Bounded Interval for Dummies

Bounded Interval is a term used in mathematics to describe a specific range or set of numbers. Let's break it down and see if we can understand it together. Picture a number line in your mind, stretch it infinitely in both positive and negative directions. Now, imagine there are two dots on this number line, let's call them point A and point B.

A bounded interval is like a little segment of this infinite number line, where we only focus on the numbers between point A and point B. It's like looking at a specific area of the number line, instead of considering all the numbers that exist.

Let me give you an example to make it clearer. Imagine you have a jar filled with colorful marbles, all of various sizes. But you want to pick out only the marbles that are bigger than 1 cm and smaller than 5 cm. In this scenario, 1 cm is your point A and 5 cm is point B. The marbles that fall within this specific range are the ones that form the bounded interval.

So, a bounded interval is simply a range of numbers that is limited or "bounded" by two specific points. These two points define the range, and the numbers between them are the ones we are interested in.

There are two different types of bounded intervals:

1. A closed bounded interval is one where the two points, point A and point B, are included in the range. It's like having fences at both ends of a playground. The kids can play within the playground area, and the boundaries of the playground are the fences. Similarly, the two points act as the boundaries of the closed bounded interval, and all the numbers between them are included in the range.

2. An open bounded interval, on the other hand, is one where the two points, point A and point B, are not included in the range. It's like looking at a section of a road without considering the endpoints. For example, if we say the temperature outside is between 20 degrees Celsius and 30 degrees Celsius, with both 20 and 30 not included in the range, then we have an open bounded interval.

Let's put this into practice with a real-life scenario. Imagine you and your friends are organizing a basketball tournament. The minimum age requirement is 10 years old, and the maximum age allowed is 15 years old. In this case, the bounded interval is from 10 to 15, with both 10 and 15 included. Anyone who is 10, 11, 12, 13, 14, or 15 years old can participate in the tournament. That is an example of a closed bounded interval.

So, in summary, a bounded interval is like a specific section or range of numbers between two defined points. It can be either a closed bounded interval, where the endpoints are included in the range, or an open bounded interval, where the endpoints are not included. It's similar to looking at a specific area of a number line, playground, or road section without considering the numbers or points outside that range.