How to Write Interval Notation: A Beginner’s Guide
Expressing Intervals Using Parentheses, Brackets, and Infinity Symbols
When writing interval notation, it’s important to use the correct symbols to represent the endpoints of the interval. The three main symbols used in interval notation are parentheses, brackets, and infinity symbols.
Parentheses are used to represent an endpoint that is not included in the interval. For example, the interval (1, 5) represents all numbers greater than 1 and less than 5, but does not include the endpoints 1 and 5.
Brackets, on the other hand, are used to represent an endpoint that is included in the interval. For example, the interval [0, 3] represents all numbers greater than or equal to 0 and less than or equal to 3.
Infinity symbols can also be used in interval notation to represent an endpoint that goes on forever in one direction. For example, the interval (−∞, 4) represents all numbers less than 4, while the interval [3, ∞) represents all numbers greater than or equal to 3.
It’s important to use these symbols correctly, as they can change the meaning of the interval. For example, (1, 5] represents all numbers greater than 1 and less than or equal to 5, while [1, 5) represents all numbers greater than or equal to 1 and less than 5.
Combining Multiple Intervals Using Union and Intersection Symbols
Sometimes it is necessary to combine multiple intervals into a single notation. This can be done using the union and intersection symbols.
The union of two or more intervals represents the set of all numbers that are in at least one of the intervals. The union symbol is represented by the symbol ∪. For example, the union of the intervals [0, 2] and (3, 5] can be written as [0, 2] ∪ (3, 5].
The intersection of two or more intervals represents the set of all numbers that are in all of the intervals. The intersection symbol is represented by the symbol ∩. For example, the intersection of the intervals [0, 2] and (1, 5) can be written as (1, 2].
It’s important to note that when combining intervals, the resulting notation may not always be a single interval. It’s also important to use the correct symbols and to ensure that the intervals being combined are compatible.
Converting Interval Notation to Graphs and Vice Versa
Interval notation can be useful for representing ranges of numbers, but it can sometimes be difficult to visualize what the interval actually represents. In these cases, it may be helpful to convert the interval notation into a graphical representation or vice versa.
To convert interval notation to a graph, you can plot the interval on a number line. For example, the interval (−3, 5] would be graphed as a closed circle at 5 and an open circle at −3, with a line connecting the two circles.
Conversely, to convert a graph to interval notation, you simply need to identify the endpoints of the interval and determine whether they are included or excluded. For example, the graph of a closed circle at 2 and an open circle at 8 would be represented in interval notation as [2, 8).
It’s important to remember that interval notation and graphs are just different ways of representing the same information, and both can be useful depending on the situation.
Common Mistakes to Avoid When Writing Interval Notation
Interval notation can be tricky, and there are some common mistakes that people make when writing it. Here are a few mistakes to watch out for:
Switching the order of the endpoints: It’s important to write the smaller endpoint first, followed by the larger endpoint. For example, [2, 6] is correct, while [6, 2] is incorrect.
Mixing up the parentheses and brackets: Parentheses represent endpoints that are not included, while brackets represent endpoints that are included. Using the wrong symbol can completely change the meaning of the interval. For example, (2, 6] and [2, 6) represent different intervals.
Forgetting to use the infinity symbol: When an interval extends to infinity, it’s important to use the infinity symbol to indicate that. For example, (−∞, 5) represents all numbers less than 5.
Combining incompatible intervals: When combining intervals using the union or intersection symbols, it’s important to ensure that the intervals being combined are compatible. For example, [0, 2] and (3, 5] cannot be combined using the intersection symbol, because they have no numbers in common.
By avoiding these common mistakes, you can ensure that your interval notation accurately represents the range of numbers you are trying to describe.
Conclusion
Interval notation is a powerful tool for representing ranges of numbers, and it’s important to use it correctly in order to avoid confusion or errors. By understanding the symbols used to represent endpoints, combining multiple intervals using the union and intersection symbols, converting interval notation to graphs and vice versa, and avoiding common mistakes, you can become proficient in writing interval notation. Whether you are studying mathematics, science, or engineering, interval notation is an essential part of communicating mathematical concepts and ideas.
