The number 0.555555... can be written as the fraction ____, and is a ____ number.
The number 0.555555... can be written as the fraction 5/9, and is a repeating decimal number.
The number 0.555555... can be written as the fraction 5/9, and is a rational number.
To understand why 0.555555... can be written as the fraction 5/9, we can use a method called geometric series. In this case, the repeating decimal 0.555555... can be expressed as an infinite sum of a geometric series.
The first step is to observe that 0.555555... is the same as 0.5*1 + 0.05*1 + 0.005*1 + ... and so on. We can see that each term after the first is obtained by dividing the previous term by 10.
Now, let's call the sum of this infinite series "S". We have:
S = 0.5*1 + 0.05*1 + 0.005*1 + ...
To find S, we multiply S by 10, which gives us:
10S = 5*1 + 0.5*1 + 0.05*1 + ...
Now, subtracting the original equation from this new equation, we get:
10S - S = 5 + 0.5 + 0.05 + ...
Simplifying the left side, we get:
9S = 5.55555...
Dividing both sides by 9:
S = 5.55555... / 9
Now, notice that the right side of this equation is just 5/9, which means that 0.555555... is equal to 5/9.
Therefore, 0.555555... can be written as the fraction 5/9. This number is considered a rational number because it can be expressed as a ratio of two integers.