Q1(i):

Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$

Initial Setup & Given Variables

We are tasked with evaluating the rational number provided in fractional form:

$x = \frac{36}{100}$

The objective is to convert this fraction into its equivalent decimal representation and mathematically classify the nature of its decimal expansion.

Step 1: Conversion to Decimal Form

To convert a fraction to a decimal, we perform the division of the numerator by the denominator. When the denominator is a perfect power of $10$ (i.e., $10^k$), the division can be executed by shifting the decimal point of the numerator $k$ places to the left.

• Numerator: $36$ (which can be written as $36.0$)
• Denominator: $100 = 10^2$

Since the exponent $k = 2$, we shift the decimal point two places to the left:

$x = \frac{36}{100} = 0.36$

Step 2: Analyzing the Nature of the Decimal Expansion

A decimal expansion is classified based on its termination behavior:

• Terminating Decimal: The division process yields a remainder of zero after a finite number of steps, resulting in a finite number of digits after the decimal point.
• Non-Terminating Repeating Decimal: The division process never yields a zero remainder, but a digit or a block of digits repeats infinitely.

Because the value $0.36$ ends exactly after the hundredths place (two decimal digits), it is classified as a terminating decimal expansion.

Step 3: Theoretical Justification via Prime Factorization

[Per the Fundamental Theorem of Arithmetic and the properties of rational numbers], a rational number expressed in its simplest form $\frac{p}{q}$ will have a terminating decimal expansion if and only if the prime factorization of its denominator $q$ is strictly of the form:

$q = 2^n \times 5^m$

where $n$ and $m$ are non-negative integers.

Let us analyze the denominator $q = 100$:

$100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2$

Here, $n = 2$ and $m = 2$. Because the prime factors of the denominator consist exclusively of $2$ and $5$, the mathematical theorem guarantees that the fraction $\frac{36}{100}$ must resolve into a terminating decimal.

Step 4: Visual Representation of the Decimal

To conceptualize $0.36$ geometrically, we can map the fraction onto a $10 \times 10$ Cartesian grid representing $100$ equal units. Shading $36$ of these units provides a precise visual truth of the decimal's magnitude.

Final Solution: The decimal form of $\frac{36}{100}$ is $0.36$, and it has a terminating decimal expansion.