0,999= 1 HOW

I beg for an answer.

#Calculus

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Comments

Let $x = 0.999 \dots$ . This means that $10x = 9.999 \dots$ .

If we subtract one from another, we get $10x - x = 9.999 \dots - 0.999 \dots \implies 9x = 9$ . This because all of the trailing decimal $9$ 's cancel out with each other.

Then, we can divide by $9$ to see that $x = \frac{9}{9} = 1$ so therefore $0.999 \dots = 1 \; \; \; \square$

James Watson - 7 months, 1 week ago

A simple explanation I find really intuitive is that -

$\dfrac{1}{9} = 0.111111 \ldots$

$\dfrac{2}{9} = 0.222222 \ldots$

And so on...

So $\dfrac{9}{9} = 0.999999 \ldots$

But we know that $\dfrac{9}{9} = 1$

Thus,

$0.999999 \ldots = 1$

@Deniz İpek Zeynioğlu

A Former Brilliant Member - 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$