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谈了本刊1984年第二期《为什么复数不能比较大小?》一文,感到还可作些补充。先看看实数集中大小的概念有些什么基本性质,显然它应满足: 1) 自反性:a≥a; 2) 传递性:若口≥b,b≥c,则a≥c; 3) 非对称性:若a≥b,b≥a同时成立,则a=b。考虑到要与线性运算相适应,故还有 4) 加法保序性:若a_1≥a_2,b_1≥b_2,则a_1+b_1≥a_2+b_2; 5) 乘非负实数保序性:若a≥b,λ为非负实数,则λa≥λb。
After talking about the article in the second issue of 1984, “Why can’t a plural number be compared?”, I feel I can add some more. First look at the basic nature of the concept of the size of the real number, it is clear that it should meet: 1) Reflexivity: a ≥ a; 2) Transmissibility: If ≥ b, b ≥ c, then a ≥ c; 3) Non Symmetry: If a≥b, b≥a holds at the same time, then a=b. Taking into account the need to adapt to linear operations, so there is 4) Additive ordering: if a_1 ≥ a_2, b_1 ≥ b_2, then a_1 + b_1 ≥ a_2+b_2; 5) Multiply non-negative real numbers ordering: if a ≥ b, λ is a non-negative real number, then λa ≥ λb.