Number Systems Converting rationals to decimals and vice versa
1. Changing a rational number into a decimal: This can be done by division or by using your calculator. 27
___ 32 = 0⋅84375
4 5 _
6 = 29 __
6 = 4⋅8 ˙ 3
2. Changing a terminating decimal into a rational: This can be done in two ways, as shown in Example 11 below.
EXAMPLE 11 Change 3⋅563 into a rational.
Solution 3⋅563 = 3 × 1 + 5 × 1 __
= 3 + 5 __
= 3563 _____
1000 10 + 6 10 + 6 × 1
___ 100 + 3
1000
____ 1000
= 3000 + 500 + 60 + 3 _________________
___ 100 + 3 × 1
____ 1000
or
Multiply above and below by 1000 because there are 3 digits to the right of the decimal point.
3⋅563 = 3⋅563 _____
1
× 1000 _____
1000 = 3563 _____
1000 Your calculator does this conversion for you.
1. Copy the Venn diagram below: Q
Z N
EXERCISE 3 2.
1
0
1
How many rational numbers are greater than 0 and less than 1?
If x is one of these rational numbers, describe x using mathematical symbols.
Q = set of rational numbers Z = set of integers N = set of natural numbers
(a) Fill the following numbers into the appropriate region of the Venn diagram:
−3, 0, 1 _
2 , 70 __
5 , 27, − 3 _
2 , 1
(b) (i) Is Q ∩ Z = Z? Why? (ii) Is Q ∩ N = N? Why?
(c) Shade in Q \ Z. How would you describe this set in words?
3. Plot the following rational numbers on a number line: 1
_ 3 , 5
(a) 1 _
(b) 4 _
(c) 2 _
(e) 2 _
5 + 3 _
5
7 − 1 _
7
9 − 5 _
9
(d) 2 _
3 + 3 _
3 × 4 _
5
5 − 1 _
4
_ 6 , 1
_ 6 , 2
_ 9
4. Simplify the following and check each answer on your calculator:
(f) (g)
(i)
( 2 _
(h) 2 _
(j) 2 _
5 5 ) 2
( 3 1 _
( 2 _
3
3 × 6 _
2 ) 2 7
) 3 × 9 _
8
( − 4 _
5
) + 3 _
5
21
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