Integers

A small break

This is a small section about integers. Since sets and more complex logic concepts are coming after, this is a preview of those concepts, just as a way to keep these things in mind.

In the previous section, I wrote about numbers. I did not specify what a number was, only that it was tied to a certain "quantity". This was to borrow the notions of object counting in the physical world, similar to how children learn to count in the first place.

Counting then, is the process of transforming those instances of objects into a concept of a number. While you have five objects, you wouldn't think of the concept of "fiveness" in general, and numbers help bring that concept out.

That's why, integers are the first "type" of number always taught, as it's easy to translate them to real life concepts. To be more precise, whole numbers that are positive. Even the idea of zero was not widespread, as the concept of "nothingness" is difficult to attribute to a "number that makes sense".

When talking about subtraction then, I did introduce out of the blue the concept of negative numbers. This is also a bit revolutionary, as in the natural world there isn't really an easy concept of "something is missing in this exact quantity". That's why the closest concept relating to a "negative number" is similar to antimatter particles, something that cancels out were it to be in contact with "positive" particles. Nowadays it's pretty common and accepted for these to be "integers", especially with things like bank accounts, or a better intuition of the concept of "borrowing".

More spoilers

I want to at least introduce the idea of divisibility and primes, (otherwise this section is nothing but me waxing philosophy about numbers).

Divisibility is a property (a true statement and fact about a certain value) that a number can have, regarding another number. For example, all numbers are divisible by 1.

That's because $1*a \rightarrow a$. Other examples are, 8 being divisible by 1,2,4,8, or 7 being divisible only by 1 and 7.

If we try to divide 8 by some other number other than the ones stated, we end up with a non-integer result. So divisibility is the property that the division between the two numbers can be an integer number.

Prime numbers are numbers that are only divisible by 1 and themselves (for various reasons, one itself is excluded from being a prime). Some primes include: 2,3,5,7,11,13,17,19,23,29...

The number of prime numbers is infinite. There is no limit to how many primes we can make, provided we go really large. Many questions are still unresolved about the structure of the primes, along with other conjectures like "any integer greater than four is the sum of two primes".

Any number can be factored into primes. That means expressing a number $a$ like the multiplication of primes $p0*p1*p2*p3...$ (it can be any number of primes, not restricted to four, but at least one).

For example, the number 8 can be represented as 2*2*2. A number like 12 can be represented as 3*2*2, it does not matter the order of these primes, only the composition. Therefore, 12 can be divisible by any "subgroup" of primes we multiply together. We take 3 and a 2? Since 3*2 is 6, 12 is divisible by 6. Taking 2 and the other 2? 2*2 is 4, and 12/4 is still an integer.

If you actually try to find what it divides into, 12/6 is 2, while 12/4 is 3. The division will always give the multiplication of the rest of the primes you did not pick. If you pick a single 2, 12/2 is 6, which is 2*3, the rest of the primes not picked from 3*2*2.

This is just a preview of the later topics, and will be covered in a much more detailed manner. The next topic is going to formalize a lot more of the notation we've been using and it is a step up from what I've shown here.