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# Is 0.9999.... = 1?

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Is 0.9999... equal to 1?  If so, how can I convince someone else that this is true?

Jun 21, 2020

#1
+783
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0.999... is a repeating decimal, so:

Let x represent the fraction that 0.999 is equal to.

x=0.999...

10x=9.999....

10x-x=9.999...-0.999....

9x=9

x=1

Jun 21, 2020
#2
+995
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True.. but this is more of a logical explanation(your math is definitely correct, didn't think of that!)

However, 0.99999... more of converges to 1 but never truly reaches it.

Eventually the difference is so small so we basically call it one.

eventually the thing is 0.000000....1 but... it goes on forever. So there is no end digit!

Thus the thing is actually 0.00000...0000...0000..

Soooo

:)

hugomimihu  Jun 22, 2020
#3
+8354
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0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + ... = $$9 \displaystyle\sum_{n = 1}^\infty \dfrac{1}{10^n}$$

By sum of infinite geometric series formula, 0.999... = $$9 \cdot \dfrac{\dfrac1{10}}{1-\dfrac{1}{10}} = 9\cdot \dfrac19 = 1$$

Jun 22, 2020
#5
+995
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Oh yes I forgot about that!

hugomimihu  Jun 22, 2020
#4
+8354
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Alternative method:

We know that

0.333... = 1/3

Multiplying both sides by 3:

0.999... = 1

Jun 22, 2020