Bringing relations and variables together: Mathematical Statements

The greek mathematicians used both the concept of general descriptions or definitions (like the general definition of what a cousin is in section 3.2) and the concept of variables (like "The Candy in My Hand" in section 3.4) to record very general facts about their geometric shapes.

Here's an example of something a mathematician might say (Don't expect to immediately understand what the statement means. We will get to that shortly).

"Consider the line, A, that is divided into two line segments, B and C, at a point P. It is always true that the length of A can be found by adding together the length of B and the length of C."

(if you are feeling confused or intimidated by this statement, make a note of your feeling, and keep reading).

Let's start by compare this to our general definition of a cousin from section 3.2

"A person's cousin is the child of one of their parents' siblings"


The mathematical definition is similar to the cousin definition in that, like the cousin definition, it is very general and abstract. It says "Consider a line, A."
This means that the definition isn't necessarily talking about my line or your line. it's just talking about any line out there that we care to pick out. Like the cousin definition, this makes it harder to understand than if we were using a concrete example. The advantage is that it's easier to apply this statement to many different situations. We can apply this to any line we might come across.

Take a look at the statement again:

"Consider the line, A, that is divided into two line segments, B and C, at a point P. It is always true that the length of A can be found by adding together the length of B and the length of C."


Let's also compare this statement to our discussion of the bag of candy in section 3.4. In that discussion, even though we didn't know which candy we had picked as "The Candy in My Hand", we could still come up with some true statements about it- for example, that it's sweet and I can eat it.

Similarly, in the mathematical definition above, even though we don't know which particular line in the mathematical world we are talking about, we can still come up with a true statement about it. In this case, the second part of the mathematical statement is "It is always true that the length of A can be found by adding together the length of B and the length of C." This means that if we break the line A into two pieces, piece B and piece C, the length of the original line, A, can be found by adding together the length of the two pieces, B and C.

Exercise: Try drawing a picture to help you get the meaning of the statement (once you've tried this, see Sample Diagram below).

Reread the statement one more time, and compare it to the meaning just provided.

"Consider the line, A, that is divided into two line segments, B and C, at a point P. It is always true that the length of A can be found by adding together the length of B and the length of C."

How do you feel about the statment now?

 

 

 

 

Answers

Here’s a sample diagram that a person might draw if they are trying to figure out what this statement means:

AppleMark
It is always true that the length of A can be found by adding together the length of B and the length of C.
 

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Copyright Jen Schellinck, 2006