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Show that when a rational number is written in decimal form, it either terminates in 0s or eventually repeats.

Use long division to convert 6 7/15 to a decimal.
Answer: ≈6.47

1 answer

To show that when a rational number is written in decimal form, it either terminates in 0s or eventually repeats, we need to consider a rational number defined as the ratio of two integers, \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

A decimal representation of a rational number can be determined by the prime factorization of the denominator \( q \) when the fraction is in its lowest terms:

  1. If the only prime factors of \( q \) are 2 and/or 5, then the decimal representation of \( \frac{p}{q} \) terminates. This is because you can multiply both numerator and denominator by a sufficient power of 2 and/or 5 to make the denominator a power of 10.

  2. If \( q \) has any prime factors other than 2 or 5, the decimal representation will be non-terminating and will enter a repeating cycle. This is because the long division process will eventually start producing the same remainders, leading to the same division steps repeating indefinitely.

Now, let's convert \( 6 \frac{7}{15} \) to a decimal using long division.

First, we convert \( 6 \frac{7}{15} \) to an improper fraction: \[ 6 \frac{7}{15} = \frac{6 \cdot 15 + 7}{15} = \frac{90 + 7}{15} = \frac{97}{15}. \]

Next, we perform long division of \( 97 \) by \( 15 \):

  1. \( 15 \) goes into \( 97 \) a total of \( 6 \) times (since \( 15 \times 6 = 90 \)).
  2. Subtract \( 90 \) from \( 97 \) gives \( 7 \).
  3. Bring down a \( 0 \) (the decimal) making it \( 70 \).
  4. \( 15 \) goes into \( 70 \) a total of \( 4 \) times (since \( 15 \times 4 = 60 \)).
  5. Subtract \( 60 \) from \( 70 \) gives \( 10 \).
  6. Bring down another \( 0 \), making it \( 100 \).
  7. \( 15 \) goes into \( 100 \) a total of \( 6 \) times (since \( 15 \times 6 = 90 \)).
  8. Subtract \( 90 \) from \( 100 \) gives \( 10 \).
  9. Bring down another \( 0 \), making it \( 100 \) again. Repeat steps 7 and 8.

Now you can see that the process will repeat starting from step 6:

The result of the long division gives us: \[ 6.4666... = 6.47 \text{ (approximately, rounding at the thousandths)}. \]

Thus, we find that: \[ 6 \frac{7}{15} \approx 6.47. \]

As a summary:

  • When a rational number is in decimal form, it either terminates or eventually repeats.
  • The long division shows \( 6 \frac{7}{15} \) results in a decimal approximation of \( 6.47 \).