6th Grade Math Textbook, Fundamentals

 &KDSWHU

13-3

Remember:Only one counterexample

is necessary to prove that a conjecture

is false.

Conjectures and Counterexamples

Objective To verify a conjectures or provide a counterexample• To form a conjecture and

prove that it is false or demonstrate its truth

Rory and his group must prove or disprove

the following conjecture:

The square of a number is alwaysgreater than
the original number.

Is this conjecture true or false?

A conjecture is a statement that appears to be true

but that has not yet formally been proven to be true.

It is usually made based on observations of patterns

and what you predict will happen for future cases.

A is a case that proves that the

conjecture is false. Only one counterexample

is needed to prove a conjecture false.

To see if the conjecture is true or false, test various cases.

Let na number

Case 1:The original number, n, is greater than 1.

If n2, then n^2 4. 4  2

If n9.1, then n^2 82.81. 82.81 9.1

If n20 , then n^2 ( (^20) )
2
420. 420  20
Case 1 does not lead to a counterexample.
Case 2:The original number, n, is less than 0.
If n1, then n^2 1. 1  1
If n, then n^2 ( )
2
. 
Case 2 does not lead to a counterexample.
Case 3: The original number, n, is greater than or equal to 0 and
less than or equal to 1.
If n , then n^2 ()
2
.  Disproved. The square is less than the original number.
So when nis greater than or equal to 0 and less than
or equal to 1, the conjecture is false.
Case 3 is a counterexample to the conjecture.

To test a conjecture, you need to think of all possibilities (cases).

If a conjecture is false, you need to provide at least one counterexample.

If a conjecture is true, you should explain why it is true for all cases.

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