positive rational The fact that the prime numbers forms a basis for multiplication of these numbers follows from the fundamental theorem of arithmetic, according to which every positive integer can be factorized uniquely into the product of finitely many primes or their inverses. If is a positive rational number expressed in simplest terms, then can be expressed as a finite combination of the primes appearing in the factorizations of The number of copies of each prime to use in this combination is its exponent in the factorization of , or the negation of its exponent in the factorization The polynomials of a single with integer coefficients, form a free abelian group under polynomial addition, with the powers of as a basis. As an abstract group, this is the same as (an isomorphic group to) the multiplicative group of positive rational numbers. One way to map these two groups to each other, showing that they are isomorphic, is to reinterpret the exponent of the prime number in the multiplicative group of the rationals as instead giving the coefficient of in the corresponding polynomial, or vice versa. For instance the rational number has exponents of for the first three prime numbers and would correspond in this way to the polynomial having the same coefficients for its constant, linear, and quadratic terms. Because these mappings merely reinterpret the same numbers, they define a bijection between the elements of the two groups. And because the group operation of multiplying positive rationals acts additively on the exponents of the prime numbers, in the same way that the group operation of adding polynomials acts on the coefficients of the polynomials, these maps preserve the group structure; they are homomorphisms. A bijective homomorphism is called an isomorphism, and its existence demonstrates that these two groups have the same properties. Although the representation of each group element in terms of a given basis is unique, a free abelian group has generally more than one basis, and different bases will generally result in different representations of its elements. For example, if one replaces any element of a basis by its inverse, one gets another basis. As a more elaborated example, the two-dimensional integer lattice consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis For this basis, the element can be written where 'multiplication' is defined so