Q1(v):

Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$

Initial Setup & Mathematical Objective

We are given the rational number $\frac{2}{11}$. Our objective is to convert this fraction into its decimal form using the long division algorithm and to classify the nature of its decimal expansion [Per the fundamental properties of rational numbers].

Step 1: Executing the Long Division Algorithm

To find the decimal expansion, we divide the numerator ($2$) by the denominator ($11$). Since $2 < 11$, the quotient begins with $0$. We place a decimal point in the quotient and append zeros to the dividend to continue the division process.

The step-by-step division proceeds as follows:

• Iteration 1: Bring down a $0$ to make $20$. Divide $20$ by $11$. The quotient is $1$. $11 \times 1 = 11$. Subtracting $11$ from $20$ yields a remainder of $9$.
• Iteration 2: Bring down a $0$ to make $90$. Divide $90$ by $11$. The quotient is $8$. $11 \times 8 = 88$. Subtracting $88$ from $90$ yields a remainder of $2$.
• Iteration 3: Bring down a $0$ to make $20$. Divide $20$ by $11$. The quotient is $1$. $11 \times 1 = 11$. Subtracting $11$ from $20$ yields a remainder of $9$.
• Iteration 4: Bring down a $0$ to make $90$. Divide $90$ by $11$. The quotient is $8$. $11 \times 8 = 88$. Subtracting $88$ from $90$ yields a remainder of $2$.

Step 2: Analyzing the Sequence of Remainders

Observing the division process, the sequence of remainders is $9, 2, 9, 2, \dots$. Because the remainder $2$ (which was our original dividend) reappears, the sequence of calculations will loop infinitely. Consequently, the block of digits in the quotient, $18$, will repeat indefinitely.

Mathematically, this is expressed as:

$\frac{2}{11} = 0.181818\dots$

Using standard mathematical bar notation to denote the repeating block of digits, we write this as $0.\overline{18}$.

Step 3: Theoretical Classification of the Decimal Expansion

According to the fundamental theorem of arithmetic applied to rational numbers, if the prime factorization of the denominator of a fraction (in simplest form) contains prime factors other than $2$ or $5$, the fraction will result in a non-terminating repeating decimal. Here, the denominator is $11$, which is a prime number distinct from $2$ and $5$.

Since the division never yields a remainder of $0$ and a specific block of digits ($18$) repeats infinitely, the decimal expansion is formally classified as non-terminating repeating (or non-terminating recurring).

Final Solution: The decimal form of $\frac{2}{11}$ is $0.\overline{18}$, and its decimal expansion is non-terminating repeating.