Inspired by reflection transformations and rotation transformations in plane geometry and dictionary orders in lattice theory, in this article, we explore, through exposing the possible motivation of definition of a defined linear order <inline-formula><tex-math notation="LaTeX">$\leqslant$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">${\mathbb {R}}$</tex-math></inline-formula> (the set of all interval numbers), how to find an easy-to-understand way to extend the ordinary linear order on <inline-formula><tex-math notation="LaTeX">$R$</tex-math></inline-formula> (the set of all real numbers) to a linear order on <inline-formula><tex-math notation="LaTeX">${\mathbb {R}}$</tex-math></inline-formula>. Theory and application aspects are also studied. For an arbitrary cardinal number <inline-formula><tex-math notation="LaTeX">$\kappa$</tex-math></inline-formula> (<inline-formula><tex-math notation="LaTeX">$\kappa \not=0$</tex-math></inline-formula>), we define seven operations and the point-wise order induced by <inline-formula><tex-math notation="LaTeX">$\leqslant$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">${\mathbb {R}}^\kappa$</tex-math></inline-formula> (the set of all sequences indexed by <inline-formula><tex-math notation="LaTeX">$\kappa$</tex-math></inline-formula> and consisting of interval numbers). It is found that <inline-formula><tex-math notation="LaTeX">$({\mathbb {R}},\leqslant)$</tex-math></inline-formula>, much like <inline-formula><tex-math notation="LaTeX">$R$</tex-math></inline-formula> (with the ordinary linear order), has some desirable properties; particularly, <inline-formula><tex-math notation="LaTeX">$\leqslant$</tex-math></inline-formula> can be used to describe the corrected degree of possibility of an interval number is smaller than another. Generalizing the discovery process of <inline-formula><tex-math notation="LaTeX">$\leqslant$</tex-math></inline-formula>, we provide an understandable way to extend the ordinary linear order on <inline-formula><tex-math notation="LaTeX">$R$</tex-math></inline-formula> to a linear order on <inline-formula><tex-math notation="LaTeX">${\mathbb {R}}$</tex-math></inline-formula> (even on the plane <inline-formula><tex-math notation="LaTeX">$R^2$</tex-math></inline-formula>), which meets some specific requirement, and we give a clear description on the relation between the admissible orders and those we propose. Those urge us to take a look at the possible applications of <inline-formula><tex-math notation="LaTeX">$\leqslant$</tex-math></inline-formula> (a delegation of linear orders on <inline-formula><tex-math notation="LaTeX">${\mathbb {R}}$</tex-math></inline-formula>). Precisely, to propose an ordering method (based on this linear order and the matched metric and weighted averaging aggregation operator), which can be used to deal with very complicated generalized interval-valued preference hesitant relations in group decision-making problems.