1. The Number Line and Absolute Values
Topic 1.2.3

2. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
California Standard:
1.1 Students use properties of numbers to demonstrate whether assertions are true or false.
What it means for you:
You’ll use the number line to show real numbers, and you’ll describe them in terms of absolute values.
Key words:
real numbers
absolute value

3. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
The number line is a useful way of representing numbers visually. –5 –4 –3 –2 –1 1 2 3 4 5
This Topic also includes information about absolute values, which show how far from zero numbers are on the number line.

4. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Number Lines Show Real Numbers as Points on a Line
All real numbers can be found on the number line, no matter how big or small they are, and no matter whether they are rational or irrational. 1 2 –6 –5 –4 –3 –2 –1 3 4 5 6
–4.5 Ö2
1,000,000,000
And for every point on the number line, there is a real number.
So there’s a one-to-one correspondence between the real numbers and the points on the number line.

5. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
graph of –4
coordinate = 1 –5 –4 –3 –2 –1 1 2 3 4 5
negative numbers
positive numbers
For each point on the number line, the corresponding real number is called the coordinate of the point.
And for each real number, the corresponding point is called the graph of the number.
The numbers to the left of zero on the number line are all negative — they’re less than zero.
The numbers to the right of zero are positive — they’re greater than zero. Zero is neither negative nor positive.

6. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Guided Practice
1. “Only integers can be found on the number line. Is this statement true or false?
False. All real numbers can be found on the number line.
2. Identify the corresponding real numbers of points A–E on the number line below.
A is –4, B is –1, C is 0, D is 2, E is 4.5. –5 –4 –3 –2 –1 1 2 3 4 5 6 7 A B C D E
Use a copy of this number line to answer questions 3–4.
4. –2
3. 6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7
3. Draw the graph of 6 on the number line.
4. Draw the graph of –2 on the number line.
Solution follows…

7. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
The Number Line Can Be Useful for Calculations
If you think of the number line as a road, then you can think of coordinates as movement along the road — either to the left or to the right, depending on the coordinate’s sign.
For example, –5 would indicate a movement of 5 units to the left.
While 4 would mean 4 units to the right. –5 –4 –3 –2 –1 1 2 3 4 5

8. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Example 1
Find 3 –
Solution
Rewriting this as 3 + (–2) + 4, you can interpret this as:
“Start at 3, move 2 to the left (to reach 1) and then 4 to the right (to reach 5).”
So 3 – = 5.
Solution follows…

9. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
The Absolute Value of a Number is its Distance from 0
The opposite of a real number c (that is, –c) lies an equal distance from zero as c, but on the other side of zero.
So the opposite of 4 (which lies 4 units to the right of zero) is –4 (which lies 4 units to the left of zero).
And the opposite of –7 (which lies 7 units to the left of zero) is 7 (which lies 7 units to the right of zero).

10. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
The distance from zero to a number is called the number’s absolute value.
It doesn’t matter whether it is to the left or to the right of zero — so absolute value just means the “size” of the number, ignoring its sign.
The absolute value of c is written |c|.
More algebraically…
c if c > 0
|c| = 0 if c = 0
–c if c < 0
The absolute value of a number can never be negative.

11. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Example 2
Find: a) |6| b) |0| c) |–23|
Solution
a) 6 is positive, so |6| = 6.
b) |0| = 0 (by definition).
c) –23 is negative, so |–23| = 23.
Solution follows…

12. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Guided Practice
Find the following absolute values.
5. |6| 6
6. |15| 15
7. |–3| 3
8. |–8| 8
10. 1 2 1 2
9. |0| 1
12. 3 –
11. Ö2 1 3 Ö2
Solution follows…

13. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Independent Practice
1. Choose the correct word from each pair to complete this sentence.
On a number line, positive numbers are found to the (left/right) of zero and negative numbers are found to the (left/right) of zero.
2. On a copy of the number line below, draw the graphs of the following coordinates: –
–3.5
0.5 5 –5 –4 –3 –2 –1 1 2 3 4 5 6 7
Solution follows…

14. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Independent Practice
In exercises 3–11, find x.
3. |3| = x 3
4. |–10.5| = x
10.5
5. |–2| = x
6. |3.14| = x 2
3.14
7. |–2.17| = x
8. |x| = 0
2.17
9. |465| = x
10. |–465| = x
465
465
11. |x| = (Hint: Look at exercises 9 and 10)
Two possible answers: 465 or –465
12. What is wrong with the equation |x| = –1?
It is not possible — the absolute value can never be negative.
Solution follows…

15. The Number Line and Absolute Values
Topic
1.2.3
The Number Line and Absolute Values
Round Up
You’ve seen the number line plenty of times in earlier grades, but it’s always useful.
You don’t always need to draw it out, but you can imagine a number line to work out which direction an operation will move a number.