The exact answer to '10 divided by 9 as a decimal' is $1.111...$, which is most accurately represented using a vinculum as $1.\overline{1}$. This notation signifies that the digit '1' after the decimal point repeats infinitely.
Understanding Repeating Decimals
When you divide two numbers, the result can sometimes be a decimal that goes on forever without ending. These are known as repeating decimals or recurring decimals. They occur when the division process never yields a remainder of zero, causing a sequence of digits to repeat indefinitely.
- Non-Terminating Division: Unlike terminating decimals (like 0.5 or 0.25) where the division ends, repeating decimals result from a division where the remainder consistently cycles through the same values.
- Notation: The common notation for a repeating decimal involves placing a bar (vinculum) over the digit or block of digits that repeats. For instance, $1.\overline{1}$ clearly indicates that only the '1' repeats. If it were $0.\overline{12}$, both '1' and '2' would repeat (e.g., $0.121212...$).
The Division Process Explained
Let's walk through the long division of 10 by 9 to see how $1.\overline{1}$ is derived:
- First Step: Divide 10 by 9.
- 9 goes into 10 one time (1 x 9 = 9).
- The remainder is 10 - 9 = 1.
- So far, the quotient is 1.
- Adding a Decimal: To continue the division, add a decimal point to the quotient and a zero to the remainder, making it 10.
- Second Step: Divide this new 10 by 9.
- 9 goes into 10 one time (1 x 9 = 9).
- The remainder is 10 - 9 = 1.
- The quotient now has a '1' after the decimal point (1.1).
- Infinite Repetition: If you were to continue this process, you would repeatedly add a zero to the remainder (making it 10) and divide by 9, always getting 1 as the next digit in the quotient and a remainder of 1. This unending cycle means the digit '1' after the decimal point will continue forever.
Why Precision Matters
Representing 10 divided by 9 as $1.\overline{1}$ is crucial for accuracy in mathematics and various fields. Simply truncating it to $1.1$, $1.11$, or $1.111$ introduces a small error. While these approximations might be acceptable in some practical scenarios, the exact answer is the repeating decimal.
For more information on repeating decimals and how they work, you can explore resources like Khan Academy's explanation of repeating decimals.
