Q7:

Write three numbers whose decimal expansions are non-terminating non-recurring.

Theoretical Foundation: Decimal Expansions of Real Numbers

In the real number system, numbers are broadly classified into two categories based on their decimal expansions: rational numbers and irrational numbers. [Per the fundamental classification of Real Numbers], a rational number's decimal expansion either terminates (ends) or is non-terminating but recurring (repeats a specific block of digits infinitely). Conversely, an irrational number cannot be expressed as a simple fraction $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$). Consequently, the defining characteristic of an irrational number is that its decimal expansion is strictly non-terminating and non-recurring.

To satisfy the prompt, we must provide three distinct examples of irrational numbers. We will derive these using three different mathematical approaches: pattern construction, algebraic roots, and transcendental constants.

Step 1: Constructing a Pattern-Based Irrational Number

The most direct way to generate a non-terminating, non-recurring decimal is to construct a sequence of digits that follows a predictable, yet non-repeating pattern. We can achieve this by systematically increasing the number of zeros between a repeating non-zero digit.

• Let the first decimal place be $0$.
• Follow it with a $1$, then one $0$ ($0.01\dots$).
• Follow it with a $1$, then two $0$s ($0.01001\dots$).
• Follow it with a $1$, then three $0$s ($0.010010001\dots$).

Number 1: $0.01001000100001\dots$

[Justification: Because the block of zeros expands by one with each iteration, there is no fixed finite block of digits that will ever repeat infinitely. Thus, it is non-recurring. The ellipsis ($\dots$) indicates it is non-terminating.]

Step 2: Utilizing Square Roots of Non-Perfect Squares

In algebra, the square root of any positive integer that is not a perfect square (such as $2, 3, 5, 6, 7$) is inherently an irrational number. [Per the theorem of irrationality of roots: If $p$ is a prime number, then $\sqrt{p}$ is irrational].

Let us take the square root of the first prime number, $2$.

Number 2: $\sqrt{2} = 1.41421356237309504\dots$

[Justification: The decimal expansion of $\sqrt{2}$ has been mathematically proven to extend infinitely without any repeating sequence of digits.]

Step 3: Referencing Transcendental Mathematical Constants

Certain fundamental constants in mathematics cannot be expressed as the root of any non-zero polynomial equation with rational coefficients. These are known as transcendental numbers, which are a subset of irrational numbers. The most famous example is Pi ($\pi$), which represents the ratio of a circle's circumference to its diameter.

Number 3: $\pi = 3.141592653589793238\dots$

[Justification: Johann Lambert proved in 1761 that $\pi$ is irrational. Therefore, its decimal representation goes on forever without settling into a permanently repeating pattern.]

Final Solution: Three distinct numbers whose decimal expansions are non-terminating and non-recurring are:
1) $0.01001000100001\dots$ (A constructed non-repeating pattern)
2) $\sqrt{2} = 1.41421356\dots$ (The square root of a non-perfect square)
3) $\pi = 3.14159265\dots$ (A transcendental mathematical constant)