Number Line

A number line is defined as a representation of positive and negative real numbers. You’ve probably seen a number line before. The image below shows an example of a number line.

 

negative_number_line
 

Definition Examples
Negative Numbers Below zero -2, -6, -10
Positive Numbers Above zero 5, 7, 10
Origin Middle point at zero 0
Arrows Show that number line can continue - \infty, + \infty

 

While a number line can include as many numbers as you want, it can be helpful to only include the range of numbers you’re interested in. A range is defined as all the values between two points. Take the following number lines as an example.

 

interval_number_line
 

As you can see, number lines are pretty flexible. A half line uses number lines and is discussed a bit further.

 

The best Maths tutors available
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Ayush
5
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Ayush
5
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
First Lesson Free>

Real Numbers

As stated in the previous section, a number line only takes real numbers into account. A real number can be:

 

Definition Examples
Rational Includes fraction or decimal 3.44, 5.5, \frac{1}{2}
Irrational Cannot be written as a fraction or decimal \pi, \sqrt{3}
Whole Has no decimal 2, 6, 98, 450
Integer Includes negative whole numbers 1, -4, 6, -105

 

Inequalities

When we talk about using a half-line, we usually talk about it when using it to express inequalities. Inequalities are defined as the relative position, or size, of two numbers. Relative size is measured when we compare two values and determine which one is above, below or equal to the other.

 

Sign Explanation Example
> Greater than 5 > 2
>=, \geq Greater than or equal to 3 \geq 3, 9 \geq 1
< Less than 1 < 3
<=, \leq Less than or equal to 2 \leq 2. 4 < 10

 

inequality_meanings

 

 

Greater Than

Let’s see how we can plot inequalities on a number line. Take the following example: you want to know which symbol can be used to fill in the following.

 

greater_than_or_less_than

 

Looking at a number line can help us determine the solution to this problem. Take the following number line.

 

right_greater

 

As we can see, the value of 4 is to the right of 1. Whenever a number is to the right of another on a number line, it is greater than that number.

 

Greater Than or Equal To

This concept can be confusing, however it can be helpful to think of it as the combination of two different statements.

 

sign_rule

 

If a number is greater than OR equal to another number, then we can use this symbol. As an example, think of all the numbers that are greater than or equal to 3. The number line below shows some sample responses.

 

interval_example_number_line

 

 

Less Than

You want to buy something online - one product has a 1 star review, the other has a 5 star review. In this situation, we quickly determine that 1 is less than 5, made easier by the fact that a 1 star value is less than a 5 star value.

 

You can see this on the number line below, as 1 is to the left of 5. Any time a number is on the left of a value, it is less than that value.

left_number_line

 

 

Less Than or Equal To

The concept of being less than or equal to is the same concept as \geq. It is the combination of two statements: a number that is less than OR equal to a value. Think of all possible numbers \leq to 2. The number line below shows some examples.

 

less_than_or_equal_number_line
 

Rules

There are some rules you can use to help you determine whether a number is less than, equal to or greater than some value. The table below summarizes rules when dealing with a number line.

 

Location on Number Line Inequality Written
To the left of a value < Less than
To the right of a value > Greater than
The same as a value = Equal to

 

Another rule to keep in mind is that the same inequality can be written two ways.

 

inequalities_rules

 

Here, 1 is less than 4 and, at the same time, 4 is greater than 1. The easiest way to remember this rule is to think of the sign as a hungry alligator, whose mouth is always pointing to the bigger number.

 

Infinity

Infinity is an important concept in math. Infinity is not a real number and is endless. You can think of infinity as a company, where the only job is to count and keep the count going - no matter how long you count, you never reach infinity. It is written as the following:

 

    \[ + \infty, - \infty \]

 

Union

When we write inequalities, we can write the interval in numbers instead of drawing it on a number line.

 

Interval Meaning Interval Includes Number line
(1,3) () exclude the numbers of the interval 2 An empty circle
[1,3] [] include the numbers of the interval 1,2,3 A filled in circle
(1,3] A combination of the above two 2,3 A combination of the two above

 

A union is a combination, or union, of two intervals. Take a look at the example below.

union_intervals

 

 

Intersection

An intersection is the interval where two intervals meet. In other words, when you take a union of two intervals, the intersection is the resulting interval. Here, we can understand what a half line is - a half line is lines which only move in one direction, and not both.

 

The two intervals in the union above are two examples of half-lines.

 

half_lines

 

As you can see, they go from one point and continue on from there.

 

When we take the intersection, we are interested in the interval where both intervals meet.

 

half_line_intersection
 

Example 1

Take the following interval as an example: (5, +\infty). Is this a half-line or not?

 

half_line_number_line

 

As you can see from the number line above, we can see that this interval starts at 5 and goes to infinity. Therefore, it is considered a half line.

 

Example 2

Find the intersection of the following union:

 

    \[ (3, 10] \cap [8, 15) \]

 

To solve this, we can plot the two intervals.

 

intersection_example

 

As we can see, the intersection is: [8,10].

Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 3.00/5 - 2 vote(s)
Loading...

Danica

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