# How does 0.99999....=1?

Mar 13, 2016

See explanation.

But,this question was asked by me,But I realized the proof and wanted it to be known for others...

#### Explanation:

Let $0.99999 \ldots . = x$

$\rightarrow 9.9999 \ldots . = 10 x$

Subtract $x$ both sides

$\rightarrow 9.9999 \ldots . - 0.9999 \ldots . = 9 x$

$\rightarrow 9 = 9 x$

$\Rightarrow x = 1$

Jul 17, 2016

The number $0.9999 \ldots = {\sum}_{n = 1}^{\infty} \frac{9}{10} ^ n$,

that is the sum of the series starting at $n = 1$

#### Explanation:

The number $0.9999 \ldots = {\sum}_{n = 1}^{\infty} \frac{9}{10} ^ n = 9 \cdot {\sum}_{n = 1}^{\infty} \frac{1}{10} ^ n = 9 \cdot {\left(\frac{1}{10}\right)}^{1} / \left(1 - \frac{1}{10}\right)$, since the sum of the geometric series ${\sum}_{n = 1}^{\infty} \frac{1}{10} ^ n = {\left(\frac{1}{10}\right)}^{1} / \left(1 - \frac{1}{10}\right)$

Then the series ${\sum}_{n = 1}^{\infty} \frac{9}{10} ^ n = 9 \cdot {\left(\frac{1}{10}\right)}^{1} / \left(1 - \frac{1}{10}\right) = 9 \cdot \left(\frac{1}{10}\right) \cdot \left(\frac{10}{9}\right) = 1$,

and so $0.99999 \ldots = 1$