What is a undefined term in math?

An expression in mathematics which does not have meaning and so which is not assigned an interpretation. For example, division by zero is undefined in the field of real numbers.

Geometry is a very broad subject. What makes Geometry much different from algebra is that we aren't dealing with numerous mathematical operations. In Geometry, we are honing our reasoning and critical thinking skills. We are focused more on why something is the way it is rather than focusing on numerical calculations. That's not to say that we won't be performing any calculations in the subject, because we will be needing actual values, we'll need to use certain statements to solve problems and actually prove different scenarios. This is where our reasoning and critical thinking skills come in. Calculus relies heavily on these two skills. This won't be easy.

Before we go too deep into the actual Geometry, we need to introduce some terms that we'll use in order to help solve problems and prove scenarios. Think of these terms as the building blocks of Geometry. Without these, we're stuck.

Undefined Term

An undefined term is a term that can't be defined so easily. There really isn't a definition to define such terms. Consider the word "the." We use the word "the" all of the time, but do we really know how to define the word "the?" "Am" is another word that can't be defined so easily. We can describe these terms, but we can't provide an actual definition. There are terms in Geometry that can't be defined so easily. We'll go over those later.

Defined Term

A defined term is, simply put, a term that has some sort of definition. Unlike "the" and "am", we can put a definition to the word "she." "She" just is defined as a term that represents us acknowledging that someone is female. Simple, right? In Geometry, we can use undefined terms to define a term.

Postulate

I like to call these statements the "well, duh" statements. These statements are "facts" that are accepted without proof. We can't approach proving these statements using conventional means. These statements are so basic that we can't use true technical jargon to explain them. However, if we use a little bit of critical thinking, we can use undefined and defined terms to help support a postulate.

Theorem

A theorem is completely opposite of a postulate. Theorems can be proven. We'll use different undefined and defined terms, as well as postulates to prove a certain statement. If a statement can be proven, then we have a theorem. 

There is another term called a corollary, which is just a supplement to a theorem, but we'll get into corollaries later.

Those are the terms we'll be using throughout the subject. There's a lot of them, and they will likely some at you all at once. Focus and understand each statement.

Let's start with a few undefined and defined terms. 

Simple Undefined Terms:

Point: A point just represents someone's position. That's all a point is. Where are you at? That's a point.

Line: A line is just a set of points that extends to one direction. It also works backwards. If we have points going due left, then we can also have points that go the opposite direction, straight right. Lines are indefinite, by the way. Later in the section, titles "Initial Postulates and Theorems You Should Know", we'll introduce a very important theorem that involve points and theorems.

Plane: A plane is the same as a line, except you have points everywhere, and the points don't go in one specific direction. It's a whole field of points.

Simple Defined Terms:

Line Segment: A line segment is just part of a line. Remember above when I said that lines are indefinite, and that they keep going and going? Line segments stop somewhere in both directions.

Ray: A ray is like a line, but the line takes off in one direction to infinity while the other side is like a line segment. The end of the line is called the endpoint.

Opposite Rays: Two rays that share the same endpoint that take off in opposite directions. The rays would create a line.

Angle: Two rays that share the same endpoint, however, the rays take off in different directions. The area in the middle of the two rays is the angle measure.

Ready to test your knowledge? Try a Problem Set.

Problem Sets

1. Give an example of an undefined term. What makes this term hard to define?

2. Give an example of a defined term. Give a definition of that term.

3. Give an example of a postulate. What makes your postulate hard to prove?

4. Give an example of a theorem. How did you  prove your theorem?

5. Draw a point.

6. Draw a line.

7. Draw a line segment.

8. Draw a ray.

9. Draw a set of opposite rays.

10. Draw an angle.

Video Definition Point Line Plane Set Examples

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."

Undefined Terms Definition

In 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.

Point

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.

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.

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.

Set

A 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:

• Physical objects like angles, rays, triangles, or circles
• Numbers, like all positive even integers; proper fractions; or decimals smaller than 0.001
• Other sets, like the set of all even numbers and the set of all multiples of five; the set of all acute angles and the set of all angles less than 15°

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:

• { 0.1, 0.2, 0.3 } for a set of three decimal numbers
• { 1, 2, 4, 8, 16 … } for the infinite set of powers of two
• {acute angles, obtuse angles, reflex angles, straight angles} for a set of angles found in plane geometry
• { 1, 2, 3 … } for the infinite set of whole positive integers
• { A, B, C … X, Y, Z } for the set of English alphabet letters

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:

• { 0.2, 0.1, 0.3 } or like this { 0.3, 0.2, 0.1 }
• { 4, 8, 16, 1, 2 … } or like this { 16, 2, 4, 1, 8 ... }

Undefined Terms Examples

Look on the floor of your bedroom. Mentally arrange a set of what you see. It might look like this:

• { socks, gym shorts, left shoe, geometry textbook }

Look at a calendar. Mentally (or, better, jot down) a set of Saturday and Sunday dates. It might look like this:

• { 13, 14, 6, 20, 7, 27, 21, 28 }

The order does not matter, but the set might be easier to work with in order from least to greatest:

• { 6, 7, 13, 14, 20, 21, 27, 28 }

Lesson Summary

Now 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.

Next Lesson:

Geometric Figures