Name: Date: Lesson 1.3 Introducing Irrational Numbers
Circle all the irrational numbers in each set of numbers.
1
9
1. 4 , 5, , 2 , 2 2.31, 7 2. 2
19
, 2 23 , 13,
11
8 , 3.001
Locate each positive irrational number on the number line using rational
approximation. First tell which two whole numbers the square root is between.
Example
13
should
lie between 3 and 4 because 32 5 9 and 42 5 16.
13 
3.605551275...
Step 1
Find an approximate value for
13 lies between the tenths
3.6
3.7
and
. So,
3.6  13  3.7 .
Step 2
Graph the interval from 3.6 to 3.7 on a number line.
© Marshall Cavendish International (Singapore) Private Limited.
13 by using a calculator: 13 5 3.605551275...
3.6
3.7
Step 3
The value of
3.61 is closer to
Step 4
Use 3.61 to locate 13 approximately
on the number line.
13 with two decimal places is
3.6
than to
3.7
3.61
. So,
.
13 is located closer to
3.6
3.6
.
3.7
13
Complete.
3.
14
Which two whole numbers is 14 between?
Find an approximate value of 14 by using a calculator: 14 5
14 lies between the tenths
and
and
The value of 14 with two decimal places is
Which tenth is 14 located closer to?
.
.
.
Reteach Course 2A 15
MIF_Reteach C2_Ch01.indd 15
15/12/11 1:46 AM
Name: Date: Locate each positive irrational number on the number line using rational
approximations. First tell which two whole numbers the square root is between.
4.
8
5.
6
6.
23 7.
32
Locate each negative irrational number on the number line using rational
approximations. First tell which two integers the square root is between.
2 11
Because 32 5 9 and 42 5 16, 11 is between 3 and 4, and 2 11 is between 23 and 24.
Step 1
Find an approximate value for 2 11 by using a calculator: 2 11 5 2
2 11 lies between the tenths
Step 2
Graph the interval from 23.3 to 23.4 23.3
on a number line.
163.31662479...
23.4 . So, 23.4  2 11  23.3 .
and
3.4
3.3
Step 3
Use the approximate value of 2 11 with two decimal places.
The value of 2 11 with two decimal places is
23.32 is closer to
23.3
than to
23.4
Step 4
Use 23.32 to locate 2 11 approximately on the number line.
23.32 .
. So, 2 11 is located closer to
23.3
3.4
11
.
© Marshall Cavendish International (Singapore) Private Limited.
Example
3.3
16 Chapter 1 Lesson 1.3
MIF_Reteach C2_Ch01.indd 16
15/12/11 1:46 AM
Name: Date: Complete.
8. 2 8
Which two integers is 2 8 between?
and
Find an approximate value of 2 8 by using a calculator: 2 8 5
2 8 lies between the tenths
The value of 2 8 with two decimal places is
Which tenth is 2 8 closer to?
and
.
.
© Marshall Cavendish International (Singapore) Private Limited.
Locate each negative irrational number on the number line using rational
approximations. First tell which two integers the square root is between.
9. 2 7 10. 2 21
11. 2 34 12. 2 48
Use a calculator. Locate each irrational number to three decimal places on the
number line using rational approximations.
13.
53 14. 2 117
Reteach Course 2A 17
MIF_Reteach C2_Ch01.indd 17
15/12/11 1:46 AM
22. Method 1
7. 5; 6
7
22 5 22.875
8
5.7
32
22.875 5 2.875
22 5 2 2.888... 5 2 2.8
8
9
22.8 5 2.8
2 8 522.828427125...
2 8 lies between the tenths 22.8 and 22.9.
The value of 2 8 with two decimal places is
22.83.
Which tenth is 2 8 closer to? 22.8.
7
2
8
8
2
9
2.8 units
2.875 units
3
2
0
1
2.8 2.875
2.9
From the number line, you see that 22.8 lies
farther to the left of 0 than 22.875.
9. 22; 23
So, 22.875  22.8
7
8
8
9
22  22
So, 22.875  22.8
7
8
8
9
4
22
1
4
5
7
10
9
,
2 , 22.31,
11
7
8 , 3.001
Find an approximate value of
14 by using a
The value of 14 with two decimal places is 3.74.
Which tenth is 14 closer to? 3.7
calculator:
14 5 3.741657387...
3.7
4.7
7.0
6.9
48
7.2
7.3
53
10.8
117
3.8
14
1. 
2. 
3. 
4. 
5. Use a calculator to represent each number in
decimal form with 3 decimal places.
0.831  0.832, 2 8  22.828,
1
p
 1.571,  0.143,
2 7  22.646
Ordering the numbers from least to greatest
using the symbol ,
1
p
2 8  2 7   0.831 
14 lies between the tenths 3.7 and 3.8.
4. 2; 3
5.8
34
Lesson 1.4
14 between?
2.4
6. 4; 5
5.9
14. 210.817
3. Which two whole numbers is
3 and 4
2.8
5. 2; 3
4.5
21
10.9
1
2. 2 19 , 2 23 , 13,
4.6
Lesson 1.3
2.6
7
13.7.280
3
4 , 5,
12. 27; 28
23. 2  2 24. 2  23
1.
2.7
11. 25; 26
The number with the greater absolute value is
22.8 and it is farther to the left of 0. Hence it is
the lesser number.
22  22
2.8
8
10. 24; 25
Method 2
22.875  22.8
© Marshall Cavendish International (Singapore) Private Limited.
5.6
8. Which two integers is 2 8 between? 22 and 23
Find an approximate value of 2 8 by using a
calculator:
7
2.9
8
2.5
6
23
7
2
8 3
2
1
7
7
2
1
0
0.831
1
π
2
2
4.8
Reteach Course 2A 157
MIF_Reteach C2_Ch01-05_Ans.indd 157
15/12/11 5:18 AM