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.

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Contents
Table of contents

    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 Number line example StudySmarterNumber 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.

    Number Line Number line characteristics StudySmarterCharacteristics 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.

    Number Line Arrowheads on a horizontal line StudySmarter

    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.

    Number Line Tick marks on a number line StudySmarter

    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.

    Number Line Positive numbers on a number line StudySmarter

    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.

    Number Line Positive and negative numbers on a number line StudySmarter

    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.

    Number Line Three on the number line StudySmarter

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

    Number Line Negative one on the number line StudySmarter

    c) Represent the number 15 on the number line.

    Number Line Fifteen on the number line StudySmarter

    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.

    Number Line 2 and 3 on the number line StudySmarter

    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.

    Number Line 2 and 3 on the number line with tick marks StudySmarter

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

    Number Line 2.5 on the number line StudySmarter

    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.

    Number Line -3 and -4 on the number line StudySmarter

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

    Number Line -3 and -4 on the number line with tick marks StudySmarter

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

    Number Line -3.7 on the number line StudySmarter

    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.

    Number Line 0.7 and 0.8 on the number line StudySmarter

    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.

    Number Line 0.7 and 0.8 on the number line with tick marks StudySmarter

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

    Number Line 0.75 on the number line StudySmarter

    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 12 on the number line.

    The fraction 12 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.

    Number Line 0 and 1 on the number line StudySmarter

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

    Number Line 0 and 1 on the number line with tick marks StudySmarter

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

    Number Line 1/2 on the number line StudySmarter

    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 -1+4=?

    Number Line Addition on the number line StudySmarter

    -1+4=3

    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 5-7=?

    Number Line Subtraction on the number line StudySmarter

    5-7=-2

    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 3×2=?

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

    Number Line Multiplication on the number line StudySmarter

    3×2=6

    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:

    InequalityInequality on the number line
    x>1

    Number Line x greater than 1 StudySmarter

    x1

    Number Line x greater than or equal to 1 StudySmarter

    x<1

    Number Line x less than 1 StudySmarter

    x1

    Number Line x less than or equal to 1 StudySmarter

    -1<x2

    Number Line x greater than -1 and less than or equal to 2 StudySmarter

    x<-1 or x>1

    Number Line x less than -1 or greater than 1 StudySmarter

    Inequalities on the number line - StudySmarter Originals

    Please read Solving Inequalities to learn more about this topic.

    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.
    Frequently Asked Questions about Number Line

    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.

    How do you 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!

    What is an example of a number line?

    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.


    What are the characteristics of a number line?

    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.


    What are the methods of number lines?

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


    • Addition: Move towards the right to add.
    • 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.

    Test your knowledge with multiple choice flashcards

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

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

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

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