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Express 0.99999…0.99999\dots in the form (p)(q)\frac{(p)}{(q)}. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

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Full solution

Q. Express 0.99999…0.99999\dots in the form (p)(q)\frac{(p)}{(q)}. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
  1. Recognize decimal as repeating: Recognize that 0.999990.99999 is a repeating decimal, where the digit 99 repeats.
  2. Assign variable x: Let x=0.99999x = 0.99999.
  3. Multiply by 1010: Multiply both sides of the equation by 1010 to shift the decimal point one place to the right. So, 10x=9.999910x = 9.9999.
  4. Subtract original equation: Subtract the original equation x=0.99999x = 0.99999 from the new equation 10x=9.999910x = 9.9999 to get 9x=99x = 9.
  5. Divide by 99: Divide both sides by 99 to solve for xx. So, x=99x = \frac{9}{9}.
  6. Simplify fraction: Simplify the fraction 99\frac{9}{9} to 11.
  7. Realize equivalence: Realize that 0.999990.99999 as a fraction is 11, which is the same as 11.\frac{1}{1}.
  8. Discuss with teacher: Discuss with your teacher and classmates why this makes sense, because it seems weird but it's true that 0.999990.99999 is just another way to write 11.

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