Grade 8 - Number Theory #1

Know that the definition of rational number is:

$\text{"A number that can be expressed as }\frac{a}{b}\text{, where }a\text{ and }b\text{ are both integers. (}b\neq0\text{)"}$

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Andrew:

Counterexample: $0.33333\cdots=\dfrac{1}{3}$ .

$\therefore\boxed{FALSE}$

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Bob:

Let ${a_n}$ satisfy $0<a_i<10$ and $a_i\in \mathbb{N}$ for all natural numbers $i$ .

Finite decimal: $0.\overline{a_1a_2a_3a_4\cdots a_n}=\dfrac{\overline{a_1a_2a_3a_4\cdots a_n}}{10^n}$

Recurring decimal: $0.\overline{a_1a_2a_3a_4\cdots a_n}=\dfrac{\overline{a_1a_2a_3a_4\cdots a_n}}{10^n-1}$

$\therefore\boxed{TRUE}$

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Catherine:

Natural number $n$ can always be expressed as $\overline{(n-1).99999\cdots}$ .

Also, negative integer $m$ can always be expressed as $\overline{(m+1).99999\cdots}$ .

However, the integer $0$ cannot be expressed as such form.

$\therefore\boxed{TRUE}$

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David:

Already proven wrong while solving Bob .

$\therefore\boxed{FALSE}$

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Emma:

Counterexample: $\dfrac{\sqrt{2}}{10^2}=0.014142135\cdots$ .

$\therefore\boxed{FALSE}$

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From Andrew~Emma, we now know that only Bob and Catherine said a correct statement.

$\therefore~2+4=\boxed{6}$ .