Classification of Numbers

The real number system is divided into rational and irrational numbers, with integers forming a subset of rational numbers. Which of the following numbers is classified as a rational number but is NOT an integer?

If $P$ is the smallest prime number and $C$ is the smallest positive composite number, what is the value of $P + C$?

Evaluate the mathematical validity of the three theoretical propositions listed below, which pertain to the fundamental properties of real numbers:

I. For two distinct irrational numbers $x$ and $y$, it is mathematically possible for both their sum $(x + y)$ and their product $(x \cdot y)$ to evaluate to rational numbers simultaneously.
II. If $N$ represents any natural number, its principal square root $\sqrt{N}$ must be either a natural number or an irrational number; it can never equate to a non-integer rational fraction.
III. The integer $0$ is formally categorized as a positive even number, and the constant $\pi$ is categorized as a rational number because it represents the exact ratio of a circle's circumference to its diameter.

Which of the given propositions is/are correct?

Consider the following $8$ mathematical expressions:

I. $\pi - \frac{22}{7}$
II. $\frac{\sqrt{27}}{\sqrt{3}}$
III. $(\sqrt{3} - \sqrt{2})^2$
IV. The infinite decimal $0.101101110\dots$ (where the number of consecutive $1$ s increases by one each time)
V. $1.414$
VI. $\sqrt{10} \times \sqrt{2.5}$
VII. $e^0$
VIII. $\sqrt{2} + \sqrt{8}$

How many of the above expressions evaluate to a rational number?

Analyze the following three mathematical assertions concerning the classification and properties of numbers:

Assertion I: The expression $\sqrt{32} - 16 \div 4 \times \sqrt{2}$ evaluates to an irrational number.
Assertion II: For any prime number $p$ that is strictly greater than $2$, the value of $p^2 - 1$ is always a multiple of $8$.
Assertion III: The number $0$ is classified as a rational number and an even integer, but it is neither a prime nor a composite number.

Which of the assertions provided above are logically correct?

Analyze the given mathematical claims regarding the classification and properties of numbers:

I. If $p$ is a prime number strictly greater than $3$, then $p^2 - 1$ is always a composite number divisible by $24$.
II. The least common multiple (LCM) of the rational numbers $\frac{6}{5}$ and $\frac{12}{7}$ is a composite integer, while their highest common factor (HCF) is a non-integer rational number.
III. The numerical value of the expression $36 \div 6 \times 3$ belongs to the set of prime numbers.
IV. According to standard Euclidean division, the remainder when $-22$ is divided by $7$ is $-1$, which is classified as a negative integer.

Which of the above claims are mathematically correct?

Consider the following set of 8 numbers:

Let:
$P =$ the number of integers in $S$
$Q =$ the number of rational numbers in $S$
$R =$ the number of irrational numbers in $S$
$T =$ the number of whole numbers in $S$

Calculate the exact value of $(Q \times R) + (P \times T)$.

A cryptography algorithm generates a numerical verification key, $V$, through a specific two-step mathematical process. First, it determines the value of $K$, which is strictly defined as the positive remainder when $-26$ is divided by $7$. Second, it calculates the final key using the expression $V = K \times (18 \div 6 \times 3)$. Based on the fundamental classification of numbers, which of the following accurately describes the final value of $V$?

A teacher provides a list of eight numerical values on the board for a classification exercise:

(I) $\sqrt{144}$
(II) $\frac{22}{7}$
(III) $\pi$
(IV) $0.\overline{36}$
(V) $\sqrt{12}$
(VI) $\frac{\sqrt{3}}{\sqrt{27}}$
(VII) $3.14159$
(VIII) $0.121221222...$ (where the number of 2s increases by one each time)

What is the exact count of rational numbers in this list?