A certain famous, fictional spy always describes his favorite beverage as "shaken and not stirred." Four concepts in geometry can best be thought of as "described and not defined." Show What you'll learn:After working your way through this lesson and video, you will be able to:
Undefined Terms DefinitionIn all branches of mathematics, some fundamental pieces cannot be defined, because they are used to define other, more complex pieces. In geometry, three undefined terms are the underpinnings of Euclidean geometry: A fourth undefined term, set, is used in both geometry and set theory. Even though these four terms are undefined, they can still be described. Mathematicians use descriptions of these four terms and work up from them, creating entire worlds of ideas like angles, polygons, Platonic solids, Cartesian graphs, and more. Simply because these terms are formally undefined does not mean they are any less useful or valid than other terms that emerge from them. These four undefined terms are used extensively in theorems, proofs, and defining other words. A point in geometry is described (but not defined) as a dimensionless location in space. A point has no width, depth, length, thickness -- no dimension at all. It is named with a capital letter: Point A; Point B; and so on. Points in geometry are more like signal buoys on the vast, infinite ocean of geometric space than they are actual things. They tell you where a spot is, but are not the spot itself (even though we show them with a dot). When you place two points on a plane, you can create a line. A line is described (not defined) as the set of all collinear points between and extending beyond two given points. A line goes out infinitely past both points, but in geometry we symbolize this by drawing a short line segment, putting arrowheads on either end, and labeling two points on it. The line is then identified by those two points. It can also be identified with a lowercase letter. With only three points, you can create three different lines, and you can also describe a plane. A plane is described as a flat surface with infinite length and width, but no thickness. It cannot be defined. A plane is formed by three points. For every three points in space, a unique plane exists. A symbol of a plane in geometry is usually a trapezoid, to appear three-dimensional and understood to be infinitely wide and long. A single capital letter, or it can be named by three points drawn on it. Modeling a plane in everyday life is tricky. Nothing will accurately substitute for a plane, because even the thinnest piece of paper, cookie sheet, or playing card still has some thickness. Also, all of these objects end abruptly at their edges. Planes do not end, and they have no thickness. SetA set can be described as a collection of objects, in no particular order, that you are studying or mathematically manipulating. Sets can be all these things:
In geometry, we use sets to group numbers or items together to form a single unit, like all the triangles on a plane or all the straight angles on a coordinate grid. Sets are shown by using braces, { } on either side of the set:
A set does not need to have a limit. The ellipsis ( … ) can indicate more terms between the start and end of the series, or it can indicate that the set between the braces continues on, infinitely. A set does not need to ordered, like an array. You can write the first two sets shown above like this:
Undefined Terms ExamplesLook on the floor of your bedroom. Mentally arrange a set of what you see. It might look like this:
Look at a calendar. Mentally (or, better, jot down) a set of Saturday and Sunday dates. It might look like this:
The order does not matter, but the set might be easier to work with in order from least to greatest:
Lesson SummaryNow that you have navigated your way through this lesson, you are able to identify and describe three undefined terms (point, line, and plane) that form the foundation of Euclidean geometry. You can also identify and describe the undefined term, set, used in geometry and set theory. Is an line a defined term?In geometry, we have three undefined terms, and those are a point, a line, and a plane. These are coined as undefined, because we cannot define them using other geometrical terms.
Is line segment A undefined term?Line segment: A line segment which extends indefinitely in both directions gives a line. Other term involved is line. We keep the line as undefined term. The only thing is that we can represent in intuitively or explain it with the help of physical model.
Why is a line considered undefined?A line has a slope that is considered undefined when it does not have any horizontal run. It will be a straight vertical line.
What terms are undefined?In geometry, formal definitions are formed using other defined words or terms. There are, however, three words in geometry that are not formally defined. These words are point, line and plane, and are referred to as the "three undefined terms of geometry".
|