A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property CI_Cache::$apc is deprecated

Filename: libraries/Driver.php

Line Number: 188

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 22
Function: driver

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 17
Function: initiateCache

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 12
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property CI_Cache::$file is deprecated

Filename: libraries/Driver.php

Line Number: 188

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 22
Function: driver

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 17
Function: initiateCache

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 12
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Contents_model::$best_cnt is deprecated

Filename: ver3/Contents_model.php

Line Number: 17

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
Line: 17
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 12
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Contents_model::$popular_cnt is deprecated

Filename: ver3/Contents_model.php

Line Number: 18

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
Line: 18
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 12
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Article_model::$best_cnt is deprecated

Filename: ver3/Article_model.php

Line Number: 18

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Article_model.php
Line: 18
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 13
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Article_model::$popular_cnt is deprecated

Filename: ver3/Article_model.php

Line Number: 19

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Article_model.php
Line: 19
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 13
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: urldecode(): Passing null to parameter #1 ($string) of type string is deprecated

Filename: ver3/Member_model.php

Line Number: 279

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Member_model.php
Line: 279
Function: urldecode

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 123
Function: getMemberView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: Trying to access array offset on null

Filename: ver3/Article.php

Line Number: 124

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 124
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: Trying to access array offset on null

Filename: ver3/Article.php

Line Number: 125

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 125
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: is_dir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 201

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 201
Function: is_dir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_exists(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/e748b326f0931861cc3f0be7c6f926a3.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 212

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 212
Function: file_exists

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/e748b326f0931861cc3f0be7c6f926a3.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/e748b326f0931861cc3f0be7c6f926a3.gif): Failed to open stream: Operation not permitted

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: is_dir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 201

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 201
Function: is_dir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_exists(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/7ae299307840bffb023846051f354778.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 212

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 212
Function: file_exists

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/7ae299307840bffb023846051f354778.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/7ae299307840bffb023846051f354778.gif): Failed to open stream: Operation not permitted

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: is_dir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 201

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 201
Function: is_dir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_exists(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/ac0cbc7d2fa2725a832b7384f3a9d3d1.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 212

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 212
Function: file_exists

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/ac0cbc7d2fa2725a832b7384f3a9d3d1.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/ac0cbc7d2fa2725a832b7384f3a9d3d1.gif): Failed to open stream: Operation not permitted

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: is_dir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 201

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 201
Function: is_dir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_exists(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/c0ac4222dc74c962540572b8cce9a070.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

Filename: libraries/Common.php

Line Number: 212

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 212
Function: file_exists

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 249
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Article.php
Line: 131
Function: initBoardView

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
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[개념 기사] 거북이를 따라잡거나 앞지를 수 없다고?
호박고구마 2021.02.05 21:22

 제논의 역설을 아시나요? 제논의 역설은 고대 그리스의 엘리아의 제논이 '만물은 흐른다'라는 이론을 반대하기 위해 만들어 낸 역설로 요약하자면 '만물은 언제나 정지해 있다'라는 것입니다. 제논의 역설은 우리가 실제로 알고 있는 결과와 다른 결과를 주장하고, 그 주장에 대한 근거를 제시했기 때문에 역설이 되었습니다. 제논의 역설의 대표적인 패러독스를 보겠습니다. 한 사람과 거북이가 경주를 할 때 그 사람은 절대 거북이를 따라잡을 수 없습니다. 네? 실제로는 이 사람이 거북이를 따라잡고 앞지르는데요? 자, 이게 무슨 말인지 더욱 자세히 알아봅시다.

 

 

 먼저, 우리가 거북이와 경주를 한다고 생각해 봅시다. 그냥 경주하면 거북이가 100% 지게 되므로 거북이는 100m 앞에서 출발한다고 합시다. 우리가 100m를 갈 동안 거북이는 10m를 갑니다. 이렇게 되면 우리는 절대 거북이를 따라잡거나 추월할 수 없습니다. 네? 이게 무슨 말이냐고요? 그러면 지금 제가 설명하는 이야기를 논리적으로 반박해 보세요. 우리가 거북이가 있는 곳까지 100m를 갑니다. 이때 거북이도 10m를 가죠. 우리가 또 거북이가 있는 10m를 더 갑니다. 이때 거북이도 1m를 가죠. 그리고 우리가 다시 거북이가 있는 1m를 갑니다. 이때도 마찬가지로 거북이도 0.1m를 갑니다. 이렇게 된다면 우리는 거북이와 가까워질 수는 있어도 영원히 거북이를 앞지르거나 따라잡을 수는 없습니다. 이것이 제논의 역설입니다. 물론 이 패러독스가 틀렸다는 것을 논리적으로 설명할 수 있는 분도 있을 겁니다. 그래도 이제부터 이 패러독스을 반박해봅시다.

 

 우리가 거북이와 경주할 때 우리가 가는 거리를 써보면 아래처럼 쓸 수 있습니다.

 

 100+10+1+0.1+0.01=111.111......

 

 자, 그렇다면 무한이 유한이 될 수 있을까요? 즉, 무한히 많은 숫자를 더했을 때 유한의 결과가 나올 수 있을까요? 무한등비급수를 활용한 다음 식을 봅시다. 참고로 무한등비급수는 \frac{a}{1-r} 입니다.

 

 111.111...... = \frac{100}{1-\frac{1}{10}}=\frac{1000}{9}

 그러므로 우리는 거북이를 \frac{1000}{9}m만에 따라잡을 수 있습니다. 이렇게 무한등비급수로 이 패러독스를 반박할 수 있습니다. 이 패러독스 외에도 화살의 패러독스, 이분법의 패러독스 등 제논의 역설의 패러독스들도 있습니다.

 

 < 오늘의 퀴즈 >

 한 사람과 거북이가 경주할 때 그 사람은 거북이를 절대 따라잡을 수 없다는 패러독스는 무엇의 대표적인 패러독스일까요?

 1. 제우스의 역설

 2. 제논의 주장

 3. 제논의 역설

 4. 제우스의 주장

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