Question

is -9.121212121... rational
Answer
is -1.500000000... rational
Answer
is -5.690134489... rational
Answer
is -7.855555555... rational
Answered by GPT-4o mini
Yes, the number \(-9.121212121...\) is a rational number. A rational number is defined as any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.

The decimal \(-9.121212121...\) is a repeating decimal, which can be expressed as a fraction. The repeating part is \(12\), indicating that it can be represented as:

\[
-9.121212121... = -9 - 0.121212121...
\]

The repeating decimal \(0.121212121...\) can be converted to a fraction:

Let \(x = 0.121212121...\).

Multiplying both sides by 100 (since the repeating block is two digits long):

\[
100x = 12.121212121...
\]

Now, subtracting the first equation from this new equation:

\[
100x - x = 12.121212121... - 0.121212121...
\]
\[
99x = 12
\]
\[
x = \frac{12}{99} = \frac{4}{33}
\]

So, \(0.121212121... = \frac{4}{33}\).

Now we can express \(-9.121212121...\) as:

\[
-9.121212121... = -9 - \frac{4}{33} = -\frac{297}{33} - \frac{4}{33} = -\frac{301}{33}
\]

Since \(-\frac{301}{33}\) is a fraction of two integers, \(-9.121212121...\) is indeed a rational number.
Answered by GPT-4o mini
Yes, the number \(-1.500000000...\) is a rational number.

A rational number is any number that can be expressed as the quotient of two integers (a fraction), where the denominator is not zero.

The decimal \(-1.500000000...\) is essentially \(-1.5\), which can be expressed as a fraction:

\[
-1.5 = -\frac{15}{10} = -\frac{3}{2}
\]

Since \(-\frac{3}{2}\) is a fraction with both the numerator and denominator as integers, \(-1.500000000...\) is indeed a rational number.
Answered by GPT-4o mini
Yes, the number \(-5.690134489...\) is a rational number.

The decimal \(-5.690134489...\) is a non-repeating decimal, but rational numbers also include terminating decimals, which can be expressed as fractions.

The decimal \(-5.690134489...\) can be considered as a finite decimal when rounded to a certain number of decimal places. If we only consider it as \(-5.690134489\), it can be expressed as:

\[
-5.690134489 = -\frac{5690134489}{1000000000}
\]

Here, \(5690134489\) and \(1000000000\) are both integers, and the denominator is not zero. Since it can be expressed as a fraction of two integers, \(-5.690134489...\) is indeed a rational number.

If the decimal were to extend infinitely without repeating, it could still be considered rational if it can be approximated effectively to a fraction as shown above. Therefore, in general, it can be said that \(-5.690134489...\) is a rational number.
Answered by GPT-4o mini
Yes, the number \(-7.855555555...\) is a rational number.

The decimal \(-7.855555555...\) can be expressed as:
\[
-7.855555555... = -7.85 - 0.000000000...
\]
The decimal part \(-0.000000000...\) simply indicates that the 5s continue indefinitely. We can express this number in a more manageable form:

Let \(x = 0.855555555...\).

To convert this repeating decimal into a fraction:

1. Multiply both sides of the equation by 10 to shift the decimal point:
\[
10x = 8.555555555...
\]

2. Now, subtract \(x = 0.855555555...\) from \(10x\):
\[
10x - x = 8.555555555... - 0.855555555...
\]
\[
9x = 7.7
\]

3. So,
\[
x = \frac{7.7}{9}
\]

4. The decimal \(7.7\) can be expressed as a fraction:
\[
7.7 = \frac{77}{10}
\]
Therefore,
\[
x = \frac{77/10}{9} = \frac{77}{90}
\]

Now that we have \(x = 0.855555555...\) as \(\frac{77}{90}\), we can write:
\[
-7.855555555... = -7 - 0.855555555... = -7 - \frac{77}{90}
\]

Now to combine:
1. Convert \(-7\) into a fraction:
\[
-7 = -\frac{630}{90}
\]
2. Then combine the fractions:
\[
-7.855555555... = -\frac{630}{90} - \frac{77}{90} = -\frac{707}{90}
\]

Since \(-\frac{707}{90}\) is a fraction of two integers (where the denominator is not zero), this confirms that \(-7.855555555...\) is indeed a rational number.