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File:0.999... = 1 CircleGraphProof.svg

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Original file (SVG file, nominally 625 × 690 pixels, file size: 6 KB)

Summary

Description
English: A circle of unit area as geometric proof of the equality that 0.999…= 1.

Explanation: A circle of unit area can be sliced up in any way and all of its parts added together will always equal 1. This 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 changes the total area of this circle with an area of 1 (both prior to the slicing and after), while the amount of circle not included in the sum goes to 0, the conclusion is that
0.9999… = 1.

Mathematica code that produced this image (before hand editing):

diagramBlue = RGBColor[0.4, 0.8, 1];
coord[n_Integer] := {
  Cos[(1/4 + 10^(-n)) 2 \[Pi]],
  Sin[(1/4 + 10^(-n)) 2 \[Pi]]
  }
radius[n_Integer] := Line[{{0, 0}, coord[n]}]
pointLabel[label_String, points_List : {x_, y_}] := Text[
  TextCell[
   label,
   "Text",
   CellFrame -> True,
   CellFrameMargins -> 0.1,
   CellFrameColor -> diagramBlue,
   Background -> White
   ],
  points
  ]
diagram = Show[
  Graphics[{
    {diagramBlue, EdgeForm[Black], Disk[]},
    radius[0],
    radius[1],
    radius[2],
    radius[3],
    pointLabel["0.9\n36°", coord[1] + {-0.05, 0.11}],
    pointLabel[
     "0.09\n3.6°", {coord[2][[1]] - 0.05, coord[0][[2]] + 0.12}],
    pointLabel[
     "0.009\n0.36°", {coord[3][[1]] + 0.05, coord[0][[2]] + 0.12}]
    }],
  ImageSize -> 500
  ]
Export[FileNameJoin[{Directory[], "Downloads", 
   "0.999... = 1 CircleGraphProof.svg"}], diagram]
Date
Source Own work based on: 0.999... = 1 CircleGraphProof.png
Author CalendulaAsteraceae
SVG development
InfoField
 
The SVG code is valid.
 
This vector image was created with Inkscape, and then manually edited
.

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
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You are free:
  • to share – to copy, distribute and transmit the work
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Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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16 March 2021

image/svg+xml

b33974d8002aafd39ad248d2e42a76c5b333c2cf

6,563 byte

690 pixel

625 pixel

File history

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Date/TimeThumbnailDimensionsUserComment
current20:29, 14 September 2023Thumbnail for version as of 20:29, 14 September 2023625 × 690 (6 KB)CalendulaAsteraceaeimprove code
02:29, 17 March 2021Thumbnail for version as of 02:29, 17 March 2021450 × 471 (27 KB)CalendulaAsteraceaeActually changed background
02:28, 17 March 2021Thumbnail for version as of 02:28, 17 March 2021450 × 471 (27 KB)CalendulaAsteraceaeAdded white background
12:29, 16 March 2021Thumbnail for version as of 12:29, 16 March 2021450 × 471 (27 KB)CalendulaAsteraceaeUploaded own work with UploadWizard

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