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Algebra: Intervals and the Real Number Line

Intervals and the Real Number Line

In this math video tutorial, we present the real number line and show how to graph point sets on it. Furthermore, we present the interval as a common number set, infinity, and unbounded intervals.

The real number line is drawn as a line segment with arrows at the end to show that it extends outward indefinitely. Tick marks are added to indicate scale and what portion of the real numbers is visible.

With this representation, we can plot points or even sets of points on the number line as dots. The distance between two points on the number line is just the larger number minus the smaller number.

Two points can also be used to define an interval. An interval is the set of real number between any two numbers on the number line. In terms a sets, an interval can be defined like this: {x | xR, axb}. This set can be specified in interval notion as [a, b]. This interval includes the endpoints a and b, therefore it is called a closed interval.

As an example of a closed interval, we have {x | xR, 0 ≤ x ≤ 3} = [0, 3]. Above, we show this interval graphed on the real number line. Notice that the dots at the endpoints we solid. This is how we show that the endpoints are included in the interval.

If the interval does not included it's endpoints, it is called an open interval. As an example of an open interval we have {x | xR, 0 < x < 3} = (0, 3), which is graphed above. Notice that we use parentheses to show that the endpoints are excluded in the interval notation and we use empty circles in the graph to show that the endpoints are excluded. If we include one endpoint and exclude the other endpoint, we have what is called a half-open interval.

We can also have an unbounded interval, which extends to infinity or negative infinity, like the one shown above. This interval is given in set notation and interval notation as {x | xR, 0 < x < 3} = (-∞, 2].