Someone said that \( 0. \overline{9} = 1 \)

Calculus Level 3

Statement One: \(\text{Let} \ 0.4\overline{9} = x \)

Statement Two: \(\implies 10x = 4.\overline{9} \)

Statement Three: \(100x = 49.\overline{9} \)

Statement Four: \(\implies x = 0.4\overline{9} = \cfrac{45}{90} = \cfrac{1}{2}\)

Since from Statement Four: , \( 0.4\overline{9} = \cfrac{45}{90} = \cfrac{1}{2} \) , multiplying both sides by 2,

Statement Five: \(\implies \left[ 0.4\overline{9} \right ] \cdot 2 = \left[ \cfrac{1}{2} \right ] \cdot 2 = 1 \)

Upon multiplying, we see that

Statement Six: \( \left[ 0.4\overline{9} \right ] \cdot 2 = 0.\overline{9}8 = 1 \)

Statement Seven: But upon searching, famously, \( 1 = 0.\overline{9} \)

Statement Eight: \( \therefore 0.\overline{9}8 = 0.\overline{9} \)

But, it will break the "reflexive property of equality" .

Which Statement Is The Start Of The Mistake In The Said Argument?

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