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What is wrong in this?

1 = (1 x 3)/3
1 = (0.333333333....33) x 3
1 = 0.99999..999
=> 0.0000....001 = 0
=> 1/10^ ∞ = 0
=> 1 = 0

Note by Ksg Sarma
5 months, 2 weeks ago

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First off, \(0 \times \infty\) isn't defined. For more info, refer to indeterminate forms.

Secondly, the last equation should be \[ \lim_{n \to \infty} \frac{1}{10^n} = 0 \]

Rather than \[ \frac{1}{10^{\infty}} = 0 \]

Also, note that \[ \lim_{n \to \infty} 0 \times 10^n = 0 \]

But, \[\lim_{n \to \infty} \frac{1}{10^n} = 0 \not \Rightarrow \lim_{n \to \infty} 0 \times 10^n = 1\] Deeparaj Bhat · 5 months, 2 weeks ago

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@Deeparaj Bhat 1/10^ ∞ is not exactly equal to 0 its value is too small and greater than 0 so when i multiply it with 0 it should be 0 . And according to your 1st statement 1 is undefined Ksg Sarma · 5 months, 2 weeks ago

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@Ksg Sarma \(\dfrac{1}{10^{\infty}}\) isn't even a number! Deeparaj Bhat · 5 months, 2 weeks ago

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moreover the main problem is with 3rd step where i am getting 1 is exactly equal to 0.999999..9999 Ksg Sarma · 5 months, 2 weeks ago

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@Ksg Sarma There are so many errors that you've made in the above note.

First error

The \(.33333....333\) you wrote implies that the \(3\)s will end since the last 3 isn't followed by \(...\) to imply that it repeats infinitely

Second error

This follows from your first and is very similar.Again.you're implying that the \(9\)s will end sometime,which they don't because the 9s repeat infinitely.(occurs in line 3)

Third error

This doesn't even make sense.If something equals something,subtracting them from each other will result in exactly \(0\).This follows from your other errors too.(occurs in line 4)

Fourth error

You can't put \(.000..1\) equal to \(\frac{1}{10^{\infty}}\) because you can't do arithmetic with infinity,it's a concept not a number.

Fifth error

Let's assume you used the correct way to express it which is \[\lim _{n \rightarrow \infty}{\frac{1}{10^{n}}}=0\],you can't multiply each side by \(10^n\) since it's in the limit,and it's not a constant. Hummus A · 5 months, 2 weeks ago

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@Hummus A in the 1st error you said , tell me how many no. of 3's are there in the middle then I agree that it will end and all the errors are similar Ksg Sarma · 5 months, 2 weeks ago

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@Ksg Sarma the 3s don't end,they repeat infinitely Hummus A · 5 months, 2 weeks ago

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@Ksg Sarma \(0.999\ldots = 1\) exactly! Deeparaj Bhat · 5 months, 2 weeks ago

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You can't play with infinity digit number like that! Darmawan Putra Wijaya · 5 months, 2 weeks ago

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why don't u just make wiki Darmawan Putra Wijaya · 5 months, 2 weeks ago

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