What is a number

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I want to show methods how we can conceptualize a number in order to actually understand what it is and how it is used in real life or in pure mathematics.

This introductory book should provide multiple benefits for anyone interested in a deeper, more fundamental understanding of mathematics or for a reader who wants to refresh or upgrade her knowledge in mathematics at school or at workplace. If you ever have wondered where all those theorems come from, can I create a theorem, how come math is so wide in scope, what are the roots of mathematical thinking, this book is for you. The book will, hopefully, put all your previous mathematical knowledge on firm and correct footing. It can provide answers about what exactly is the subject of investigation and research in mathematics and what mathematics is all about. The book should provide firm, in depth intellectual tools for understanding a quantification process that can be used virtually in any area of human activity such as economics, finance, engineering, space sciences, physics, sales, planning, etc.

The explanation of a number concept, of a number definition can be a useful and effective starting point for all those creative minds who ask “why mathematics?” “what for we have to calculate all that?”, and “what is the number actually?”.

So, let’s start. Let’s say that we can conceptually, visually, in our minds, distinguish objects among themselves, and that we can, and then that we want, to count those distinct objects.

When we, for whatever reason, group objects into some collections (they don’t have to be of the same kind nor similar at all), we can be in a position to determine which collection has more objects, if we want to. Don’t forget, we don’t have defined yet any concept of numbers, i.e. labels, names for counts that reflect the size of a collection, its number of elements.

How, then, we can determine which set, which collection has more elements, objects?

The essential method is to compare collections of objects by pairing, matching, elements from one collection, with elements, objects from another collection. Pairing has to be, obviously, one to one.  

Let’s say, we have a set of pens, set of apples, and a set of bananas, as shown in the Picture 1.

We can start first by determining which, if any, set has more elements.

In order to do that,  we need to match, to pair, one object from one set with only one object  from another set. If no objects are left unmatched then two sets have the same number of elements. As we can see in the Picture 1, by pairing apples and pens, and apples and bananas, no objects are left unmatched, hence these three sets have the same number of elements. We do not have the name yet for that count, for that number, but, the good thing is we know what we are talking about! We are talking about certain number of elements, defined by the exact match of these three collections. That property is what we are after! That “numberness” is what we are looking for to capture and define.

Now, note one very interesting thing. If we replace bananas with cars, and pair, match every car with every pen, we can see that the match is again achieved! The pairs are again complete and no cars or pens or apples are left unmatched. Hence, these two collections, two sets, have the same number of elements.

If we add a car then we can see that sets of cars has more elements than the set of apples or set of pens. Or, if we add a pencil, we can see that set of pens has more elements than the other two sets.

It is this property, this “numberness”, common in pairing two or more sets, that we call a number. You see, no matter what kind of objects are in two sets, if they match, that property, is the actual number. That is the concept of a number! It is, at this point, completely arbitrary how we are going to label this numerical concept we have just discovered and defined. Word, symbol, reference for it is completely arbitrary. We talk about number three here. In English, it is called number three, and the numeric symbol is 3. In other languages it can have different name and nomenclature for it may be different as well. Label for our new concept is really of very minor significance at this point. It is the concept of “numberness” that adjective, that property the sets have, the property sometimes used as a noun number,  is way more important than the tag, than that curvy trail of ink on the paper we use to reference it. What is important is what the trail of ink on paper represents and not ink itself.

Let’s go back to our quantification adventure.

Of course, we could start with three object and obtain number five, or seven objects, and obtain number seven, etc. Notice how we, now, have this set property to work with. set property related to its number of elements, the quantity of objects, the pure count of objects, that can be abstracted (because that count is the same for all sets having that number of elements). We have abstracted a property, quantity from any two sets of objects, that we can call a number, or a count! That’s the actual concept of a number.

Notice, also, how “number” in its essence, is not even a noun, but more like an adjective, that describes “quantitative” aspect of two sets, the number of paired elements of two collections.

Any time you have number in your mind first, and only then you decide what you are going to count, you defined the number what it is.

As long as we know that these labels, symbols, 1,2,3,…represent that property of one to one pairing between the elements of two sets (with the goal to determine if they have the same number of elements or not) we are on a good path to work with numbers and quantification.

You may ask at this point “well, I don’t always see two sets when I count objects of one set, I just count them without any pairing with the elements of another set”. Good question! What you actually do, by, say, counting CDs, or lemons, in your collection, is matching them with the set of natural numbers in your mind, which is completely ok. But, note, you have natural numbers at your disposal to use them for counting other objects, while in our previous explanation we were actually just defining the numbers! We were after very definition of a number. Once the numbers are defined, as we did for number three, 3, you can use that number three and other numbers in counting any kind of objects!

Look at that number five, say, the universal count of five, for any kind of objects. One more interesting conclusion follows. You see, how at this point, we can deal with counts only! We can deal with a count 3 and a count 5, which we call a number 3 and a number 5, regardless which objects they may represent count of! When we want to add them, it will always be 3 + 5 = 8, no matter what we have counted! Apples, pears, cherries, lemons, their taste, texture, color, cannot change the numerical result 3 + 5 = 8. That’s one of the beautiful sides of mathematics. The search for truth about pure quantitative relations. And this is exactly what pure math is about! Beauty of the applied math, on the other hand, is in the challenge to find all kinds of relationships between objects and concepts that need to be quantified.

From the application point of view, we, now, can use our generalized knowledge that 3 + 5 = 8, and utilize it any real world situation. For instance, if we have 3 cars and we buy 5 more cars, we will have 8 cars.

Note that “purity” of math is just related to the fact that we do not care what we have just counted. We were only interested in adding, subtracting, dividing pure counts, pure numbers we have abstracted from real world objects counted.

Mathematicians have an exotic term for the number of elements in a set, for the set’s size – it’s cardinality of a set.

You may ask “I can just start counting and continue counting objects without putting them into any set. I can even stop counting at any arbitrary time and still get a count, without specifying any set”. It is actually completely true. You don’t have to have defined set first then count elements within it to obtain a number. Technically, you are forming set “on the go”, set whose elements can be defined as “anything I can see around me I can put in set and count right then and there”.

This question is also interesting from another point of view. In slightly different approach, you can start with number 5 and then count any object you see around until you complete five counts. You see, at this point, you used pure math in real world scenarios, perhaps even unconsciously! You started with a pure number five, and it was up to you what are you going to count. Of course in physics, engineering, economics, trading, it matters what you count! That’s why we have to drag units beside pure numbers to remind us what we have counted and what we may want to count when we go back from pure mathematical calculations to the real world. More about that in a second.

If we want to mix pure math and real world scenarios and objects we are counting, it’s easy, but, of course, it has to be carefully done! What we need to do is to put a small letter beside the number, to keep track what we have counted. Hence, in physics we have 3m + 5m = 8m, for distance, length in meters. Then, also we can put 3 apples + 5 apples = 8 apples in agriculture studies. We essentially do two steps here, during the additions of real objects. When we want to add 3m + 5m, we actually separate pure numbers from the meters counted, we enter with these two numbers the world of pure math, where we do calculation of pure numbers only, 2 + 3 = 8. Then we go back to real world of meters (because we have those small letters to remind us what we have counted) with the result 8, and associate the name of the object, in this case it’s a physical unit of length, or distance, which is meter, (m), to the number 8. And, voila, we have just used pure math in the real world application!