1. Repeating
Decimals
and
Irrational
Numbers

2. Repeating Decimals
•Can be represented as a fraction
•The denominator of the simplified form of the
fraction has prime factors other than 2 or 5
Writing repeating decimals
0.3 = 0.3333…
3.465 = 3.4656565…

3. Write 5 as a decimal
22
0. 22727
22 5 0
44 0.227
60
44
160
154
60
44
160

4. Order from smallest to largest:
3.6, 3.6, 3.76, 3.63
1 3.6 = 3.60000…
3 3.6 = 3.66666…
4 3.76 = 3.76666…
2 3.63 = 3.63636…
3.6, 3.63, 3.6, 3.76

5. Write 0.12 as a fraction in simplest form
100x = 12.121212…
x = 0.121212…
99x = 12
99 99
x = 12
99
= 4
33

6. Write 3.6 as a fraction in simplest form
10x = 36.66666…
x = 3.66666…
9x = 33
9 9
x = 11
3

7. Real Numbers
Counting Numbers: 1, 2, 3, 4, …
Whole Numbers: 0, 1, 2, 3, …
Integers: …, -3, -2, -1, 0, 1, 2, 3, …
Rational Numbers: can be written as fractions
Irrational Numbers: non- repeating, non-
terminating decimals
Some irrational Numbers:
π 2 3
10 7.272272227...

8. What are the square roots of 36?
6 because 62 = 36
-6 because (-6)2 = 36
Every positive number has two square roots.
The principal square root of a number is its
positive square root.

9. Find:
a) The square roots of 25 5 and -5
b) The principal square root of 25 5
c) 25 = 5
d) 25 = -5

10. How long is 2?
Square’s Length of
area side
1 1
2 2
4 2
2 9 3
2 is between 1 and 2

11. Simplifying roots
1 1
4 2
9 3
18 2 33 3 2
75 35 5 5 3
3 3
8 2 2 2 2
3 3 3
24 2 2 2 3 2 3

12. Order from smallest to largest:
31
8.9, 65%, 65 , 4
4 8.9 = 8.9
1 65% = 0.65
3 65 = A little larger than 8
2 31 = 7 3
4 4
31
65%, , 65 , 8.9
4

13. MAP 8th-grade
released item, 2006

14. Pythagorean Theorem
•Relates the sides of a right triangle
b c
a2 + b2 = c2
a
x2 + 42 = 62
x 6 x2 + 16 = 36
-16 -16
4 x2 = 20
x = 20 or 2 5

15. DESE Sample MAP item, grade 8