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In each area of mathematics, there are axioms – a set of statements about objects and their properties. Then there is the concept of a model – a specific invented “thing” of objects and their properties that satisfy the axioms.
There are many models of natural numbers. For example, a sequence of sets (specially constructed), sequences of the form aN….a2a1, where ashki are numbers (that is, the characters 0,1,2,3,4,5,6,7,8,9), while the first ashka is not 0.
There are also quite a few models of real numbers. This is a straight line (geometric), the sequence aN….a2a1, b1b2..bK…… (they are finite or infinite), dodekind sections, and many others.
In a specific task, the most convenient model is selected. If you are interested in the definition of real numbers, take a look here wikipedia.org in the axiomatic approach graph.
I'll answer your question with a quote, or rather, I won't answer it.
“What is wholeness? The question can now irritate a philosopher who is preoccupied with important problems. This is integrity, of course! No, still? This is unity! But what is unity? Unity is a property of the one, the one. What is the one? In fact, what is one? Number. What is a number? Is unknown. There is no definition of a number. A number is defined by quantity, quantity by count, count by unit, and unit� is a number. You can formalize this procedure in any way you want, for example, you can define a number as “a concept that serves to quantify objects”, with reference to the term “quantity”, and quantity is defined as “a display of the common and the single”, the single, in turn,� – as “the beginning of the set”, ” set ” as “class”, “class” as “a set of elements”, i.e. again, the unity of units, a number. To put it even more abstractly, a number is “the class of all equilumerated classes”, and “equilumerated classes” are classes that have a one-to-one correspondence between all their elements, i.e. an element of one class can be matched to an element of another class without ambiguity. But “class element” cannot be defined otherwise than as a unit; a unit is a number. When defining a number, we do not get out of the tautology in any formalization, we only achieve that we confuse ourselves in order to avoid meeting face to face with this unknown�— number, unity. Unity is one, unity is number, number is the count of units […]”
Excerpt from the book: Vladimir Veniaminovich Bibikhin. “Mir.” Aquarius, 1995. iBooks.
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Numbers initially appear as an abstraction of the number of items or events that have occurred. Even animals, in the process of developing a reflex, or searching for food, can bring the number of objects requested by the experimenter (voice the required number of times); if this number is small enough.
Further, people in ancient times noticed general patterns of situations when many individual items are added(subtracted).: that when two more items are added to two items , the resulting number of items is always four. The ancients realized that counting allowed them to predict certain situations or figure out the past. For example, to understand whether the loot brought will be enough to pay tribute to the leader. Or whether all the tributaries paid off =). Later, agriculture formed the idea of fractions, trade and exchange-of negative numbers (debt and property).
The development of geometry introduced irrational numbers and made the concept of number an object of pure science
Strictly speaking, there is no such concept of a “number” in mathematics: there are separately defined concepts of a natural number, an integer, a rational number, a real number, and a complex number, where each of the following is a generalization of the other; p-adic numbers stand separately. Further generalizations are not commonly referred to as numbers, but rather as sets with operations-groups, rings, modules, fields, algebras, linear spaces – that generalize sets of “numbers”in one way or another.
Natural numbers can be constructed as indeterminate objects whose behavior is described by axioms, for example, the Peano system. We can try to “naively” define a natural number as the equivalence class of finite sets with respect to the equivalence relation: two sets are called equivalently powerful if each element of one can be uniquely associated with an element of the other, then we will call the class of sets that are equally powerful to each other a “number”. (If we do not restrict ourselves to finite sets, we will get, in addition to natural ones, a wider class of objects called “cardinal numbers of sets”). But with such definitions, you need to be more careful, you can easily run into paradoxes. On the other hand, it is quite possible that the concept of number arose in human life in this way: there is something in common between two sticks, two apples and two stars, and not the same thing that is common between an apple and a star. This is where mathematics begins as a method of thinking as abstractly as possible.
The following sets of numbers are constructed using equivalence classes quite strictly: an integer can be defined as the equivalence class of pairs of natural numbers with a common difference, a rational number is the equivalence class of pairs of integers with a common quotient, and a real number is the equivalence class of sequences of rational numbers converging to the same limit. A complex number is defined as a pair of real numbers with certain rules of addition and multiplication.