Generic Intervals
An interval in math is a range of numbers between two assigned numbers and includes all of the real numbers between those two numbers. If you already know, real numbers are pretty much any number you can imagine about: 4.67, 151, √6, -0.143, π, etc. Intervals can be written with the help of inequalities, a number line, or in interval notation! There are also unique techniques to indicate whether the two assigned numbers, called endpoints, are included in the interval.
Example of Intervals
You must have heard that. A weather forecaster just made a prediction that there is going to be a rainfall with at least 4 but less than 9mm of rain! Thus, what are the different amounts of rain that we could anticipate based on those numbers? Well, go get your wipers ready and then let's find out.
When the forecaster said that there would be at least 4 but less than 9 millimeters of rain, he described the amount of rain in an interval!
Using Inequalities in Identifying Intervals
Inequalities in math are the symbols that represent mathematical signs such as greater than, and greater than or equal to, less than, and less than or equal to. Now considering these symbols, let’s decompose our weather forecast down and write the interval with the help of inequalities:
The 1st part is that it will rain at least 4mm. That implies that the amount of rain, which we will depict with the variable x since its unknown, will be at least equal to 4 but could be greater than 4. The 2nd part of our interval is that the amount of rain, x, will be less than 9mm.
Thus, we will read this inequality in intervals like this: 4 is less than or equal to x, which is less than 9. Thus, this interval is including endpoint 4 (since it's equal to), but not including endpoint 9. Therefore, x is any real number from 4 all the way to the last real number prior to 9.
Identifying Intervals Using the Number Line
To describe an interval on a number line, you have to first construct two circles at the two endpoints of the interval. So, we will construct two circles at 4 and 9. Now, construct a line to join the two circles! The last step is to color inside the circles only if the endpoint is included in the interval.
(Image will be uploaded soon)
Types of Intervals
1. Finite Interval
A finite interval (bounded interval) is an interval, whose both endpoints are numbers (also variables, which as you know describe unknown numbers).
Example: (-5,2]
Inequality:
-5 < x ≤ 2
These kinds of endpoints (numbers) are what we call finite endpoints.
2. Infinite Intervals
An infinite interval is that whose minimum one endpoint is infinity. The infinity interval is represented by the symbol ∞. You would be surprised to know that there is also minus infinity denoted as -∞ as well as the plus infinity denoted as +∞.
3. Open and Closed Intervals
Usually, it’s all about the types of brackets: a square bracket or a parenthesis.
You might already be familiar when there is a square bracket – the endpoint is related to the interval. Whereas, when there is a parenthesis - the endpoint does not pertain to the interval. That being said,
In the 1st possibility (a square bracket) the endpoint is closed,
In the 2nd possibility (a parenthesis) – the endpoint is open.
The interval can be referred to as:
Closed interval – Both brackets include the square brackets, for example: [2, 9].
Left-Closed Interval – Only the left side bracket is a square bracket, for example: [1, 5).
Right-Closed Interval – Only the right side bracket is a square bracket, for example: (-4, 7].
Open Interval – Both brackets include parentheses, for example: (-6, -2).
Left-Open Interval – Only the left side bracket is a parenthesis, for example: (-2, 8].
Right-Open Interval – only the right side bracket is a parenthesis, for example: [7, 11).
For most intervals, two descriptions are proper at the same time.
FAQs on Intervals
Q1. What is Meant by Perfect Intervals?
Answer: Perfect intervals of all intervals consist of only one basic form. The first (also known as prime or unison), 4th, 5th and 8th (or octave) are all perfect intervals. These intervals are known as "perfect" possibly because of the way that these types of intervals sound and that their frequency ratios are basic whole numbers.
To identify the interval between 2 notes just determine the pitch of the lowest number and begin counting until you reach the top number. When counting intervals you always begin from the bottom number and count both notes.
Q2. What is Meant by an Inverted Interval?
Answer: An inverted interval, when one of the notes is scaled up or down an octave. There is also a term called interval notation. An interval notation is a technique to define continuous sets of real numbers by the numbers which bound them. An interval, when mathematically written, seems somewhat like ordered pairs. But, they are not meant to represent a particular point. Instead of, they are meant to be a shorthand technique to write an inequality or system of inequalities.
Q3. What is Writing Interval Notation?
Answer: Intervals are mathematically written with rectangular brackets or parentheses, and two numbers delimited with a comma. The two numbers are known as the endpoints of the interval. The number on the left represents the lower bound or least element. The number on the right represents the upper bound or greatest element.
