Arm yourself with the tools to be the king or queen of heady mathematical debates, like the age old question of whether 0.999.... = 1.

What is the value of

\[ \large \sqrt{ ( \color{red} {- 8} ) ^ 2 } \hspace{.15cm} ? \]

\[\large {\color{purple}a}{\color{red}x} + {\color{green}5}{\color{purple}b} = {\color{red}2}{\color{purple}a} + {\color{purple}b}{\color{green}q} \\\\\\ \large \text{implies} \\\\\\ \large {\color{red}x} = {\color{red}2} \text{ and } {\color{green}q}={\color{green}5}.\]

**True or false?**

For all real numbers \({\color{purple}a}, {\color{purple}b}, {\color{red}x}, \text{ and } {\color{green}q}\), the statement above must be true.

\[\begin{equation} \begin{split} \sqrt{-1}=&i \quad&\ldots(1) \\ \dfrac{1}{\sqrt{-1}}=&\dfrac{1}{i}\quad& \ldots(2) \\ \dfrac{\sqrt {1}}{\sqrt{-1}}=&\dfrac{1}{i} \quad &\ldots(3) \\ \sqrt{\dfrac{1}{-1}}=&\dfrac{1}{i} \quad &\ldots(4) \\ \sqrt{\dfrac{-1}{1}}=&\dfrac{1}{i}\quad &\ldots(5) \\ \dfrac{\sqrt{-1}}{1}=&\dfrac{1}{i} \quad &\ldots(6) \\ i=&\dfrac{1}{i} \quad& \ldots(7) \\ i^2=&1 \quad & \ldots (8) \\ -1=&1\quad & \ldots(9) \end{split} \end{equation}\]

Consider these steps above.

In which step is the (first) mistake committed?

Given any two periodic functions, \(\color{red}{f(x)}\) and \(\color{blue}{g(x)}\), the function \(\color{purple}{h(x)} = \color{red}{f(x)} + \color{blue}{g(x)}\) will also be periodic with a period that is, at most, the product of the original two periods (the period of \(f(x)\) times the period of \(g(x)\)).

What is the value of

\[ \large \sqrt{1\text{%}}\, ? \]

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