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Rational Numbers

Can all numbers be written as fractions? Don't be irrational - understand some of the fundamental classifications of numbers.

Level 3

         

\[\large \text{Is } 0.9999 \ldots = 1? \]

Note. The "\(\ldots\)" indicates that there are infinitely many 9's.

Find the number of rational numbers \(r, ~~0<r<1\), such that when r is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.

\[\large{A = 0.\overline{19} + 0.\overline{199} \qquad , \qquad B = 0.\overline{19} \times 0.\overline{199}}\]

Recall that \(0.\overline{19}\) stands for the repeating decimal \(0.19191919...\) for example, and that the period of a repeating decimal is the number of digits in the repeating part. In this case, the period of \(0.\overline{19}\) is 2. Find the sum of the periods of \(A\) and \(B\).

Let \(a_k\) represent the repeating decimal \(0.\overline{133}_k\) for \(k \geq 4\). The product \(a_4 a_5 \cdots a_{99}\) can be expressed as \(\frac{m}{n!}\) where \(m, n\) are positive integers and \(n\) is as small as possible. \(\frac{m}{n}\) can be expressed as \( \frac{p}{q}\) where \(p, q\) are coprime integers. What is \(p+q\)?

If the following infinite series \(S\) is evaluated as a decimal,what is the 37th digit to the right of the decimal place?

\[ \large S=\frac { 1 }{ 9 } +\frac { 1 }{ 99 } +\frac { 1 }{ 999 } +\ldots + \frac { 1 }{ { 10 }^{ n }-1 } + \ldots \]

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