Download Rational and Irrational numbers weeks 4,5

Rational Numbers
converting fractions to decimals
Every number has a decimal expansion
1
12
1
4
1
2
2
5
6
3
16
1
10
1
6
14
27
Rational Numbers
A rational number is the ratio of 2 integers
Rational Numbers
a
b
rational number
Rational numbers can terminate in 0 or repeat.
Which of these numbers terminate in 0?
Which of these numbers repeat?
3
4
5
6
12
7
10
11
20
4
fraction
8
decimal
1
7
terminating
recurring
ends in 0
repeats
converting decimals to rational
converting decimals to rational
Recurring decimals
Terminating decimals
0.7
0.7
let x equal the
reccuring decimal
let x = 0.777777..... (1)
10x = 7.777777.....
0.045
multiply (1) by 10 {tenths}
subtract (2) - (1)
2.34
0.625
(2)
10x = 7.777777.....
x = 0.777777.....
9x = 7
x= 7
9
converting decimals to fractions
0.564
Recurring decimals
let x =
EASY
(1)
0.564564564.....
1000x = 564.564564.....
(2)
multiply (1) by 1000 {thousandths}
Change 0.3 into fraction form
MEDIUM
Change 0.45 into fraction form
TRICKY
Change 0.4714285 into fraction form
subtract (2) - (1)
1000x = 564.564564.....
x = 0.564564564.....
999x = 564
x = 564
999
Change 0.317 into fraction form
CHALLENGE
Change 2.16 into fraction form
= 188
333
Change 0.333 into fraction form
x = 0.333
10x = 3.333
9x = 3
x = 3= 1
9 3
Change 0.45 into fraction form
x = 0.4545...
100x = 45.4545...
99x = 45
x = 45 = 15
99 33
Change 0.31717... into fraction form
x = 0.31717...
100x = 31.717...
99x = 31.4
x = 31.4 = 314
99 990
Change 2.166... into fraction form
Change 0.4714285 into fraction form
x=
0.4714285...
10000000x = 4714285.4714285...
9999999x = 4714285
x = 4714285
9999999
0.4
0.45
0.432
0.409
0.428571
0.461538
EXTENSION
4
/9
5
/11
16
/37
9
/22
3
/7
6
/13
x = 2.1666...
10x = 21.6666...
9x = 19.5
x = 19.5 = 195 =
9
90
21/6
Pick any terminating decimal. Label it D.
Count how many places there are after the decimal point and label this
number P.
Now multiply D by 10P
10P
Reduce the fraction.
What have you demonstrated?
Decide whether each of the following numbers is rational
or irrational.
If it is rational, explain how you know.
If 12 = 1
Then what does √1 equal?
√36?
What about √4?
If the square root of a number is an integer then it is a perfect square.
What about √2?
It's not an integer.
It's not a terminating decimal.
It's not a repeating decimal.
Because it does neither of the above it is an irrational number.
What about -√2?
A
B
C
D
E
F
G
0.333
√4
√2
π = 3.141592..
11
1/7 = 0.142857
12.3456565656
What other irrational numbers can you think of?
Graph √½ and ½ on a number line.
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0
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Complete this statement.
√½
½
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Graph √ and
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0
< = >
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on a number line.
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Complete this statement.
√
<
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= >
Ordering real numbers
Complete this statement.
We are going to order these decimals from least to greatest.
4
√8
< =>
0.47,
0.474,
0.47, √0.23
First, decide if these numbers are rational or irrational
4
5
√ 45
< = >
1
4
√41
< =>
3
< = >
√7
Ordering real numbers
We are going to order these decimals from least to greatest.
0.52,
0.525,
0.525, √0.276
First, decide if these numbers are rational or irrational
Write them to 6 decimal places
Write them to 6 decimal places
0.477777...
0.474474...
0.474747...
0.479583...
Textbook
Page 477
Exercise 3-15
Page 478
Ex 16, 19, 32, 34, 36, 38, 40
0.525252...
0.525525...
0.525555...
0.525357...
For each pair of numbers, decide which is larger without using a calculator. Explain your choices.
Without using your calculator, label approximate locations for the following numbers on the number line.