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# Maths loves approximation

Why 0.9999........... is equal to 1.0 ; 1.9999.......... is equal to 2.0 and so on?

Note by Shobhit Singh
2 years, 2 months ago

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By definition, the infinite decimal $$\overline{a_0.a_1a_2a_3}\ldots$$ is the the limit of the sequence $$\overline{a_0},\overline{a_0.a_1},\overline{a_0.a_1a_2},\overline{a_0.a_1a_2a_3},\ldots$$. So

$$0.9999\ldots=\lim_{n\to\infty}0.9999\ldots9 \text{ (n 9's)}=\lim_{n\to\infty}1-\frac{1}{10^n}=1.$$

Finite decimals are easy - you just add up powers of $$10$$. Like $$.99=9\times10^{-1}+9\times10^{-2}$$. But you can't add infinitely many numbers together - so infinite decimals are defined with limits. You'll learn about series (limits intuitively giving the sum of infinitely many numbers) later on in school! · 2 years, 2 months ago