Determine if 0, point, 8, 6, 8, 6, 6, 8, 6, 6, 6, 8, 6, 6, 6, 6, 8, 6, 6, 6, 6, 6, 8, point, point, .0.868668666866668666668... is rational or irrational and give a reason for your answer.
To determine whether the given number is rational or irrational, we need to analyze its decimal representation.
The given number is 0.868668666866668666668..., and it appears to have a repeating pattern. This pattern consists of the digits 8 and 6 repeating.
A rational number is defined as a number that can be expressed as a fraction of two integers. Irrational numbers, on the other hand, cannot be expressed as a fraction.
By looking at the decimal representation, we can see that the repeating pattern indicates that the number can be expressed as a fraction. To convert it into fraction form, let's denote it as x:
x = 0.868668666866668666668...
To get rid of the repetitive part, we can subtract an identical number multiplied by a power of 10:
1000x = 868.668666866668666668...
Now let's subtract the original equation from this:
1000x - x = 868.668666866668666668... - 0.868668666866668666668...
This simplifies to:
999x = 868
Dividing both sides by 999:
x = 868/999
Since x can be expressed as a fraction of two integers, it is a rational number.
Therefore, 0.868668666866668666668... is a rational number.
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To determine if 0.868668666866668666668... is rational or irrational, we need to analyze the pattern of the repeating decimals.
Step 1: Let's identify the pattern in the given decimal. We can see that the pattern repeats after the first three digits, which are 868.
Step 2: Let's denote the repeating part of the decimal as x. Therefore, x = 868.
Step 3: Now, let's express the given decimal as an algebraic expression. The given decimal can be expressed as 0.868 + x/1000 + x/1000000 + ...
Step 4: Simplifying the expression, we get 0.868 + x/1000 + x/1000000 + ... = 0.868 + 0.001x + 0.000001x + ...
Step 5: Further simplifying the expression, we get 0.868 + 0.001x + 0.000001x + ... = 0.868 + x(0.001 + 0.000001 + ...)
Step 6: The sum of the infinite geometric series 0.001 + 0.000001 + ... can be calculated as a/(1 - r), where a is the first term (0.001) and r is the common ratio (0.000001).
Step 7: Applying the formula, we get 0.868 + x(0.001 + 0.000001 + ...) = 0.868 + x(0.001/(1 - 0.000001))
Step 8: Simplifying further, we get 0.868 + x(0.001/(1 - 0.000001)) = 0.868 + 0.001x/(0.999999)
Step 9: Now, let's express the repeating part of the decimal as a fraction. Since x = 868, putting the value in the expression above, we get 0.868 + 0.001(868)/(0.999999) = 0.868 + 0.868/(0.999999)
Step 10: Simplifying the expression, we get 0.868 + 0.868/(0.999999) = 0.868 + 0.868869
Step 11: Combining the decimal and the repeating part, we get 0.868 + 0.868869 = 1.736869
Step 12: Now, let's examine whether the resulting number is rational or irrational. If the resulting number can be expressed as a fraction of two integers, then it is rational. Otherwise, it is irrational.
Since the resulting number 1.736869 cannot be expressed as a fraction of two integers, it is irrational.
Therefore, the number 0.868668666866668666668... is irrational.
To determine whether the given number, 0.868668666866668666668..., is rational or irrational, we need to analyze its pattern.
First, let's simplify the pattern by grouping the repeating decimal as follows: 0.8686(686668)...
Now, let's define the terms in the pattern:
- Term 1: 0.8686
- Term 2: 686668
We notice that Term 2 repeats indefinitely.
To determine if the number is rational or irrational, we need to check if the repeating decimal can be written as a fraction (ratio) of two integers.
To do this, let's assign a variable to our number. Let's call it x:
x = 0.868668666866668666668...
Now, let's multiply our number by a power of 10 to eliminate the repeating part:
1000x = 868.668666866668666668...
Next, let's subtract the original number from the expanded one:
1000x - x = 868.668666866668666668... - 0.868668666866668666668...
Simplifying the equation:
999x = 868
Now, divide both sides of the equation by 999:
x = 868 / 999
At this point, we have expressed the number as a fraction, specifically 868/999, which means it can be written as a ratio of two integers. Therefore, the number is rational.
To summarize, the number 0.868668666866668666668... is rational because it can be represented as the fraction 868/999.