show that the following recurring decimals are rational.
(1) 0,4 (the dot is on the 4)
(2) 0,21 (the dot is on both 2&1)
To show that a recurring decimal is rational, we need to show that it can be expressed as a ratio of two integers.
(1) 0,4:
Let x = 0,4. Since the decimal part is repeating only one digit, we can write x as follows:
10x = 4,4 (multiply both sides by 10)
Subtracting the equation x from 10x, we get:
10x - x = 44 - 0,4
9x = 4 (subtract the decimals)
Dividing both sides of the equation by 9:
x = 4/9
Therefore, 0,4 is rational and can be written as the fraction 4/9.
(2) 0,21:
Let x = 0,21. Since the decimal part is repeating two digits, we can write x as follows:
100x = 21,21 (multiply both sides by 100)
Subtracting the equation x from 100x, we get:
100x - x = 2121 - 21
99x = 2100 (subtract the decimals)
Dividing both sides of the equation by 99:
x = 2100/99
Reducing the fraction, we get:
x = 100/33
Therefore, 0,21 is rational and can be written as the fraction 100/33.