To convert the repeating decimal [tex]\(0.2\dot{8}\)[/tex] into a fraction, let's follow these steps:
1. Represent the Decimal:
The decimal [tex]\(0.2\dot{8}\)[/tex] means [tex]\( 0.288888... \)[/tex], where "8" is the repeating part.
2. Construct an Equation:
Let [tex]\( x = 0.2\dot{8} \)[/tex].
Then [tex]\( 10x = 2.\dot{8} \)[/tex] because multiplying by 10 shifts the decimal point one place to the right.
Also, [tex]\( 100x = 28.\dot{8} \)[/tex], because multiplying by 100 shifts the decimal point two places to the right, placing "28" before the repeating decimal part.
3. Subtracting to Remove the Repeating Part:
We subtract the first equation from the second to eliminate the repeating part:
[tex]\[
100x - 10x = 28.\dot{8} - 2.\dot{8}
\][/tex]
Simplifying this gives:
[tex]\[
90x = 26
\][/tex]
4. Solving for [tex]\(x\)[/tex]:
Divide both sides by 90:
[tex]\[
x = \frac{26}{90}
\][/tex]
5. Simplifying the Fraction:
We need to simplify [tex]\(\frac{26}{90}\)[/tex]. The greatest common divisor (GCD) of 26 and 90 is 2. So, we divide the numerator and the denominator by 2:
[tex]\[
\frac{26 \div 2}{90 \div 2} = \frac{13}{45}
\][/tex]
Thus, the fraction representation of [tex]\(0.2\dot{8}\)[/tex] is [tex]\( \frac{13}{45} \)[/tex].
In terms of floating-point precision, this fraction [tex]\( \frac{13}{45} \)[/tex] approximates to the decimal [tex]\( 0.288888... \)[/tex], which corresponds to the original repeating decimal [tex]\(0.2\dot{8}\)[/tex].
6. Further Precision:
For more precision, using the result:
[tex]\[
0.2888888888888889 \equiv \frac{5204159569405907}{18014398509481984}
\][/tex]
Hence, the exact fraction for enhanced precision of the repeating decimal [tex]\(0.2\dot{8}\)[/tex] is [tex]\(\frac{5204159569405907}{18014398509481984}\)[/tex].
