Here is a continuation of this post. Long story short, I am learning number theory from a book [1]. The author of this book [1] begins his own analysis of number theory with the “system of rational numbers, i.e. of numbers integral and fractional, positive and negative, including zero.” There is a brief mention that the rational numbers follow from operations on “the ordered sequence of natural numbers 1,2,3,…” Note that the word “system” is another word for “set,” in the context of number theory. In this reference [1], a variable for a rational number is given by a small italic character such as $a, b, x,$ and $y$.

One important point is that division is well defined for the set of rational numbers with the one exception of division by zero, which is undefined.

The author then details the Fundamental Laws of Arithmetic. Three laws for addition seem worth noting, so I copy them here [1].

Commutative Law: $a + b = b + a$

Associative Law: $(a+b) + c = a + (b + c)$

Law of Monotony: $a < b$ always implies $a + c < b + c$

I think these three laws are fairly self-evident; given a rational number of objects, the order in which I count them–or group them–does not matter. I can think of the addition process of two numbers as preceding by default in an order from left to right in the absence of other symbols such as parentheses. Furthermore, parentheses indicate the order in which arithmetic operations should be performed–quantities in parentheses take precedence by construction. Finally, if I have two collections $A$ and $B$ with the same rational number of objects and I add to collection $B$ a greater rational number of objects than to collection $A$, then collection $B$ now has more objects than $A$.

Alternatively, here are three geometrical proofs of these three laws for addition.

Similarly, four laws for multiplication are important, so I copy them here [1].

Commutative Law: $ab = ba$

Associative Law: $(ab)c = a(bc)$

Distributive Law: $(a+b)c =ac + bc$

Law of Monotony: $a < b$ implies, provided $c$ is positive, $ac < bc$

I now include geometrical proofs for these four laws.

Now, multiplication is just repeated addition. For example, $(+2) (+3)$ is the addition of $+2$ three times: $0+(+2)+(+2)+(+2)$. With this notation reduced to the simple case of a nonzero number, the negative number $-2$ can be interpreted as $(+2) (-1) = 0-(+2)$ which is the subtraction from zero of the positive number $+2$. Alternatively, $-2$ can be interpreted as $(-2) (+1) = 0+(-2)$ which is the addition to zero of the negative number $-2$. The usage of the number $0$ allows for a nice interpretation of multiplication in terms of addition.

Also, since $(+2)(-1)$ is the subtraction from zero of the positive number $+2$, it follows that $(-1)(-1)$ is the subtraction from zero of the negative number $-1$, which is $0-(-1)$.

To evaluate $(-1)(-1)$, I use an argument from reference [2]:

First note that

$-1 * 0 = 0$.

Next,

$-1 * (-1 + 1) = 0$.

Apply the distributive law.

$(-1)(-1) + (-1)(+1) = 0$.

Simplify.

$(-1)(-1) = 1$.

This argument can be generalized for any positive, real numbers $a$ and $b$.

First note that

$-a * 0 = 0$.

Next,

$-a * (-b+b) = 0$.

Apply the distributive law.

$(-a)(-b) + (-a)(+b) = 0$.

Simplify by adding $ab$ to both sides.

$(-a)(-b) = ab$.

Hence a negative number multiplied by a negative number is a positive number.

References

[1] Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.

[2] https://gdaymath.com/lessons/powerarea/1-5-why-is-negative-times-negative-positive/