equal pieces) each of these 'new' numbers

$\tag 1 S_n = \{\frac{k}{n} \text{ with } k \text{ an integer and } 0 \lt k \lt n \}$

But every positive rational number that is not an integer has a mixed number representation,

$\tag 2 m + s \text{, with integer } m \ge 0 \text{ and } s \in S_n $

So, a positive rational can be looked at as so many whole pies, plus, perhaps, a number of slices from a pie (unity) that has been equally divided into so many pieces.