浅谈数学推广——以若干不等式为例

Brief Exploration on Mathematical Generalization—— Take Several Inequalities as an Example

推广是数学知识发展的一种基本方式,也是数学家研究和创造的一种基本手段和策略,没有推广就没有数学的发展,推广是引导数学发现的一种重要手段.本文以两道习题的解答与推广为例,展示了数学推广的基本形式与过程.并提出:好的推广应具有满意的广泛性,而又保留了有趣的实例的味道.好的数学推广是一项综合性较强、难度较大的创造性研究工作,需要具备一定的经验和运用科学的思维方法.

Generalization is a kind of basic way of mathematical knowledge development and is also a common approach and strategy to research and even create mathematics by mathematicians. Without the generalization, there will be no development of mathematics. Generalization is an important means to guide the mathematical discovery. It is demonstrated in this paper that the basic form and process of the mathematical generalization based on the generalization of two mathematics problems. And the excellent generalization with satisfactory generality and full of the character of living examples. All in all, a perfect mathematical generalization is a kind of creative research work with stronger comprehensiveness and greater difficulty, thus such a job needs a certain experience as well as a scientific method of thinking.