Inductive Reasoning
Consider this list:
- Houses have doors
- Office buildings have doors
- Schools have doors
- Hospitals have doors
What conclusion can you draw based on this list of facts? (Hint: Think about into what general category houses, office buildings, schools, and hospitals can be classified, then think about what is common about them.)
| One of the common methods used in geometric proofs is called inductive reasoning. Inductive reasoning is a way of drawing a general conclusion from special cases by examining and exploring their common properties. This type of reasoning is also called mathematical investigation. |
By this method, we reach general conclusions using known facts, similar clues, and common patterns. In other words, we take special cases and generalize the results so that they are valid for all similar cases.
Conjecture
| A conclusion developed by generalizing an idea using the inductive method is called a conjecture. |
Conjectures are usually valid unless we can provide a typical case that disproves it.
For example, given the list:
- pigeons fly
- parrots fly
- robins fly
- crows fly
- hawks fly
You may make the conjecture that all birds fly. Is that always true? While it is true that most birds fly, penguins do not. Penguins would be an example of a typical case that disproves our conjecture. This is called a counterexample. You will learn more about these examples in a few pages.
As another example, using different right triangles we can develop a conjecture that states:
In a right triangle, the square of hypotenuse is equal to the sum of the square of it legs.
This conjecture about right triangles is valid because we can't find any right triangle whose sides do not comply with this statement.
Do you know the name of the theorem described above?
