Q1(ii):

Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$

Initial Setup & Mathematical Objective

The objective is to express the rational number $\frac{1}{11}$ in its decimal form and classify the nature of its decimal expansion. To achieve this, we apply the standard long division algorithm, dividing the numerator ($1$) by the denominator ($11$).

Step 1: Execution of the Long Division Algorithm

We set up the division $1 \div 11$. Since the dividend ($1$) is strictly less than the divisor ($11$), we place a decimal point in the quotient and append zeros to the dividend to proceed with the fractional parts.

• Tenths place: Bring down a $0$ to make the dividend $10$. Since $10 < 11$, the quotient digit is $0$. The remainder is $10$.
• Hundredths place: Bring down another $0$ to make the dividend $100$. Divide $100$ by $11$. The maximum multiple is $11 \times 9 = 99$. The quotient digit is $9$, and the remainder is $100 - 99 = 1$.
• Thousandths place: Bring down a $0$ to the remainder $1$, making it $10$. Since $10 < 11$, the quotient digit is $0$. The remainder is $10$.
• Ten-thousandths place: Bring down another $0$ to make it $100$. Divide $100$ by $11$. The quotient digit is $9$, and the remainder is $100 - 99 = 1$.

The visual representation of this algorithmic process is detailed below:

Step 2: Analysis of the Remainder and Quotient Sequences

By tracking the division process, we can observe a strict mathematical pattern in the remainders and the resulting quotient digits.

[Per the properties of rational numbers], because the remainder sequence $(10, 1, 10, 1, \dots)$ begins to repeat, the sequence of digits in the quotient will also repeat infinitely. The repeating block of digits in the quotient is $09$.

Step 3: Classification of the Decimal Expansion

Because the remainder never becomes zero, the decimal expansion is non-terminating. Furthermore, because a specific block of digits ($09$) repeats infinitely, the expansion is repeating (or recurring). Mathematically, this is denoted by placing a vinculum (bar) over the repeating block of digits:

$ \frac{1}{11} = 0.090909\dots = 0.\overline{09} $

Final Solution: The decimal form of $\frac{1}{11}$ is $0.\overline{09}$, and it has a non-terminating repeating (recurring) decimal expansion.