## The Number of Points on Two Line Segments

We say that a set is countably infinite if we can pair the elements with set of counting numbers 1, 2, 3, and so on. Believe it or not, the number of positive integers and the number of integers (both negative and positive including 0) have the same number of elements. It is because we can pair them in a one-to-one correspondence such as shown in the below.

As shown on the table, if we continue indefinitely, we know that we can pair each counting number with an integer in a one-to-one correspondence without missing any element.

Using this concept, we show intuitively that the number of points on two line segments is equal even if they have different lengths. We can do this by showing that for each point on segment $\overline{AB}$, there is a corresponding point on segment $\overline{CD}$. » Read more

## Are all infinities created equal?

A one-minute explanation that not all infinities are equal. The video shows that the number of real numbers from 0 to 1 is  greater than the number of all natural numbers 1, 2, 3, 4, and so on.

If the video is too fast or too short, you can read a more detailed explanation in  Counting the Real Numbers.

## Is 0 a natural number?

When I was still teaching,  several of my students asked me this: Is 0 a natural number?

Short Answer

It depends on the convention. In some references, the set of natural numbers includes 0. In other references, the set of natural numbers does not include 0.

Long Answer

When we define a particular mathematical concept, we come to some sort-of agreement. For example, we might want to call the sequence  1, 4, 7, 10, … “cool numbers.” If all of us agree on this, we will have no problem.  However, other group of uncool guys might give it another name. Or, they might keep the name, but change the sequence.  This was what happened to the set of  natural numbers. » Read more

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