In a geometry course, once you decide whether a conjecture appears to be true, you need to prove (or disprove) it. Fortunately, there are reliable procedures that you can use for proving conjectures. These procedures are generally known as geometric proofs.
In order to prove that a conjecture is true, you must show it to be true for all possible situations or examples. Suppose you were asked to prove or disprove the following statement.
All integer multiples of 5 end in either a 0 or 5 (0 or 5 in the unit's place).
What steps might you take to show that this statement is true? Look carefully at the approach taken on each of these slides.
|
Does it appear to be true? Let's pick several multiples of 5 and see if this conjecture appears to be true.
The conjecture All integer multiples of 5 end in either a 0 or 5 (0 or 5 in the unit's place) appears to be a true statement. The appearance of "truthiness" is not the same as proof, however. We have not shown that the statement is true for all possible multiples of 5. What about even Multiples of 5? All integers are either even or odd. Therefore, we must prove that our conjecture is true for even and odd multiples of 5. An even integer can be written as 2x, where x is any integer. To show that even multiples of 5 end in a zero, we could write: 5×(even number) = 5(2x) = 10x Note that final answer will always end in a 0. What about odd Multiples of 5? An odd integer can be written as 2x + 1, where x is any integer. To show that odd multiples of 5 end in 5, we could write: 5×(odd number) = 5(2x + 1) = 10x + 5 Note that any answer to this equation will end in 5. Conjecture Proved Step by step, we proved that both even and odd multiples of 5 end in either a 0 or a 5. Since all integers are either odd or even, we have shown that all multiples of 5 end in either a 0 or a 5.
The evidence we used here to prove the conjecture was not a list of examples, as in an informal proof, but a set of logical statements. That makes this process an example of a formal proof. |
 |
Question
If a conjecture appears to be true, what must be done in order to prove it is true?
