About this video
Geometry builds new ideas from what we already know, starting with definitions. Some basic ideas, like point and line, are left as undefined terms because defining them would be circular or too complicated. These undefined terms are then used to define new terms, such as a line segment.
You'll be able to
- explain what an undefined term is and name point and line as examples.
- explain what makes a definition circular.
- define a line segment using undefined terms.
Key words
- undefined term
- A basic word, like point or line, that we don't formally define but assume everyone understands intuitively.
- circular definition
- A definition that ends up using the word being defined, for example defining point using location and location using point.
- point
- A location in space with no size, area, volume or dimensions.
- line
- A straight path with no thickness that extends forever in both directions.
- line segment
- Part of a line with two distinct endpoints, including all the points on the line between them.
Transcript
- [Instructor] One thing we will find out repeatedly in geometry, is that it's all about really building from what we know to find out new things. And some of the most basic things we're going to think about are definitions in geometry. But as we'll see, there's certain things that either would be so complex to define or so circular to define, and I'll show you in a second what circular means, that we leave them undefined.
We just assume that people intuitively know what we mean by them. For example, if I were to define what a point is, and I literally, drew a point here, how would you define that? Pause this video and think about it. Well, you might want us call it a location in space.
A location in space. Maybe you can say what it doesn't have. So, it has no size, or area, or volume, maybe shape, maybe dimensions. But is this really a satisfying definition? Well, how would you define a location? Well, a location, maybe you have to define that using a point.
You might be able to say place or something like that. But at the end of the day, you might be referring back to the thing that you're trying to define to begin with, and this is what we call a circular definition. If we define a point as a location and a location as a point, then we are kind of using them to define each other.
Now, there are likely ways to do this without referring to each other, but it might get very, very complicated. So, in geometry, especially in an introductory geometry course, we tend to assume the point is an undefined term. That is intuitive, that we can actually talk about, well, it is a location space.
And we can talk about what it doesn't have and we can see that right over here. But this isn't really a definition, because it could be viewed as a little circular. Similarly, we could think about a line. A line, something like that. And you might define a line as maybe it is straight.
But then we have to think about, well, how do you define straight? Maybe you'd have to use a line somehow to do that. Maybe you could use something about extends infinitely. You might want to think about what does it mean to extend in both directions? In both directions.
You could once again say that it has no thickness. So, this gives you a sense of what a line is, but once again, maybe to define straight, you have to use a line again. Or maybe it just becomes too complicated. So usually, in an introductory geometry course, we assume things like points and lines.
Let me write the word line here, 'cause that's what I was defining the whole time. We assume that these are undefined terms that you can intuitively know what they are, but then we can use these things to define more and more things. For example, something we will use a lot in geometry is a line segment.
And we could use these undefined terms that we can intuitively know what they are to define what a line segment is. We could define this as part of a line. Part of a line. That has two distinct endpoints. That has two. And the reason why I have to say two distinct endpoints, because if there were the same endpoint, then we're just talking about a point here not a line segment.
Two distinct. We could say different end points. End points. And includes all the points in between. And includes all points on line in between. So, this right over here, this is a defined term and we are defining it using some of these undefined terms or these terms that we just assume we intuitively know.
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