For an ordered pair of real numbers, if both of its two entries are contained in the open interval $(0,1)$, it is called an Ordered Pair of Normalized real numbers (OPN). In this paper, a comprehensive theory is presented, which provides a novel mathematical tool for dealing with OPNs. Specifically, the eight arithmetic operations and trigonometric functions of OPNs are given. The set of OPNs is closed under these operations. Later, a total order defined on the set of OPNs is shown, and Cauchy–Schwarz inequality of OPNs’ version is proved to be true. Finally, a decision making example is given to demonstrate the feasibility and rationality of the OPNs theory proposed.

对于有序实数对，如果其两个条目都包含在开放时间间隔中 $（0，\mathrm{1个}）$，称为标准化实数有序对（OPN）。本文提出了一种综合理论，为处理OPN提供了一种新颖的数学工具。具体来说，给出了OPN的八个算术运算和三角函数。在这些操作下，这组OPN被关闭。后来，显示了在OPN集合上定义的总顺序，并且证明了OPN版本的Cauchy–Schwarz不等式是正确的。最后，给出一个决策实例来说明所提出的OPNs理论的可行性和合理性。