Express 0.99999 ... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Question
Express $0.99999 \ldots$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Answer
To express $0.99999\ldots$ (also written as $0.\overline{9}$) in the form $\frac{p}{q}$, we use a simple algebraic trick.
Let $x = 0.\overline{9}$. Then:
$$ 10x = 9.\overline{9} $$
Subtracting the original equation from this, we get:
$$ 9x = 9 $$
Therefore, $x = 1$. This means that $0.\overline{9}$ can be expressed in the form $\frac{p}{q}$ as $\frac{1}{1}$.
Many people find this result surprising because it's unintuitive at first that an infinite series of $9$s after the decimal point is exactly equal to $1$. It makes sense when you consider the properties of limits and infinite series in mathematics, particularly in how we understand and define real numbers. The concept that $0.\overline{9} = 1$ is a well-established and accepted principle in mathematics, underscored by our algebraic manipulation here and in the principles of calculus and analysis. It challenges our initial intuitions about numbers and infinity but is firmly rooted in mathematical theory.
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