Interval Definition

Intervals are a very important idea in mathematics. The basic definition of an interval is the numbers inside of a range. A range is usually made up of two numbers. Take a look at the image below, which gives some real life examples of intervals.

 

interval_example
Interval Type
A From 14:00 to 15:00 Time
B From a grade of F of 0 score to an A of 90 points Test Scores
C From 18 years old to 64 Age

 

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Interval properties

As you saw in the previous section, intervals can be almost anything: time, age, test scores and more. However, there are a few things you should keep in mind when creating your own interval. The most important property of an interval is that it includes all real numbers.

 

real_number_examples

 

The image above shows an example of real numbers. These are explained in the table below. 

Type Explanation
A Whole number No decimals, positive
B Rational number Positive numbers with decimals or fractions
C Rational number Can also be negative
D Irrational number Can’t be written as a decimal

 

Including Ends

The most important thing to remember is the notation of intervals on the line itself and when written in numbers. On a number line, we usually depict intervals as a straight line with two circles at the end. When the circle is filled in, this means that we want to include that number in our interval.

 

interval_number_line

 

Take the image above as an example, where we are between 20 and 40. Because both circles are filled in, we know that we start at 20 and end with 40, rather than starting at 21 and ending at 39. 

Because drawing a number line every time we want to express this is inconvenient, we can also simply write:

 

include_interval

 

Notice that the brackets mean the same thing as the filled in circle: it includes 20 and 40. 

Excluding Ends

When we want to exclude the ends, we can simply leave the circles empty. Let’s take the same example above, only this time, say we only want to include 40.

 

exclude_number_line

 

Here, the number line is saying we want to exclude 20 and include 40. This means that we start counting our interval at 21 and finish with 40. When we want to exclude a number from our interval, we simply use parenthesis.

 

exclude_example

 

This means the same thing as our number line: exclude 20, include 40. 

Inequalities

Inequalities deal with two ideas: greater and less than. The table below shows the definitions of both.

 

greater_than_number_line

 

A Less Than < When a number, x, is to the left of another number, y, on a number line 2 < 10
B Greater Than > When a number, x, is to the right of another number, y, on a number line 10 > 2

 

We can also combine inequalities with an equals sign. Check out the possible combinations below.

 

less_than_number_line

 

A Less than OR equal to \leq When a number is to the left of or equal to another on a number line 2 \leq 10, 10 \leq 10
B Greater than OR equal to \geq When a number is to the right of or equal to another on a number line 10 \geq 2, 10 \geq 10

 

Infinity

Infinity is something that has no end. Infinity is endless and is a very helpful concept in maths. When we talk about infinity in inequalities, we always exclude infinity. This is because we never reach infinity - it is endless after all!

 

infinity_interval

 

As you can see, because we exclude infinity, it is always used with parenthesis. Let's take [3, +\infty) as another example. When we draw this on a number line, instead of using an empty circle, we simply use an arrow.

 

infinity_number_line

Notation

Now that we have learned about intervals and inequalities, we can combine both to understand statements of inequalities. Say, for example, that you have a birthday coming up. You know you will get between 100 to 200 pounds from your family and friends. This can be written as:

 

less_than_greater_than

 

Notice that x is between 100 and 200 because it can be anywhere: 101, 115, 199, and so on. Take a look at the image below and the table that describes the image. 

A a < x < b (a,b) Exclude a and b
B a > x > b (a,b) Exclude a and b
C a \leq x \leq b [a,b] Include a and b
D a \geq x \geq b [a,b] Include a and b

 

Union

A union can be thought of as combining several expressions of inequality. The easiest way to understand this is to use infinity. For example, Antoine and Marie are playing a video game where you have positive and negative points. The table describes their possible results.

 

Antoine (- \infty, 10] We know Antoine scored 10 points or less
Marie (7, + \infty) We know Marie scored above 7 points

 

If we wanted to find the region where they could have tied, we can make a union of these two intervals. A union is written as:

 

union_example
To understand this, we simply draw it on the number line.

 

union_number_line

 

To find the region where they tied, we need to find the intersection. 

Intersection

An intersection is exactly as it sounds: it is where two things meet. In this case, it is where our two intervals meet. Lets continue from the example above.

 

Here are the possible scenarios of this game.

 

finde_union_intersection
 
 

Antoine (A) wins A = 10 If Antoine scores 10 points
Marie (M) wins A > 10 If Marie scores more than 10 points
Antoine and Marie tie 7 < A leq 10

7 < M leq 10

If both Antoine and Marie score above 7 points and 10 points or less

 

We can break the table down using the number line.

 

A A < 7 Antoine loses, as his interval is the only one going below 7
B 7 < A leq 10

7 < M leq 10

Both Antoine and Marie win, as their intervals both intersect each other between these two points
C M geq 10 Marie wins, as her interval is the only one at 10 or above

 

As you can see, the region where they tie is the intersection because it is the interval where both Antoine and Marie’s intervals meet each other. Taking away the A and C parts of the number line, we are left with:

 

intersection_number_line
 
 

Which gives us the following:

 

Notation Intersection Description
7 < x leq 10 (7,10] Exclude 7 and include 10

 

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.