# 0.999 … = 1

It really does.

Many people have a problem accepting this equality.

The ellipsis denotes that the nines continue forever; there is no last nine.

Jen the Blue brought this up one night recently.  I didn’t see it at the time and was asked by Upset if it was true.

We can go through some arguments.  There are many.

$$1)$$ An Arithmetical Argument.

Do you accept that $$1/3 = 0.333 …$$?

Do you accept that $$1/9 = 0.111 …$$?

Multiply (an arithmetic operation) the first by $$3$$ :

Thus, $$1 = 0.999 …$$.

Multiply the second by $$9$$ :

Thus, $$1 = 0.999 …$$.

Still not convinced?

$$2)$$ An Algebraic Argument.

Define :

Let $$x = 0.999 …$$

Multiply both side by $$10$$ :

$$10x = 9.999 …$$

Separate the RHS :

$$10x = 9 + x$$

Subtract $$x$$ from both sides :

$$9x = 9$$

Divide both sides by $$9$$ :

$$x = 1$$

$$3)$$ A Pre-Calculus Argument

A series is the sum of the terms of an infinite sequence of numbers.

A series is convergent if the sequence of its partial sums tends to a limit.

We can express $$0.999 …$$ as

$$0.999 … = \frac{9}{10} + \frac {9}{100} + \frac{9}{1000} + \frac{9}{10000} +…$$

A geometric series is of the form

$$a + ar + ar^2 + ar^3 + …$$

where $$a$$ is the first term and $$r$$ is the common ratio.  When $$-1 < r < 1$$ the series converges to a limit given by the formula  $$\displaystyle \frac{a}{1-r}$$.

Our number $$0.999 …$$ is a geometric series with first term $$\frac{9}{10}$$ and common ratio $$\frac{1}{10}$$.

Substituting these into the formula we have

$$\displaystyle \frac{9/10}{1-1/10} = \frac{9/10}{9/10} = 1$$

$$4)$$ A Wolfram Alpha Argument.

Type $$0.999 …$$ into Wolfram Alpha and it returns a value of $$1$$.

© OldTrout $$2018$$

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