Final answer:
The number 0.59595959... is rational because it can be written as the ratio 59/99, which is a fraction representing a recurring decimal as a ratio of two integers.
Step-by-step explanation:
To show that the number 0.59595959... is rational, we can write it as a ratio of two integers. First, let's denote the recurring decimal 0.59595959... by the symbol x.
Let x = 0.59595959...
To isolate the repeating part, we multiply x by 100, which gives us:
100x = 59.59595959...
Now, let's subtract the original x equation from the 100x equation:
100x - x = 59.59595959... - 0.59595959...
99x = 59
To find the value of x, we divide both sides by 99:
x = 59 / 99
Therefore, 0.59595959... can be expressed as the ratio 59/99, which is a fraction and thus shows that 0.59595959... is a rational number.