Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
Throughout Geometry, students write definitions and test conjectures using counterexamples. When writing definitions, counterexamples are useful because they ensure a complete and unique description of a term. If a counterexample does not exist for a conjecture (an if - then statement), then the conjecture is true.
A key term in geometry is counterexample. the way we define counterexample is an example that makes a definition or conjecture incorrect. The reason why this is important is because if you can find a counterexample for a definition, let's say a teacher asks you to write the definition of a rectangle. If you can find a counterexample to your definition you don't have a good definition. You need to be more specific.
So let's look at a couple of examples and see if we can find a counterexample. The opposite of any number is smaller than the original number. So I guess we could say pick an original number, which I'm going to abbreviate ON. And then we'll have the opposite. So I'm going to write OPP.
So let's say our original number was 2. The opposite of 2 is negative 2. In which case the original number is larger than the opposite number, or the way of stating it here is the opposite is smaller than the original.
But is this always true? If I picked an original number that was negative. Let's say negative 1. The opposite of negative 1 is positive 1. So I'm going to say that the opposite is 1. In which case the opposite is not smaller than the original. It is larger.
So a counter example to the statement could be negative 1. But any negative number will make this statement not true.
Another statement is if a molecule is H20 then it is a liquid. We know that's not true because water could come in two other forms. It could come in ice, which is the solid form. Or it could come in as vapor or steam. So this is your gaseous form. So depending on temperature and pressure, water could also be ice or vapor. So both of these are statements or examples that make this statement incorrect.
Let's look at one more. In this statement we're talking about multiples of a number. If every multiple of 20 is divisible by 4. So I guess we could think about well multiplies of 20, we could say 20, 40, 60, 80 and so on. Those three dots mean and so on. Well, 20 is divisible by 4. 40 is divisible by 4. 60 is divisible by 4. You'd get 15 if you divided by 4 and 80 is divisible by 4.
The reason they're all divisible by 4 is because if we look at 20, if I break it down into its factors, I could write 20 as 4 times 5. So if I multiply 20 by 2, then notice I'm going to have 4 as a factor. If I multiply 20 by 3, then I'm still going to have this 4 as a factor.
No matter what I multiply 20 by, this 4 will be here which means it will be divisible by 4. There is no counter example that will make this statement false, which means this statement is always true.
So the key to a good counter, to a good definition or a conjecture is to make sure you cannot find a counterexample that makes it false.