The decimal expansion of 43/162

A Deeper Look into the Decimal Expansion of 43162\frac{43}{162}16243​

The decimal expansion of 43162\frac{43}{162}16243​ offers an interesting insight into repeating decimals, which is a common phenomenon when dividing certain fractions. Let’s explore this in more detail.

Why Does Repetition Occur?

A repeating decimal occurs when the long division process results in a pattern that repeats indefinitely. In this case, when we divide 43 by 162, we eventually reach a point where the remainder starts repeating. This leads to the same sequence of digits being generated over and over again in the decimal part of the number.

The fraction 43162\frac{43}{162}16243​ is a rational number, meaning it can be expressed as a fraction of two integers. Rational numbers, when expressed as decimals, either terminate or repeat. In this case, the division doesn’t terminate but repeats after a certain number of digits, specifically “265432.”

Step-by-Step Breakdown of the Long Division

If we revisit the division process in more detail, we see how the pattern emerges:

1. First Division (430 ÷ 162):
   • 162 goes into 430 two times (resulting in 324).
   • Subtract 324 from 430, leaving a remainder of 106.
   • Bring down a zero to get 1060.
2. Second Division (1060 ÷ 162):
   • 162 goes into 1060 six times (resulting in 972).
   • Subtract 972 from 1060, leaving a remainder of 88.
   • Bring down a zero to get 880.
3. Third Division (880 ÷ 162):
   • 162 goes into 880 five times (resulting in 810).
   • Subtract 810 from 880, leaving a remainder of 70.
   • Bring down a zero to get 700.
4. Fourth Division (700 ÷ 162):
   • 162 goes into 700 four times (resulting in 648).
   • Subtract 648 from 700, leaving a remainder of 52.
   • Bring down a zero to get 520.
5. Fifth Division (520 ÷ 162):
   • 162 goes into 520 three times (resulting in 486).
   • Subtract 486 from 520, leaving a remainder of 34.
   • Bring down a zero to get 340.
6. Sixth Division (340 ÷ 162):
   • 162 goes into 340 two times (resulting in 324).
   • Subtract 324 from 340, leaving a remainder of 16.
   • Bring down a zero to get 160.

At this point, when dividing 160 by 162, we get a remainder of 160, which brings us back to the beginning of the cycle. The repeating sequence “265432” begins to reappear.

How to Express Repeating Decimals

When dealing with repeating decimals like this one, mathematicians use a notation to represent the repeating part of the decimal. In the case of 43162\frac{43}{162}16243​, the repeating part is “265432,” so we write:

43162=0.265432‾\frac{43}{162} = 0.\overline{265432}16243​=0.265432

The bar above the repeating digits signifies that the sequence repeats indefinitely.

Decimal Expansion and Approximation

While the exact decimal expansion is infinite, it’s often practical to approximate the value by truncating the decimal after a certain number of digits. For example, if we round to six decimal places, we get:

43162≈0.265432\frac{43}{162} \approx 0.26543216243​≈0.265432

This approximation is often sufficient for most real-world calculations. However, it’s important to recognize that the true value of the decimal never ends, and the digits continue repeating forever.

Converting Repeating Decimals Back into Fractions

A key feature of repeating decimals is that they can always be converted back into fractions. For instance, if we have a decimal like 0.265432‾0.\overline{265432}0.265432, we can use algebraic methods to convert it back into the original fraction 43162\frac{43}{162}16243​.

Here’s a simplified process for converting repeating decimals into fractions:

1. Let x=0.265432‾x = 0.\overline{265432}x=0.265432.
2. Multiply both sides by a power of 10 that shifts the decimal point far enough for the repeating part to align.
3. Subtract the original equation from the shifted equation to eliminate the repeating decimal.
4. Solve the resulting equation for $x$, which will give you the fraction.

For 0.265432‾0.\overline{265432}0.265432, this method would confirm that the fraction is indeed 43162\frac{43}{162}16243​.

Conclusion

The decimal expansion of 43162\frac{43}{162}16243​, which is 0.265432‾0.\overline{265432}0.265432, provides a clear example of a repeating decimal. Understanding how repeating decimals arise from long division can deepen our understanding of rational numbers. Moreover, knowing how to approximate, represent, and even convert these decimals back into fractions makes them a useful tool in mathematics.