Last time, we saw how to construct natural numbers from the ground up. Starting from $0$ and the increment operation and then establishing some intuitive axioms. In particular the principle of induction is powerful and goes a long way, other axioms lead to many symmetries/properties between the new operations we defined – addition and multiplication which occur as fundamental examples of recursive definitions. We also saw that the natural numbers are completely ordered.
Integers come along with natural numbers. They come as a way to keep track of the deficit between two naturals. So structurally they are made of a pair of natural numbers, modulo an equivalence condition. By modulo we mean we consider a set of pairs of natural numbers to be equal or equivalent if the rule that “their difference is equal” is satisfied. We see the pair of natural numbers through this “lens” where two natural number pairs appear same (cf. the animation below). For example we put $3 \setminus 5 = 7 \setminus 9 = 11\setminus 13 .$ But the operation “$\setminus $” is not yet defined. This is just an auxiliary symbol to represent the pair, guided by our intuition. In fact, we will define the subtraction, using this very abstraction of integers soon. Integers, to start with will be viewed as a pair of natural numbers written formally (in the form) as $a \setminus b.$
We define the addition and multiplicaation operations on integers using our framework of integers as pairs of natural numbers. The symmetries/properties follow immediately like in natural numbers. The additional structure in integers (namely, the negation) helps us define the subtraction operation. There is again a linear order on integers, which extends the order on naturals.
Given the equivalence condition on the integers, the above definition is subject to a well-definedness check. If two different representations of the same integer are chosen, whether the the results of addition and multiplication are still equivalent. Indeed it turns out that’s the case!
$n \setminus 0$ behaves in the same way as the natural number $n$: $$ (n \setminus 0) + (m\setminus 0) = (n+m) \setminus 0; ~ (n\setminus 0)\times (m \setminus 0) = nm \setminus 0.$$ Further, $(n \setminus 0) = (m\setminus 0)$ if and only if $n = m.$ This allows us to identify the natural number $n$ with the integer $n \setminus 0$! In this sense, all natural numbers can be considered as integers.
Now to the structure which makes it very clear that the Integers are an extension to Natural numbers.
If $a\setminus b$ is an integer, we define the negation $-(a\setminus b)$ to be the integer $(b\setminus a)$. In particular, if $n = n \setminus 0$ is a positive natural number, we can define its negation $-n = 0 \setminus n$.
Intuitively, that's all the integers we know. And indeed, it precisely follows that there is a trichotomy in integers: they are either $0$, a positive natural number or a negation of a positive natural number.
This, follows from the order in Natural numbers. If $a \setminus b$ is an integer, then either $a = b$, or $a > b$ or $a < b$. If $a> b$, then $a = b + c$ for some positive natural number $c$, and hence $a \setminus b = c \setminus 0$ is a positive natural number. If $a < b$, then $b=a + c$ for some positive natural number $c$. Then, $ a \setminus b=0 \setminus c$ by our equivalence condition. This shows $a \setminus b=-c.$ If $a=b$, then $a \setminus b=0 \setminus 0=0$, by our equivalence condition again. The other case is similar and we thus have integers as an extension to the natural number in the sense of negation. One can also show that no two of the above relations in the trichotomy can be satisfied at simultaneously! It will soon be clear that the Integers are an extension to Naturals in the negative direction.
The following symmetries follow from the definitions and properties of natural numbers which can be proven ny using the pair form of integers.
Drumrolls please… We finally define the subtraction operation on integers (or naturals), which was the whole point of introducing integers.
Most importantly, if $a, b$ are two natural numbers we see that: $a - b = a + -b = (a \setminus 0) + (0 \setminus b) = a \setminus b!$ So, the integers which were defined to be only in the form of $a \setminus b$ in the beginning is indeed $a - b$ funtionally too! We thus ditch the "$\setminus$"" notation and start using the notation of subtraction. Given an understanding that, the integers are first defined abstractly just using the natural number pairs, rather than any operation. That is, one must define integers first to make sense of subtraction. Subtraction then becomes a functional equivalence to the original abstract definition of integers. Note that subtraction is applicable more broadly -- we can subtract two integers, while the notation we started with defines (as we now know) integers as difference between two natural numbers. Indeed we defined all the operations in a way that captures the intuition of subtraction. It's really a no surprise, other than the demonstration of the work of mathematics, and our goal to construct a rigorous theory of numbers.
This complete the architecture of integers. We see that the integers are an extension to natural numbers in the negative direction. The integers are again completely ordered. The operations of addition, multiplication and subtraction are defined and satisfy the laws of algebra. The integers further have a natural structural trichotomy.
Next up: Rational Numbers.
PS: Note that the natural numbers and integers have no zero divisors. And integers fall into a class of algebraic objects called commutative rings. Details may be some other time.