Natural Numbers
Natural numbers, denoted by the symbol ℕ, are the numbers used to count. As such, they represent the most immediate connection that a mathematical abstraction, the number, has with the objects of the physical world
They are used to count, but do they start from 1 or 0?
Precisely because of their close connection with material things that can be counted (goods, animals, people, etc.), some mathematicians hold that the series of natural numbers must begin with 1. Others think, however, that it must start with 0, partly because this choice allows a rigorous formalization of the concept of natural numbers, such as that provided by the so-called Peano axioms.
Because of this ambiguity, the symbol used to denote the set of natural numbers is often accompanied by a clarifying character in subscript or superscript:
- ℕ₀ indicates that the series starts from 0; in this case, the set of natural numbers corresponds to the set of non-negative integers (i.e., all positive numbers plus 0);
- ℕ₁, ℕ₊, ℕ* indicate that 0 is excluded and that the series starts from 1; in this case, the natural numbers correspond to the set of positive integers.
Natural numbers form an ordered and infinite set
Whether they begin with 0 or with 1, natural numbers form an ordered set. Each of them has a successor, that is, a just larger natural number. And except the first of the series (0 or 1, depending on the case), each has a predecessor: a just minor natural number. So, for example, 72 is the successor of 71 and the predecessor of 73.
Moreover, given any two natural numbers, it is always possible to establish which of the two is greater and which is smaller. For example, given the numbers 13 and 17, 13 is smaller than 17. Symmetrically, 17 is greater than 13. This relationship is expressed mathematically with the symbols 13 < 17 or 17 > 13.
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The set of natural numbers is infinite:
- ℕ₀={0,1,2,…}
- ℕ₁={1,2,3,…}
In set notation, the three dots inside the curly brackets indicate precisely that the enumeration of natural numbers can continue indefinitely. It means that there is no natural number, however large it may be, which does not have a successor. This property can also be seen in the following way: every natural number, except the first in the series, is the successor of another natural number.
Algebraic properties
Natural numbers have some properties related to addition and multiplication:
- Closure of the set. For any natural number a and b, the sum 𝑎+𝑏 and the product 𝑎×𝑏 are themselves natural numbers.
- Associative property. For any natural number a, b and c, 𝑎+(𝑏+𝑐)=(𝑎+𝑏)+𝑐 and 𝑎×(𝑏×𝑐)=(𝑎×𝑏)×𝑐.
- Commutative property. For any natural number a and b, 𝑎+𝑏=𝑏+𝑎 and 𝑎×𝑏=𝑏×𝑎.
- Distributive property of multiplication over addition. For any natural number a, b and c, 𝑎×(𝑏+𝑐)=(𝑎×𝑏)+(𝑎×𝑐).
- Multiplication by zero. If a and b are natural numbers such that 𝑎×𝑏=0, then either 𝑎=0 or 𝑏=0 or a and b are both 0.
