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Chapter 7: Problem 20

Express $$ 0.859859859 \ldots $$ as a fraction; here the digits 859 repeat forever.

Short Answer

Expert verified
The repeating decimal value \(0.859859859 \ldots\) can be expressed as the fraction \(\frac{859}{999}\).

Step by step solution

01

Assign a Variable

Let x represent the given repeating decimal value, which can be written as: \(x = 0.859859859 \ldots\)
02

Shift the Decimal Point

Since the repeating part consists of 3 digits (859), we'll multiply both sides of the equation by a power of 10, in this case \(10^3\), to shift the decimal point 3 places to the right: \(1000x = 859.859859859 \ldots\)
03

Subtract the Original Equation from the Shifted Equation

Now, we'll subtract the original equation (x) from the shifted equation (1000x), which will eliminate the repeating decimal: \(1000x - x = 859.859859859 \ldots - 0.859859859 \ldots\) Simplify the equation: \(999x = 859\)
04

Solve for x

Next, we'll solve for x by dividing both sides of the equation by 999: \(x = \frac{859}{999}\) So, the repeating decimal value \(0.859859859 \ldots\) can be expressed as the fraction \(\frac{859}{999}\).

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Most popular questions from this chapter

Evaluate the arithmetic series. $$ 1001+1002+1003+\cdots+2998+2999+3000 $$

Assume \(n\) is a positive integer. Find the coefficient of \(w^{198}\) in the expansion of \((w+3)^{200}\).

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Explain why the polynomial \(p\) defined by $$ p(x)=\frac{x^{4}-10 x^{3}+39 x^{2}-50 x+24}{4} $$ is the only polynomial of degree 4 such that \(p(1)=1\), \(p(2)=4, p(3)=9, p(4)=16,\) and \(p(5)=31\). The graph of \(\frac{x^{4}-10 x^{3}+39 x^{2}-50 x+24}{4}\) on the interval [-1,5] .

Express $$ 5.1372647264 \ldots $$ as a fraction; here the digits 7264 repeat forever.
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