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1.3 Prove that [tex]1.\dot{3}[/tex] is a rational number.
To prove that [tex]\(1.\overline{331}\)[/tex] is a rational number, we proceed as follows:
1. Define the repeating decimal:
Let [tex]\( x = 1.\overline{331} \)[/tex]. This notation means that the sequence "331" repeats infinitely after the decimal point.
2. Shift the decimal point to the right three places:
Since the repeating part has 3 digits, we multiply [tex]\( x \)[/tex] by [tex]\(1000\)[/tex]:
[tex]\[
1000x = 1331.\overline{331}
\][/tex]
3. Set up the equation to eliminate the repeating decimal:
Now, we have two equations: [tex]\[
x = 1.\overline{331} \quad \text{(Equation 1)}
\][/tex] [tex]\[
1000x = 1331.\overline{331} \quad \text{(Equation 2)}
\][/tex]
By subtracting Equation 1 from Equation 2, we eliminate the repeating part: [tex]\[
1000x - x = 1331.\overline{331} - 1.\overline{331}
\][/tex]
4. Simplify the subtraction:
On the left side we have: [tex]\[
1000x - x = 999x
\][/tex] On the right side we have: [tex]\[
1331.\overline{331} - 1.\overline{331} = 1330
\][/tex]
Therefore: [tex]\[
999x = 1330
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], divide both sides by 999: [tex]\[
x = \frac{1330}{999}
\][/tex]
6. Simplify the fraction:
To confirm that [tex]\(\frac{1330}{999}\)[/tex] is in its simplest form, we can check for common factors. Both the numerator and the denominator can be divided by their greatest common divisor (GCD), which is 1 in this case:
Notice that the fraction is already in its simplest form; it does not simplify further.
Therefore, [tex]\(1.\overline{331} = \frac{1330}{999}\)[/tex], proving that [tex]\(1.\overline{331}\)[/tex] is a rational number because it can be expressed as a ratio of two integers.
By transforming a repeating decimal into a fraction, we've shown explicitly that [tex]\(1.\overline{331}\)[/tex] is indeed rational.
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