Before we start talking about number representation systems, bases, hexadecimal systems, and tol pescao 🐠, we should briefly discuss what a number is.
In modern mathematics, defining a number is very complicated. There are complex numbers, matrices, tensors, vector subspaces… in short, the topic is extensive.
Fortunately, for this course, a more traditional explanation will suffice. Let’s say that:
A number is an abstract concept that serves to identify quantities and measures.
Or, put colloquially, a number is what appears naturally when someone started counting sheep 4,000 years ago (and possibly fell asleep).
Then the concept of zero was added, negative numbers, fractional numbers… and well, more or less, we all know this story.
Number Representation
Let’s say that “more or less” we already know what a number is, and that it is an abstract concept. But, we need to find a way to represent this number (for example, because I have to store it, write it, or transmit it).
Here I emphasize, and I’m going to be a bit annoying, that we should not confuse a number with its representation. A number has no form; it is a concept, it is abstract.
To understand this:
This is NOT a house. It is an image that represents a house. Nor is the word house a house.
The real house is where you live. It’s a big thing, with walls, that you can go inside, you pay a mortgage for it,… that’s the real house (the other things are just representations of the house).
In the same way, 1,270,000 is not a number; they are symbols that represent a number. But we are so used to it that we end up thinking it IS the number. But it is not; they are just strokes to write the number (just like the image of the little house or the word house).
How to Represent a Number
Let’s imagine that no one has represented a number before, and we have to invent a way to represent them. There are infinite ways to store numbers. We could start inventing forms and literally never finish.
This problem was already encountered thousands of years ago by the first people who dedicated themselves to this. In fact, we know that humanity has used ways to represent numbers (for example, Roman numerals or Babylonian numbering).
But let’s continue with inventing our own way of representing numbers. Let’s start with the most obvious. Let’s put marks for each number. For example, “little dots”. So 1, one dot. 3, three dots.
Okay, but it wouldn’t be very practical to have so many dots. Imagine having to count 5,374 on a sheet to know what number it is. This doesn’t work very well, does it?
So let’s better use symbols. Let’s create a different symbol for each number. That way, we don’t have to count so many dots. But we have another problem… we aren’t going to memorize 5,374 symbols, are we?
Here comes the problem with numbers; there are so many of them! (infinite).
- If I use positions, I will have many positions
- If I use symbols, I will have many symbols
Neither option works alone. So they had to invent something intermediate.
Positional Notation
How do we avoid having too many positions and too many symbols? With positional notation, which is the system you are used to using.
Positional notation uses both symbols and their position to represent the number
Let’s see how it works,
- We define a limited series of symbols
- When we increment the numbers, we move from one to another
- If we run out of symbols, we put the same symbol on the left.
For example, let’s imagine we took these symbols 🔵,🔺, 🟩,⭐. Our system would work like this.
I have provided the equivalent in the decimal system below, because it is easier for us to understand. But the number is the same; only the representation is different.
Concept of Base
A little theory. We call “base” the set of symbols that our positional notation system will use. In reality, we generally refer only to the number of symbols we handle (because the specific symbols… I don’t really care).
Thus, our previous system has this base 🔵,🔺, 🟩,⭐, that is, we say it is Base 4.
The base is related to how “long” your numbers will be
- A large base means occupying fewer positions. But it requires learning more symbols
- A small base requires learning fewer symbols. But the numbers will occupy more positions.
Decimal System
Finally, we arrive at the decimal system, which is the positional notation system we are used to. As we know, the base of the system is 0, 1, 2, 3, 4, 5, 6, 7, 8 and (that is, 10 symbols, base 10).
Why was 10 symbols chosen and not 7 or 12? Because potato, absolutely for no reason. Well, yes, look at your hands and count your fingers; that’s where the thing goes 🖐️.
Moreover, 10 is a number of symbols that is comfortable for us to memorize. Working with 500 symbols would make it very difficult to operate, due to our capacity for memorization.
But aside from this, we could have taken any other base or any other set and number of symbols. Now we would see it as the most normal thing in the world (just like we do with the decimal system).
What happens is that we are so used to working in the decimal system that any other representation “confuses us”. But keep in mind that any of these representations is as valid as any other.