Question

Write \(2 / 7\) as a repeating decimal.

Short answer

The repeating decimal representation of the fraction \(2/7\) is \(0.\overline{285}\), where the line above the digits 285 indicates that these digits repeat indefinitely.

Step-by-step solution

Set up long division

Set up the division, with 2 as the dividend and 7 as the divisor: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{0} & . & 2 & 8 \\ \cline{2-5} 7 & 2&.&8 & 5 \\ \cline{2-3} \multicolumn{2}{r}{2} & 1 & & \\ \cline{2-3} \multicolumn{2}{r}{0} & 5& & \\ & & &\downarrow & \\ & & - \ | &3 &\\ & & 3 | & &\\ & & + | &5&\\ & & \downarrow | && \\ & & \downarrow | & 2&\\ \end{array} $$

Perform long division

Next, perform long division - focusing only on the decimal part - until we observe a repeating pattern in the digits. $$ \hspace{15mm} \begin{array}{c|cc ccc} \multicolumn{2}{r}{0} & . & 2 & 8 & 5 \\ \cline{2-6} 7 & 2 & . & 8 & 5 & \\ \cline{2-4} \multicolumn{2}{r}{2} & 1 & & & \\ \cline{2-4} \multicolumn{2}{r}{0} & 5 & 7 & & \\ \cline{3-5} %\multicolumn{3}{r}{1} & . & & \\ \multicolumn{1}{r}{} & - &\ 3 & 5 & \\ \cline{3-5} \multicolumn{1}{r}{} & & 1 & 1 & 0\\ \cline{4-5} \multicolumn{1}{r}{} & & & 7 &\\ \cline{4-5} \multicolumn{1}{r}{} & & & 3 & 9&\\ \cline{4-6} \multicolumn{1}{r}{} & & & & 2&\\ \end{array} $$

Observe the repeating pattern

After obtaining additional decimal places, we can see that the decimal part starts repeating from the third digit after the decimal point. The quotient is now approximately 0.285, and we can observe that the pattern of the quotient will continue to repeat every 3 decimal places.

Write the repeating decimal representation

Finally, we can write the repeating decimal representation of the fraction \(2/7\) as \(0.\overline{285}\), where the line above the digits 285 indicates that these digits repeat indefinitely.

Key concepts

Understanding Long Division

Long division is a method for dividing two numbers to obtain a decimal representation. When solving the problem \(2 \div 7\), the number 2 is the dividend and 7 is the divisor. Since 2 is less than 7, we start with a zero before the decimal point. This process helps us find how many times 7 fits into 20, then 200, and so on.

• Place a decimal point in the quotient.
• Append zeros to the dividend to continue the division.
• Continue until a pattern is observed in the quotient or desired accuracy is achieved.

Using long division allows us to break down the process systematically and identify repeating patterns in the decimal.

Exploring Decimal Representation

Decimal representation involves expressing fractions as decimals. For fractions like \(\frac{2}{7}\), this process often results in repeating decimals. These are decimals in which a pattern of digits repeats infinitely.

• When performing long division, if the remainder repeats, so will the decimal.
• The repeating sequence in \(\frac{2}{7}\) is \(285\).
• Represent repeating decimals using a line or bar above the repeating part, e.g., \(0.\overline{285}\).

Decimal representation allows us to express seemingly complex fractions in a more accessible numerical form.

Identifying Pattern Recognition in Decimals

Pattern recognition is crucial when working with repeating decimals. It involves spotting repeated sequences within the decimal. In the case of dividing \(2\) by \(7\), the decimal sequence \(0.285\) starts repeating after a few steps.

• Look for repeated remainders during the long division process.
• Observe the sequence of digits after the decimal point.
• Confirm the length of the repeating pattern—we see it here as 3 digits: \(285\).

Understanding these patterns helps in accurately writing and working with repeating decimals in mathematics.