User:Tdadamemd/Visual proof of 0.999...=1

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Visual proof of 0.999... = 1
The visual proof that 0.999... = 1 starts with an image that goes along with the algebraic proof. One form of the algebraic proof is:



\begin{align} \frac{1}{3}          & = 0.333\dots  \\ 3 \times \frac{1}{3} & = 3 \times 0.333\dots \\ 1                    & = 0.999\dots \end{align} $$

The first step uses the application of simple fractions and long division. The second step is simple multiplication, and extending the concept of simple multiplication to the infinitely repeated decimal. The third step concludes the algebraic proof.

The decimal representation 0.999... can be represented by an infinite series:

0.999... = 0.9 + 0.09 + 0.009 + ...

An infinite series is also represented in the image to the right, where a square of unit area has been infinitely quartered. This summation of the area of all of the highlighted squares equals: $$\frac{1}{3}$$. This fact is readily proven by examining the 'L'-shape formed by the three squares left untouched after the first quartering and noting that exactly one-third of that shape has been highlighted. The summation of the infinite series is proven by noting that the symmetry of the 'L'-shape with its one-third ratio of highlighting remains unchanged throughout successive quarterings, ad infinitum. Exactly 1/3 of the unit square has been highlighted.

Verification that 0.999... = 1 is done simply by noting that 1/3 = 0.333..., and that if the other two-thirds of the 'L'-shape were to be highlighted, then it would result in: 3 x 0.333... = 0.999... And by observing that the infinite progression leaves absolutely none of the square unfilled, it is shown that 0.999... = 1.

This equality can be shown directly in the image showing a blue circle of unit area. Like the unit square above it, this circle can be sliced in any way imaginable, and the parts when added all back together will sum to exactly 1. The slicing does nothing to change the area. It is one before, and it is one after. In this case, the circle is sliced up into ever thinner pie slices where the first slice leaves 90% untouched (0.9), the second slice leaves 90% of the remaining wedge untouched (0.09), the third slice leaves 90% of that remaining wedge untouched (0.009), and this process continues without end. The summation of all slices produces the infinite series:

0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.9999...

And because the slicing never changed the total area of this circle with an area of 1 (both prior to the slicing and after), the conclusion is that: 0.9999... = 1.

Note
For the quartered-square image, the area highlighted is directly illustrating the Base 4 infinite series:

0.14 + 0.014 + 0.0014 + ... = 0.111...4 = 0.333...10 = 1/3

When the full 'L'-shape is highlighted it proves that 0.333...4 = 1.