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File:GeometricSquares.svg

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Description

An illustration of geometric series. Each of the purple squares is 1/4 of the area of the previous square (1/2×1/2=1/4, 1/4×1/4=1/16), with the total area being



The area summation of all highlighted regions adds up to exactly 1/3. While not necessarily an intuitive result, this can be easily verified by examining the 'L' shape made by the largest three squares (the three untouched after the first quartering). Exactly 1/3 of the area of these three squares is highlighted. This shape and proportion repeats infinitely unchanged throughout subsequent quarterings. So by inspection of the symmetry, the infinite series maintains the exact total of 1/3.

This square serves as a verification that the numbers 0.999... and 1.000... are exactly equal. While the one-third of the square that is highlighted has the decimal notation of 0.333..., filling in the other two-thirds of the 'L' shape results in 3 x 0.333... = 0.999... which results in the entire unit square being filled in, so 0.999... = 1 exactly.

SVG redraw of original image.
Date
Source en:Image:GeometricSquares.png
Author en:User:Jim.belk (original); Pbroks13 (talk) (redraw)
Other versions File:Square-InfinitelyQuartered-EndlessZoom.gif Square-InfinitelyQuartered-EndlessZoom.gif


Public domain This work has been released into the public domain by its author, Jim.belk at English Wikipedia. This applies worldwide.
In some countries this may not be legally possible; if so:
Jim.belk grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Captions

Geometric series

Items portrayed in this file

depicts

30 June 2008

image/svg+xml

File history

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Date/TimeThumbnailDimensionsUserComment
current21:58, 30 June 2008Thumbnail for version as of 21:58, 30 June 2008359 × 350 (14 KB)Pbrks{{Information |Description=An illustration of geometric series. Each of the purple squares is 1/4 of the area of the previous square, with the total area being :<math>\frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots \;=\; \frac{1}{3}.</math

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