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# Number Line

Visual representations are always useful to help you understand concepts more easily. What about numbers? What can we use to represent numbers to help us understand how they relate to other numbers? A very simple tool that we can use is a number line.

In this article, we will define what a number line is, and its characteristics, and we will explain how to create one. We will also show you how to use the number line to do basic mathematical operations and represent inequalities.

Let's start by defining what we mean by number line.

## What is a number line?

A number line is a visual representation of numbers, using a horizontal line, which includes equally spaced divisions representing each number.

### Characteristics of a number line

A typical number line will look like the one shown below.

Number line example - StudySmarter Originals

Looking at the number line above, we can identify the following characteristics:

• A horizontal line with arrowheads at each end, which means that it extends to positive infinity on the right side, and to negative infinity on the left side.

• Equally spaced tick marks, or divisions, that represent each integer number.

• Zero is normally at the centre of the number line, but not necessarily.

• The numbers on the right-hand side of zero are the positive numbers.

• The numbers on the left-hand side of zero are the negative numbers.

• If you move towards the right on the number line you will find bigger numbers. If you move toward the left on the number line you will find smaller numbers.

Characteristics of a number line - StudySmarter Originals

### How to create a number line?

The steps to create a number line are as follows:

1. Draw a horizontal line with arrowheads at each end, long enough to represent the range of numbers that you require. You can use a ruler to help you draw it, if you are doing it manually.

2. Include equally spaced tick marks along the horizontal line. If using a ruler, you can draw a tick mark on each centimetre as a guide. However, the number line is not the same as a ruler, so the space between tick marks does not have to be exactly 1 cm, but try to be consistent.

3. Label each tick mark below the line with an integer number, starting with zero below the tick mark in the middle of the line.

4. Label the tick marks to the right of zero with positive numbers starting from 1, and add one each time to label the following tick mark moving towards the right.

5. Label the tick marks to the left of zero with negative numbers starting from –1, and subtract one each time to label the following tick mark moving towards the left.

And there you have it, a number line that you can customise depending on what number or range of numbers you need to represent on it!

## Examples of a number line

To represent or plot a specific number on the number line, you can draw a circle on the tick mark corresponding to that number.

The labels on a number line normally correspond to integer numbers, but you can actually represent any number on the number line, including fractions and decimals.

If you need to represent bigger numbers on the number line, then you can choose a different scale. For example, if you need to represent the number 40, then you can choose a scale of 5, 10 or 20.

### Integers on the number line

First of all, let's see a few examples representing integers on the number line.

a) Represent the number 3 on the number line.

b) Represent the number –1 on the number line.

c) Represent the number 15 on the number line.

Notice that in this case, we have used a scale of 5.

Representing integers on the number line - StudySmarter Originals

### Decimals on the number line

When representing decimal numbers on the number line, you need to think about it as zooming into it, or looking at it closer. To allow you to represent the decimal part of this type of numbers more accurately, we need to create more tick marks or divisions as required.

To represent a decimal number on the number line, you can follow these steps:

• Identify what two integer numbers the decimal number is in between, and label them on the number line.

• Identify the decimal part of the decimal number. Draw 9 equally spaced tick marks in between the two tick marks already labelled, to create 10 new intervals that will allow you to represent the digit in the tenths place.

• Starting on the number corresponding to the part of the decimal number before the decimal point,

• If the decimal number is positive, move towards the right as many tick marks as the value of the decimal part of the decimal number.

• If the decimal number is negative, move towards the left as many tick marks as the value of the decimal part of the decimal number.

• Draw a circle on the tick mark where you end up to represent the decimal number.

Let's see some examples to help you understand the process more clearly.

a) Represent the number 2.5 on the number line.

The decimal number 2.5 is between 2 and 3, so let's label them on the number line.

The decimal part of 2.5 is 5 tenths, so let's draw 9 equally spaced tick marks between 2 and 3. Each tick mark represents an increment of 0.1.

Now starting from 2, we move towards the right 5 tick marks to find 2.5.

b) Represent the number –3.7 on the number line.

The decimal number –3.7 is between –3 and –4, so we need to label them on the number line.

The decimal part of –3.7 is 7 tenths, so let's draw 9 equally spaced tick marks between –3 and –4.

Starting from –3, we move towards the left 7 tick marks to find –3.7.

Suppose you need to represent decimal numbers with more than one decimal. In that case, you can follow a similar process as before, but labelling the two decimal numbers that the given decimal number is in between.

c) Represent 0.75 on the number line.

The decimal number is between 0.7 and 0.8, so let's label them on the number line.

The decimal part of 0.75 is 75 hundredths, so let's draw 9 equally spaced tick marks between 0.7 and 0.8. In this case, each tick mark represents an increment of 0.01.

Starting from 0.7, we move towards the right 5 tick marks to find 0.75.

Representing decimals on the number line - StudySmarter Originals

### Fractions on the number line

The easiest way to represent fractions on the number line is to convert the fraction into a decimal number by dividing the numerator by the denominator, and then following the same steps as before. Please read Convert between Fractions and Decimals, if you need to refresh the basics.

a) Represent the fraction on the number line.

The fraction is equal to the decimal number 0.5. Now we can follow the same steps as before to represent 0.5 on the number line.

The decimal number 0.5 is between 0 and 1. So let's label them on the number line.

The decimal part of 0.5 is 5 tenths. Let's divide the number line between 0 and 1 into 10 equal intervals.

Starting from 0, we move towards the right 5 tick marks to find 0.5.

Representing fractions on the number line - StudySmarter Originals

## Number line Maths

You can perform mathematical operations like addition, subtraction and multiplication with the help of the number line. Let's see what methods you need to use in each case.

### Number line methods

#### Addition on the number line

To add numbers on the number line you can follow these steps:

• Draw a number line using an appropriate scale to represent the numbers required.

• Start on the first number in the sum.

• Move towards the right as many tick marks as the value of the second number in the sum.

• The tick mark where you end up will be the result of the sum.

Use the number line to solve

Addition on the number line - StudySmarter Originals

#### Subtraction on the number line

To subtract numbers on the number line you can follow these steps:

• Draw a number line using an appropriate scale to represent the numbers required.

• Start on the first number in the subtraction.

• Move towards the left as many tick marks as the value of the second number in the subtraction.

• The tick mark where you end up will be the result of the subtraction.

Use the number line to solve

Subtraction on the number line - StudySmarter Originals

#### Multiplication on the number line

To multiply numbers on the number line you need to remember that multiplication is the same as repeated addition. Keeping that in mind, we can use the skip count method following these steps:

• Draw a number line starting from zero, long enough to represent the required numbers.

• Move towards the right as many times as indicated by the value of the first factor in the product, in equal intervals indicated by the second factor in the product.

• The tick mark where you end up will be the result of the product.

If either of the factors in the multiplication is negative, but not both, then you need to move towards the left of zero (negative numbers) to represent the solution of the multiplication.

Use the number line to solve

Starting from zero, we move towards the right 3 times in intervals of 2.

Multiplication on the number line - StudySmarter Originals

## Number line inequalities

Let's recall the definition of inequalities.

Inequalities are algebraic expressions that, instead of representing how both sides of an equation are equal, represent how one term is less than (<), less than or equal (≤), greater than (>), or greater than or equal (≥) to the other.

Inequalities can also be represented on the number line. The rules that you need to keep in mind when representing inequalities on the number line are as follows:

• The symbols > (greater than) and < (less than) exclude the specific value as part of the solution. The symbols ≥ (greater than or equal) and ≤ (less than or equal) include the specific value as part of the solution instead of excluding it.

• Use an empty circle to represent that the value of x is not part of the solution.

• Use a closed circle if the value of x is part of the solution.

Represent the inequalities below on the number line:

 Inequality Inequality on the number line

Inequalities on the number line - StudySmarter Originals

## Number Line - Key takeaways

• A number line is a visual representation of numbers, using a horizontal line, which includes equally spaced divisions representing each number.
• The numbers on the right-hand side of zero are the positive numbers. The numbers on the left-hand side of zero are the negative numbers.
• If you move towards the right on the number line you will find bigger numbers. If you move towards the left on the number line you will find smaller numbers.
• To represent or plot a specific number on the number line, you can draw a circle on the tick mark corresponding to that number.
• You can represent any number on the number line, including fractions and decimals.
• You can perform mathematical operations like addition, subtraction and multiplication with the help of the number line.
• Inequalities can also be represented on the number line. An empty circle is used to represent that the value of x is not part of the solution, and a closed circle is used if the value of x is part of the solution.

A number line is a visual representation of numbers, using a horizontal line, which includes equally spaced divisions representing each number.

The steps to create a number line are as follows:

1. Draw a horizontal line with arrowheads at each end, long enough to represent the range of numbers that you require. You can use a ruler to help you draw it, if you are doing it manually.

2. Include equally spaced tick marks along the horizontal line. If using a ruler, you can draw a tick mark on each centimetre as a guide. However, the number line is not the same as a ruler, so the space between tick marks does not have to be exactly 1 cm, but try to be consistent.

3. Label each tick mark below the line with an integer number, starting with zero below the tick mark in the middle of the line.

4. Label the tick marks to the right of zero with positive numbers starting from 1, and add one each time to label the following tick mark moving towards the right.

5. Label the tick marks to the left of zero with negative numbers starting from -1, and subtract one each time to label the following tick mark moving towards the left.

And there you have it, a number line that you can customise depending on what number or range of numbers you need to represent on it!

An example of a number line is a horizontal line with arrowheads at each end. The horizontal line has equally spaced tick marks, or divisions, that represent each integer number. Zero is at the middle of the number line. The scale used is 1. The range of numbers represented goes from -5 to 5.

The main characteristics of a number line are as follows:

• A horizontal line with arrowheads at each end, which means that it extends to positive infinity on the right side, and to negative infinity on the left side.
• Equally spaced tick marks, or divisions, that represent each integer number.
• Zero is normally at the centre of the number line, but not necessarily.
• The numbers on the right-hand side of zero are the positive numbers.
• The numbers on the left-hand side of zero are the negative numbers.
• If you move towards the right on the number line you will find bigger numbers. If you move towards the left on the number line you will find smaller numbers.

You can perform mathematical operations like addition, subtraction and multiplication with the help of the number line by following specific methods:

• Subtraction: Move towards the left to subtract.
• Multiplication: Use the skip count method. Starting from zero, move towards the right as many times as indicated by the value of the first factor in the product, in equal intervals indicated by the second factor in the product. If either of the factors in the multiplication is negative, but not both, then you need to move towards the left of zero to represent the solution of the multiplication.

## Final Number Line Quiz

Question

What is a number line in Math?

A number line is a visual representation of numbers, using a horizontal line, which includes equally spaced divisions representing each number.

Show question

Question

What is the meaning of the arrowheads at each end of the number line?

The arrowheads at each end of the number line mean that it extends to positive infinity on the right side, and to negative infinity on the left side.

Show question

Question

Does zero have to be at the centre of the number line?

Not necessarily, it depends on the range of numbers that you are representing.

Show question

Question

What numbers are on the right-hand side of zero on the number line?

Positive numbers

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Question

What numbers are on the left-hand side of zero on the number line?

Negative numbers

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Question

If you move towards the right on the number line, what will you find?

Bigger numbers

Show question

Question

If you move towards the left on the number line, what will you find?

Smaller numbers

Show question

Question

What type of numbers can you represent on the number line?

The labels on a number line normally correspond to integer numbers, but you can actually represent any number on the number line, including fractions and decimals.

Show question

Question

When do you need to use a different scale to create a number line?

You can do this if you need to represent bigger numbers on the number line.

Show question

Question

What methods do you use to perform addition, subtraction and multiplication on the number line?

• Subtraction: Move towards the left to subtract.
• Multiplication: Use the skip count method. Starting from zero, move towards the right as many times as indicated by the value of the first factor in the product, in equal intervals indicated by the second factor in the product. If either of the factors in the multiplication is negative, but not both, then you need to move towards the left of zero to represent the solution of the multiplication.

Show question

Question

When representing inequalities on the number line, what do you need to use to represent that the value of x is not part of the solution

An empty circle

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Question

When representing inequalities on the number line, what do you need to use to represent that the value of x is part of the solution?

A closed circle

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