Determine if 0.51515111511115111115... is rational or irrational and give a reason for your answer.
0.51515111511115111115... is __________ because ____________________________
1. It is a decimal that repeats
2. It is the square root of a perfect square
3. It is a decimal that terminates
4. It is a decimal that does not repeat nor terminate
5. It is the square root of a non-perfect square
0.51515111511115111115... is irrational because it is a decimal that does not repeat nor terminate.
0.51515111511115111115... is a decimal that repeats, so it can be represented as a fraction. This means it is rational.
Here is the reasoning:
1. The decimal representation of 0.51515111511115111115... repeats, with the repeating block being "151".
2. The pattern "151" can be written as 151/1000. (The repeating block represents the numerator, and the denominator is obtained by counting the number of digits in the block and posing them as "9"s in the denominator, i.e., 1 digit = 9, 2 digits = 99, 3 digits = 999, and so on.)
3. Therefore, 0.51515111511115111115... can be expressed as the fraction 151/1000.
4. Since it can be expressed as a fraction, it is rational.
0.51515111511115111115... is a rational number because it is a decimal that repeats.
To determine whether a decimal is rational or irrational, we need to analyze its pattern. In this case, we see that the digits "51" repeat continuously. Whenever a decimal repeats in a pattern, it can be expressed as a fraction of two integers.
To find the fraction that represents this repeating decimal, we can designate the repeating part as "x" and write an equation to solve for it as follows:
x = 0.51515111511115111115...
We can multiply both sides of the equation by a power of 10 to eliminate the repeating part. Since there are two digits repeating, we can multiply by 100:
100x = 51.51515111511115111115...
Next, we can subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 51.51515111511115111115... - 0.51515111511115111115...
This simplifies to:
99x = 51
Now, we can solve for x:
x = 51/99
This fraction represents the repeating decimal, so we have successfully expressed it as a rational number. Therefore, 0.51515111511115111115... is a rational number because it is a decimal that repeats.