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If this system is decimal, then the symbols from 0 to 9 should receive the status of the initial numbers composed of units in particular: the number “one” is denoted as 1=1, the number “two” is denoted as 2=1+1, the number “ three ” as 3=1+1+1 etc. up to the number nine. Numbers after 9 and up to 99 adding up from tens and ones for example, 23=(10+10)+(1+1+1) and get the corresponding names: ten, eleven, twelve … ninety-nine. Numbers after 99 are made up of hundreds, tens and units, etc. Thus, the names of only the initial numbers must be preliminarily counted from units. All other numbers are named so that their quantity can be counted using only the initial numbers.39
3.2.2. Axioms of Actions
All arithmetic actions are components of the definition the essence of the number. In a compact form they are presented as follows:
1. Addition: n=(1+1…)+(1+1+1…)=(1+1+1+1+1…)
2. Multiplication: a+a+a+…+a=a×b=c
3. Exponentiation: a×a×a×…×a=ab=c
4. Subtraction: a+b=c → b=c−a
5. Division: a×b=c → b=c:a
6. Logarithm: ab=c → b=logac
Hence, necessary definitions can be formulated in the form of axioms.
Axiom 1. The action of adding several numbers (summands) is their
association into one number (sum).
Axiom 2. All arithmetic actions are either addition or derived from
addition.
Axiom 3. There are direct and inverse arithmetic actions.
Axiom 4. Direct actions are varieties of addition. Besides the addition
itself, to them also relate multiplication and exponentiation.
Axiom 5. Inverse actions are the calculation of function arguments.
These include subtraction, division and logarithm.
Axiom 6. There aren’t any other actions with numbers except for
combinations of six arithmetic actions.40
3.2.3. Basic Properties of Numbers
The consequence to the axioms of actions are the following basic properties of numbers due to the need for practical calculations:
1. Filling: a+1>a
2. The neutrality of the unit: a×1=a:1=a
3. Commutativity: a+b=b+a; ab=ba
4. Associativity: (a+b)+c=a+(b+c); (ab)c=a(bc)
5. Distributivity: (a+b)c=ac+bc
6. Conjugation: a=c → a±b=b±c; ab=bc; a:b=c:b; ab=cb; logba=logbc
These properties have long been known as the basics of primary school and so far, they have been perceived as elementary and obvious. The lack of a proper understanding of the origin of these properties from the essence the notion of number has led to the destruction of science as a holistic system of knowledge, which must now be rebuilt begi
The presented above axiomatics proceeds from the definition the essence the notion of number and therefore represents a single whole. However, this is not enough to protect science from another misfortune i.e. so that in the process of development it does not drown in the ocean of its own researches or does not get entangled in the complex interweaving of a great plurality of different ideas.
In this sense, it must be very clearly understood that axioms are not statements accepted without proof. Unlike theorems, they are only statements and limitations synthesized from the experience of computing, without of which they simply ca
Pic. 33. Initial Numbers Pyramids
3.3. The Basic Theorem of Arithmetic
3.3.1. Mistakes of the Greats and the Fermat's Letter-Testament
The earliest known version of the theorem is given in the Euclid's "Elements" Book IX, Proposition 14.
If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.
The explain is following: “Let the number A be the least measured by the prime numbers B, C, D. I say that A will not be measured by any other prime number except B, C, D”. The proof of this theorem looks convincing only at first glance and this visibility of solidity is strengthened by a chain of references: IX-14 → VII-30 → VII-20 → VII-4 → VII-2.
However, an elementary and even very gross mistake was made here. Its essence is as follows:
Let A=BCD where the numbers B, C, D are primes. If we now assume the existence of a prime E different from B, C, D and such that A=EI then we conclude that in this case A=BCD is not divisible by E.
This last statement is not true because the theorem has not yet been proven and it doesn’t exclude for example, BCD=EFGH where E, F, G, H are primes other than B, C, D. Then
A:E=BCD:E=EFGH:E=FGH
i.e. in this case it becomes possible that the number A can be divided by the number E and then the proof of the theorem is based on an argument that has not yet been proven, therefore, the final conclusion is wrong. The same error can take place also in other theorems using decomposition of integers into prime factors. Apparently, due to the archaic vocabulary Euclid's “Elements” even such a great scientist as Euler did not pay due attention to this theorem, otherwise, he would hardly have begun to use “complex numbers” in practice that are not subordinate to it.
The same story happened with Gauss who also did not notice this theorem in the Euclid's "Elements", but nevertheless, formulated it when a need arose. The formulation and proof of Gauss are follows:
“Each compose number can be decomposed into prime factors in a one only way.
If we assume that a composite number A equal to aαbβcγ …, where a, b, c, … denote different primes, can be decomposed into prime factors in another way, then it is first of all clear that in this second system of factors, there ca
This is an almost exact repeating of erroneous argument in the Euclid's proof. But if this theorem is not proven, then the whole foundation of science built on natural numbers collapses and all the consequences of the definitions and axioms lose their significance. And what to do now? If such giants of science as Euclid and Gauss could not cope with the proof of this theorem, then what we si
This letter was sent by Fermat in August 1659 to his longtime friend and former colleague in the Parliament of Toulouse the royal librarian Pierre de Carcavy from whom he was received by the famous French scientist Christian Huygens who was the first to head the French Academy of Sciences created in 1666. Here we give only some excerpts from this Fermat's letter, which are of particular interest to us [9, 36].
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So, count is the nominate starting numbers in a finished (counted) form so that on their basis it becomes possible using a similar method to name any other numbers. All this of course, is not at all difficult, but why is it not taught at school and simply forced to memorize everything without explanation? The answer is very simple – because science simply does not know what a number is, but in any way ca
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The axioms of actions were not separately singled out and are a direct consequence of determining the essence a notion of number. They contribute both to learning and establish a certain responsibility for the validity of any scientific research in the field of numbers. In this sense, the last 6th axiom looks even too categorical. But without this kind of restriction any gibberish can be dragged into the knowledge system and then called it a “breakthrough in science”.