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We give a new classification of monomial curves in the affine 4-dimensional space $A^4(K)$. It relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient conditions for the defining ideal of a monomial curve in $A^4(K)$ to have a unique minimal system of binomial generators.