# Natural number

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One of the fundamental concepts in mathematics. Natural numbers may be interpreted as the cardinal numbers of non-empty finite sets. The set $\mathbf{N} = \{1,2,3,\ldots\}$ of all natural numbers, together with the operations of addition $({+})$ and multiplication $({\times})$, forms the natural number system $(\mathbf{N},{+},{\times},1)$. In this system, both binary operations are associative and commutative and satisfy the distributive law; 1 is the neutral element for multiplication, i.e. $a \times 1 = a = 1 \times a$ for any natural number $a$; there is no neutral element for addition, and, moreover, $a + b \neq a$ for any natural numbers $a,b$. Finally, the following condition, known as the principle of mathematical induction, is satisfied. Any subset of $\mathbf{N}$ that contains 1 and, together with any element $a$ also contains the sum $a+1$, is necessarily the whole of $\mathbf{N}$. See Natural sequence; Arithmetic, formal.