Mathematics is a rigorous science with its own unique way of thinking and logical reasoning system.
Counterexamples can help you master the basic knowledge of mathematics more deeply, in many ways! Observe and think about problems from multiple angles, improve your mathematics accomplishment and cultivate your scientific research ability.
The application of counterexamples can strengthen the rigor of reasoning, cultivate the criticism of thinking, develop reverse thinking and divergent thinking, and comprehensively improve the ability to solve problems.
Usually, finding a counterexample requires more imagination and creativity than finding a proof. The process of giving counterexamples is the process of gradually improving our mathematical ability.
Give two examples:
The arithmetic square root of a positive number 1. must be less than this positive number.
Traditionally, we think that the arithmetic square root of 9 is 3 and the arithmetic square root of 36 is 6 ... It seems that the above statement is correct. But if you give a decimal example, for example, the arithmetic square root of 0.09 is 0.3, and 0.3 > 0.09, this counterexample shows that the above statement is actually wrong.
2. A set of quadrilaterals with parallel opposite sides and another set of quadrilaterals with equal opposite sides are parallelograms.
In fact, this is also a false proposition. It is best to use "isosceles trapezoid" as a counterexample.
At all times and in all countries, some seemingly correct propositions in mathematics are solved by counterexamples. In the history of mathematics, appropriate counterexamples often promote the development of mathematics, and there are countless such examples. You can check the history of mathematics.