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

Real Numbers: A Summary

For the past two years, we have talked a lot about real numbers.  We have talked about integers and its operations (addition, subtraction, multiplication, and division), we have discussed about rational and irrational numbers, and we have talked about their properties, structure, and wonders. In this post, we are going to summarize what we have learned about them.

Figure 1 - The Number Line

The set of real numbers is the collection of all rational and irrational numbers. By convention, real numbers are represented by a line infinitely long where the positive real numbers are situated at the right hand side of 0, while the negative are at the left hand side. It is also important to note that for each point on the number line, there exists a corresponding real number equivalent to it, and for each real number, there is a corresponding point on the line that represents it. » Read more

Military drills and the closure property of real numbers

In military drills, we are familiar with the commands that let the soldiers face or turn to certain directions. Left face (sometimes called left turn), right face, and about face are probably the most common commands used. If we are facing north, a left face would mean turning left 90 degrees, which means facing west. In the following discussion, we will agree that our starting position is  facing north. We will call this position, the standard position.

Let us represent the turns with letters:A for about face, R for right face, and L for left face. Notice that whatever combination of turns we do, LR or A, the result is confined to the four directions. An R followed by an A is equivalent to an L (facing west) with respect to the standard position. Likewise an L followed by another L is equivalent to A.We will to denote our starting position P as reference; it is  not turn command, so if the soldier is not facing north (see second figure), a  P will just mean that the soldiers remain in their current position. » Read more

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