The Real Number Line

Algebra 1 Ch 2.1 – The Real Number Line

Objective 

Students will graph and compare real numbers using a number line

Vocabulary      

Throughout this course we will be working with real numbers. Real numbers can be pictured as points on a horizontal line called the real number line The point labeled 0 is called the origin Points to the left of zero represent negative numbers Points to the right of zero represent positive numbers Zero is neither positive nor negative.

Real Number Line

Negative Numbers

-5

-4

-3

-2

-1

Positive Numbers

0

Origin

1

2

3

4

5

Vocabulary     

In the real number line the scale marks are equally spaced and represent integers. Integers are the set of numbers that include positive numbers, negative numbers and zero The real number line has points that represent fractions and decimals as well as integers. The point that corresponds to a number is called the graph of the number Drawing the point is called graphing or plotting the number

Graphing Real Numbers 

To graph a real number: 1. 2.

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Draw and label a number line Find where the number is on the number line and place a dot on the number

Let’s look at an example…

Example # 1 

Graph the numbers ½ and -2.3 on a number line

1. Draw and label a number line

-5

-4

- 2.3 -3 -2

½ -1

0

1

2

3

4

5

Find where the number is on the number line and place a dot on the number

1.

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In this instance ½ is positive so it is to the right of zero and less than 1

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-2.3 is negative and to the left of zero 2.3 units

Comparing Real Numbers 

Once points have been plotted on the number line you can compare the numbers.

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Let’s look at an example

Note: Be careful here…there is more than 1 way to read the comparison

Example # 2 Graph – 4 and – 5 on a number line. Then write 2 inequalities that compare the numbers

-5

-4

-3

-2

-1

1. Plot the point – 4

0

1

2

3

4

5

1. Plot the point – 5

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On the graph – 5 is to the left of – 4 , so – 5 is less than – 4 which is written as –5<–4

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On the graph – 4 is to the right of – 5 , so – 4 is greater than – 5 which is written as –4>–5

Comments 

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About a number line….it is expected that you already know the following: The further to the left of zero the smaller the number is…. The further to the right of zero the larger the number is….

Ordering Real Numbers  

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Often times you will be asked to order numbers from least to greatest or greatest to least. The goal here is to see if you understand the relationship of the numbers…that is… do you know which number is bigger or smaller than another number You can use the number line help you by plotting the numbers and then reordering them… A strategy you can use here is to convert the numbers to decimals and then plot and order them as requested Let’s look at an example…

Example # 3 

Write the numbers in increasing order: - 2, 4 , 0, 1.5, ½ , - 3/2

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First convert ½ and -3/2 to decimals. Which are .5 and – 1.5 respectively

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Then plot each number on the number line: - -2 3/2 or -1.5

-5

-4

-3

- 3/2 -2 -1

0 ½ or .51.5

0

½

1

1.5

4

2

3

4

5

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Since you are numbering in increasing order read the number line from left to right

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The solution is: -2, -3/2, 0, ½ , 1.5, 4

Comments 

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In the previous example we converted 2 fractions into decimals to make it easier to plot on the number line… When writing your final solution to the problem you should always use the original numbers. In this case it was ½ and – 3/2 which are displayed in the solution

Opposites 

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In a number line two numbers that are the same distance from zero but in opposite directions are called opposites… Let’s look at an example…

Example # 4 -5

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-4

-3

-2

-1

0

1

2

3

4

5

In the above number line 2 and – 2 are the same distance from zero but on opposite sides. Therefore they are considered opposites

Absolute Value 

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The absolute value of a number is simply the distance the number is from zero on a number line. Vertical bars | | are used to represent the absolute value of a number The symbol |a| is read as the absolute value of a. The next slide shows a summary of absolute value

Absolute Value Example If a is a positive number then the |a| = a

|3| = 3

If a is zero, then |a| = 0

|0| = 0

If a is a negative number, then the | - a| = a

|- 3| = 3

Notice: The absolute value of a number is NEVER negative

Comments 

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On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn 

2. 3. 4. 5.

Graph the numbers on a number line then write 2 inequalities that compare the numbers -6 and 4 -6.4 and -6.3 -7 and 2 -2.7 and 3/4

Your Turn 

2. 3. 4.

Draw a number line, plot the points and then write the numbers in increasing order 4.8, – 2.6, 0, -3, ½, - ½ 7, - ½ , 2.4, - ¾, - 5.8, 1/3 3 ½ , 3.4, 4.1, -5, -5.1, -4 ½

Your Turn  2. 3. 4. 5.

Find the opposite of the number 10 -3 3.8 -2.5

Your Turn 

Find the absolute value

3.

|7| |-4| |½| |0| + 2

4. 5. 6.

Your Turn Solutions 1. 2. 3. 4. 5. 6. 7.

-6 < 4 or 4 > -6 -6.4 < -6.3 or -6.3 > -6.4 -7 < 2 or 2 > -7 -2.7 < ¾ or ¾ > -2.7 -3, -2.6, - ½ , 0, ½ , 4.8 -5.8, - ¾ , - ½ , 1/3, 2.4, 7 -5.1, -5, -4 ½ , 3.4, 3 ½ , 4.1

1. 2. 3. 4. 5. 6. 7. 8.

-10 3 -3.8 2.5 7 4 ½ 2

Summary 

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A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about graphing, ordering and comparing numbers on the real number line as well as opposites and absolute value… Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide…

Credit 

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I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following:  Have your name, date and period as well a lesson number as a heading.  Do each of the your turn problems showing all work  Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted:  Without a complete heading  Without showing work for the your turn problems  Without a summary in your own words…