When it comes to the world of mathematics and a world with zero, we use the terms undefined and indeterminate often to describe the solution to many problems that are otherwise unsolvable. However, when do we use each one? The difference is crucial, so this is designed to help you understand it better.

First let us start with a Kahoot! Activity:

Indeterminate, Undefined, or Zero?

Now, although we understand what zero is, we can have a hard time determining when something is undefined compared to when it is indeterminate. What do each of them mean? Well, let us take a look at the definition of each:

Indeterminate:having a quantity with no definite or definable value.

Undefined:having a quantity that is not defined or does not exist.

With the examples in the Kahoot above, we have a combination of each of them. Notice how almost all of them included zero. Let us take some time to understand each question:

Slope of y=2
When y=2, this means that we have a graph of the following manner:



To determine the slope of the line y=2, we usually think of this as being rise over run, or rise/run. With this in mind, we can take any two points on this line, such as (0, 2) and (2, 2). Let us look at the rise between these two point. Do we increase on the y-axis? No! Therefore, the rise is zero. The run on the other hand is the distance between the x values of the two points. So we run from 0 to 2, or run two. Therefore, our slope is 0/2 which is 0. So the slope of any horizontal line is zero, not undefined or indeterminate.

Slope of x=2

When x=2, this means that we have a graph of the following manner:



Similarly, to determine the slope, we need to determine the proportion of the rise to run of the line. With this in mind, let us take the points (2, 0) and (2,5). The rise of this is easy, as we take the difference of the y-values. This gives us a difference between zero and five which is five. Therefore, our rise is 5. Our run, is the difference between the x-values. The difference between two and two is 0. Therefore, our rise over our run is 5/0. This makes our slope undefined because anything divided by 0 is undefined.
The explanations of why this is undefined and the other questions can be found on the page Why is Zero Evil?

The big difference between undefined and indeterminate is the relationship between zero and infinity. When something is undefined, this means that there are no solutions. However, when something in indeterminate, this means that there are infinitely many solutions to the question. Again, zero and infinity are the root to a lot of mathematics, including the difference between these two words.