Determine if 0.515115111511115111115... is rational or irrational and give a reason for your answer.
To determine if 0.515115111511115111115... is rational or irrational, we can analyze its decimal representation.
Let's define the decimal representation as follows:
0.515115111511115111115... = 0.5 + 0.015 + 0.0015 + 0.00015 + ...
The first term, 0.5, is rational since it can be expressed as the fraction 1/2.
The subsequent terms, 0.015, 0.0015, 0.00015, and so on, can be expressed as the fraction n/10^k, where n is a positive integer and k is the number of digits after the decimal point. Therefore, they are also rational.
Since 0.515115111511115111115... can be represented as a sum of rational numbers, it is rational.
To determine if the number 0.515115111511115111115... is rational or irrational, we need to analyze its pattern.
Let's denote the repeating block of digits as "x." In this case, x = 1115.
We can write the given number as follows:
0.515115111511115111115... = 0.51 + 0.001x
Now, let's try to express this number in the form of a fraction.
Multiply both sides of the equation by 1000 (to shift the decimal point):
1000(0.515115111511115111115...) = 1000(0.51 + 0.001x)
This simplifies to:
515.115111511115111115... = 510 + x
Subtract 510 from both sides:
515.115111511115111115... - 510 = x
Simplifying further:
5.115111511115111115... = x
Now, let's multiply both sides by 1000 to get rid of the decimal:
1000(5.115111511115111115...) = 1000x
This gives us:
5115.111511115111115... = 1000x
Subtracting x from both sides:
5115.111511115111115... - 5.115111511115111115... = 1000x - x
Simplifying the expression:
5110 = 999x
Dividing by 999:
5110/999 = x
This equation shows that x is rational since it can be expressed as a fraction. Hence, the number 0.515115111511115111115... is rational.
To determine if the number 0.515115111511115111115... is rational or irrational, we need to examine the pattern and determine if it repeats or not.
Let's analyze the pattern:
0.5 | 151 | 15115 | 1115111 | 15111151111 | 5111111511111 | ...
It is clear that the pattern continues with the digits "151" being repeated. However, the length of the repeating block seems to be increasing with each repetition.
Since the repetition block is not the same length, we can conclude that the number formed by repeating this pattern is not a rational number. In other words, 0.515115111511115111115... is an irrational number.
To double-check, we can also express the number as a fraction. We can observe that the first three digits after the decimal point (0.515) can be written as 515/1000. However, the remaining digits do not form a finite fraction. This means that the number cannot be expressed as the ratio of two integers, confirming that it is irrational.