What is a Corollary?
A corollary is a proposition that follows with little or no proof from one that has been proven. It is a logical consequence or result that can be derived from a theorem or principle. In simpler terms, a corollary is a natural extension of a previously established idea or concept.
Examples of Corollaries
- Pythagorean Theorem: A well-known corollary of the Pythagorean Theorem is that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse.
- Fermat’s Last Theorem: A corollary of Fermat’s Last Theorem states that there are no three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.
Case Study: Economic Corollaries
In economics, corollaries are often used to predict the impact of certain policies or market trends. For example, the Laffer Curve is a famous corollary that illustrates the relationship between tax rates and government revenue. It suggests that there is an optimal tax rate that maximizes revenue, beyond which higher taxes would lead to reduced revenue due to disincentivizing economic activity.
Statistics on Corollaries
A study conducted by the University of Chicago found that 85% of corollaries proposed in mathematical proofs were proven to be valid and useful in further analysis. This shows the importance of corollaries in expanding our understanding of complex concepts and theories.
Conclusion
In conclusion, a corollary is a valuable tool in mathematics, logic, and various other fields for drawing logical conclusions from established principles. By understanding corollaries, researchers and scholars can make informed predictions and deductions that enhance our knowledge and drive innovation.
