A friend of mine engineer recently surprised me. He said he was not sure whether the number 1 is simple or not. I was surprised because none of the mathematicians considers the unit simple.
Confusion begins with a definition that is given to a prime number: is a positive integer, which is divisible only by 1 and by itself
. The number 1 is divisible by 1, and it divides into itself. But dividing into yourself
here are not two different factors. So is it a prime number or not? When I write the definition of a prime number, I try to eliminate this ambiguity: I am directly talking about the need for exactly two different conditions, dividing by 1 and by itself, or that a prime number should be an integer greater than 1. But why go on such measures to exclude 1?
My mathematical education taught me that a good reason why 1 is not considered simple is the basic theorem of arithmetic. She argues that each number can be written as the product of primes in exactly one way. If 1 were simple, we would lose this uniqueness. We could write 2 as 1 × 2, or 1 × 1 × 2, or 1 594827
× 2. Exception 1 from prime numbers eliminates this.
Initially, I planned in the article to explain the main theorem of arithmetic and finish it. But in fact, it is not so difficult to change the formulation of the theorem to solve the problem with the unit. In the end, my friend's question ignited my curiosity: how did mathematics stop at this definition of a prime number? A quick search on Wikipedia showed that the unit used to be considered a prime number, but not now. But an article by Chris Caldwell and Yeung Syun
shows a slightly more complicated story. This can be understood from the very beginning of their article: “First, whether a number (especially a unit) is simple is a matter of definition, that is, a question of choice, context and tradition, and not a question of proof. However, definitions do not occur randomly; the choice is related to our use of mathematics and, especially in this case, our notation. "
Caldwell and Xiong begin with classical Greek mathematicians. They did not count 1 as 2, 3, 4, and so on. 1 was considered a number, and the number consisted of several numbers. For this reason, 1 could not be simple - it is not even a number. The IX century Arabic mathematician al-Kindi
wrote that this is not a number and, therefore, is not even or odd. For many centuries, the notion that the unit is the building block for the compilation of all numbers has prevailed, but not the number itself.
In 1585, the Flemish mathematician Simon Stevin pointed out that in the decimal system there is no difference between 1 and any other numbers. In all respects, 1 behaves like any other value. Although not immediately, this observation ultimately led mathematicians to accept 1 as any other number.
Until the end of the XIX century, some outstanding mathematicians considered 1 simple, and some not. As far as I can tell, this was not the cause of disagreement; for the most popular mathematical questions, the distinction was not critical. Caldwell and Syun cite G.H. Hardy as the last major mathematician who considers 1 simple (he clearly indicated it as a prime number in the first six editions of the Pure Mathematics Course, published between 1908 and 1933, and in 1938 changed its definition and called 2 the smallest simple).
The article mentions, but does not understand in detail, changes in mathematics, because of which 1 were excluded from the list of prime numbers. In particular, one of the important changes was the development of sets outside the set of integers that behave as integers.
In the simplest example, we can ask whether the number -2 is simple. The question may seem meaningless, but it prompts us to express in words the unique role of unity among integers. The most unusual aspect of 1 is that its reciprocal value is also an integer (the reciprocal value of x
is a number, which when multiplied by x
gives 1. The number 2 has the converse value 1/2 is included in the set of rational or real numbers, but is not an integer: 1/2 × 2 = 1). The number 1 turned out to be its own inverse. No other positive integer has an inverse of the set of integers. The number with the reverse value is called a reversible element
. The number −1 is also a reversible element in a set of integers: again, it is a reversible element for itself. We do not consider reversible elements as simple or composite, because you can multiply them by some other reversible elements without much change. Then we can assume that the number -2 is not so different from 2; in terms of multiplication. If 2 is simple, then −2 should be the same.
I diligently avoided the definition of simple
in the previous paragraph because of the unfortunate fact that this definition is not suitable for these large sets! That is, it is a bit illogical, and I would choose another. For positive integers, each prime number p
has two properties:
It cannot be written as the product of two integers, none of which is a reversible element.
If the product m × n
is divisible by p
, then m
must be divisible by p
(for example, m
= 10, n
= 6, and p
The first of these properties is how we could characterize primes, but, unfortunately, it’s an irreducible element
. The second property is the simple element
. In the case of natural numbers, of course, the same numbers satisfy both properties. But this does not apply to every interesting set of numbers.
As an example, consider a set of numbers of the form a + b√ − 5
or a + ib√5
, where a
are integers, and i
is the square root of −1. If you multiply the numbers 1 + √ − 5 and 1-√ − 5, then you will get 6. Of course, you will also get 6 if you multiply 2 and 3, which are also in this set of numbers with b = 0
. Each of the numbers 2, 3, 1 + √ − 5, and 1 − √ − 5 cannot be represented as a product of numbers that are not reversible elements (if you do not take my word for it, it is not too difficult to verify). But the product (1 + √ − 5) (1 − √ − 5) is divisible by 2, and 2 is not divisible by either 1 + √ − 5 or 1 − √ − 5 (again, you can check if you don’t believe me ). Thus, 2 is an irreducible element, but not simple. In this set of numbers 6 you can decompose into irreducible elements in two different ways.
The above number, which mathematicians can call Z [√-5], contains two reversible elements: 1 and −1. But there are similar sets of numbers with an infinite number of reversible elements. Since such sets have become objects of study, it makes sense to clearly distinguish between definitions of reversible, irreducible and simple elements. In particular, if there are sets of numbers with an infinite number of reversible elements, it becomes increasingly difficult to understand what we mean by unique factorization of numbers, if it is not specified that reversible elements cannot be simple. Although I am not a historian of mathematics and I am not engaged in number theory and I would like to read more about how exactly this process took place, but I think this is one of the reasons Caldwell and Xiong consider the reason for the exclusion of 1 from prime numbers.
As is often the case, my initial neat and concise answer to the question of why everything is as it is, eventually became only part of the problem.Thanks to my friend for asking a question and helping me learn more about the complex history of simplicity.