Much like there are objects with length $0$ (namely, single points), there are real gaps of length zero (namely, the absence of a single point).

Plugging variables into an expression is essential for solving many algebra problems. See how to plug in variable values by watching this tutorial.

The nested interval theorem made me feel the length of a point should be $0$...

While if the length of a point on the real number line is $0$, then I get a contradiction : Supposing we remove the point $0$ on the real number line, then there is a gap there, and the width of the gap is $0$ since the length of a point on the real number line is $0$, however I think the width of the gap being $0$ is equivalent to there being no such gap on the real number line, so this leads to a contradiction.

If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, or equivalently the absolute value of the second number minus the first one. Taking this difference is the process of subtraction.

Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number.

Thus, for example, the length of a line segment between 0 and some other number represents the magnitude of the latter number.

This is actually the way my high school geometry teacher explained this question to me when I had it a long time ago, and it stuck with me. It doesn't contradict any of the other answers given, I should say, but it may help you think about how to interpret these things.

The difference 3-2=3+(-2) on the real number line.

Two numbers can be multiplied as in this example: To multiply 5 × 3, note that this is the same as 5 + 5 + 5, so pick up the length from 0 to 5 and place it to the right of 5, and then pick up that length again and place it to the right of the previous result. This gives a result that is 3 combined lengths of 5 each; since the process ends at 15, we find that 5 × 3 = 15.

In other words: There are just to many of them, so you wont notice if a single one is missing ;).

The midpoint of a line segment is the point midway between the endpoints of the line segment. This tutorial shows you how to take two endpoints and figure out the midpoint of the line segment. Check it out!

While this is not supposed to be a complete, satisfying answer, I would like to draw your attention to an area where your "paradox" has a straightforward interpretation.

Number lines are great ways to represent a group of numbers, and in this tutorial you'll see how to graph a group of numbers on a number line

So, what do you mean by length of a point? Well, whatever it is it seems reasonable to define it to have the following properties: 1) the length of an interval $[a,b]$ is $b-a$; and 2) if the point $p$ is contained in an interval, than the length of the point is $\le $ the length of the interval. We further assume all lengths are measured by non-negative real numbers, so there are no infinitesimals around at all.

I think the width of the gap is 0 is equivalent to that there is no such gap exists on the real number line, this leads to a contradiction.

The measure of a discrete point is in fact zero ($0$).

Sometimes it is convenient to scale the numbers on the number line with a logarithmic scale, using scientific notation. For example, the number one inch to the right of 0 might be 1, then the number one inch farther to the right would be 101 (=10), then one inch to the right of that would be 102 (=100), then 1000, then 10,000, etc. This approach is useful, for example, in illustrating a sequence of events in the history of the universe or of evolution, or in comparing distances to various stars.

Point A is on the coordinate 4 and point B is on the coordinate -1.

If you remove a single point from an interval, the total length does not change. The number of points also stays the same, but only because the interval has infinitely many points to begin with.

The point that is exactly in the middle between two points is called the midpoint and is found by using one of the two following equations.

Now, from these it follows that the length of any point is $0$. So, any notion of length of a point conforming to the above must assign length $0$ to each point. That is a theorem.

These different ways of associating sets with sizes are called measures. The measure corresponding to length is called the Lebesgue measure, and the one corresponding to number of points is called the counting measure. These may or may not mean anything to you at this point, but you will encounter them later on if you continue working with real analysis.

In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.

What's wrong here? Does a point on the real number line have a width? If so, what is the length of a point on the real number line? Infinitesimal?

Think of drawing a real number from $[-1, 1]$. The probability of drawing $0$ is arbitrarily small; there is an infinity number of other numbers to draw instead. So $P(X = 0)$ is actually $0$, as is the length of the interval $[0, 0]$ - but drawing it is still not impossible. By completeness, $P(X \neq 0) = 1$ - this will almost surely happen (yet not surely).

Much like many other 'paradoxes' you'll find that carefully formalizing the problem makes it go away. Whatever counter intuitive phenomena persist is not paradoxical, but simply counter intuitive.

If we want to find the distance between two points on a number line we use the distance formula:

For more, see articles: Null set and Lebesgue measure.

What is the length of a point on the real number line

Want to find the point midway between two locations? Then you're looking for the midpoint! The midpoint of a line segment is the point located midway between the endpoints of the line segment. This tutorial tells you about the midpoint of a line segment. Take a look!

If you have two points on a number line, the midpoint is the point that is located directly midway between the two points. Take a look at this tutorial and learn about the midpoint of two points on a number line!

A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.

Consequently, the situation you describe, where removing a point having length $0$ results in a broken line is a correct description. It is not a paradox but rather a counter intuitive situation. Well, what shall we do with it? You can examine our assumptions and change them, or hone your intuition. In this case, I suggest the latter.

Since an interval is made up of an infinite number of points, I am considering the relation of the length of an interval and the length of a point, this lead me to ask what is the length of a point on the real number line ?

$$AB=\left | 4-(-1) \right |=\left | 4+1 \right |=\left | 5 \right |=5$$

When you're looking at a map, you can find the point midway between two locations by calculating the midpoint. This tutorial takes you through the process of finding the point midway between two cities.

The ordering on the number line: Greater elements are in direction of the arrow.

...however I think the width of the gap being $0$ is equivalent to there being no such gap on the real number line...

I think the width of the gap is $0$ is equivalent to that there is no such gap exists on the real number line

Take for example the rational numbers. They can be considered as having infinitely many gaps, because the don't include the irrationals. However, those gaps have no "width", because you can come arbitrarily close to each irrational from above and below with a rational number. The situation is similar if you remove a single real number from all real numbers.