01.21 Ισοδύναμα Κλάσματα

Τι είναι τα ισοδύναμα κλάσματα ;

Πώς δημιουργώ ισοδύναμα κλάσματα ;

Πώς ελέγχουμε αν δυο κλάσματα είναι ισοδύναμα ;

Τι είναι απλοποίηση ενός κλάσματος ; Ποιο κλάσμα λέμε ανάγωγο ;

επισκέπτες ως τώρα

Στην κορυφή της σελίδας

J! Info Request Session Profile Queries47 Log

38.573MB706ms

Joomla! Version	4.4.12
PHP Version	8.1.29
Identity	guest
Response	200
Template	rt_notio
Database	Server mysql Version 10.6.18-MariaDB-cll-lve Collation utf8mb4_general_ci Conn Collation utf8mb3_general_ci

$_GET: []
$_POST: []
$_SESSION: array:1 [ "joomla" => "***redacted***" ]
$_COOKIE: []
$_SERVER: array:54 [ "PATH" => "/usr/local/bin:/usr/bin:/bin" "TEMP" => "/tmp" "TMP" => "/tmp" "TMPDIR...

session: array:3 [ "counter" => 1 "timer" => array:3 [ "start" => 1743708753 "last" => 1743708753...
registry: array:3 [ "data" => [] "initialized" => false "separator" => "." ]
user: array:18 [ "id" => 0 "name" => null "username" => null "email" => null "password" => "***r...

afterLoad (4.11MB) (19.5ms)
afterInitialise (9.02MB) (52.06ms)
afterRoute (5.75MB) (29.07ms)
beforeRenderComponent com_content (34.4KB) (400μs)
Before Access::preloadComponents (all components) (466.48KB) (2.76ms)
After Access::preloadComponents (all components) (155.77KB) (701μs)
Before Access::preloadPermissions (com_content) (4.35KB) (14μs)
After Access::preloadPermissions (com_content) (1.85MB) (4.01ms)
Before Access::getAssetRules (id:8651 name:com_content.article.5593) (130.8KB) (117μs)
After Access::getAssetRules (id:8651 name:com_content.article.5593) (27.55KB) (256μs)
Before Access::getAssetRules (id:1 name:root.1) (1.38MB) (13.77ms)
After Access::getAssetRules (id:1 name:root.1) (6.95KB) (20μs)
afterRenderComponent com_content (1.58MB) (18.14ms)
afterDispatch (634.77KB) (5.46ms)
beforeRenderRawModule mod_custom (statcounter) (9.26MB) (69.34ms)
afterRenderRawModule mod_custom (statcounter) (40.59KB) (952μs)
beforeRenderModule mod_custom (statcounter) (720B) (7μs)
afterRenderModule mod_custom (statcounter) (44.45KB) (1.16ms)
afterRender (2.27MB) (488ms)

1 x afterRender (2.27MB) (69.13%)	487.72ms
1 x beforeRenderRawModule mod_custom (statcounter) (9.26MB) (9.83%)	69.34ms
1 x afterInitialise (9.02MB) (7.38%)	52.06ms
1 x afterRoute (5.75MB) (4.12%)	29.07ms
1 x afterLoad (4.11MB) (2.76%)	19.50ms
1 x afterRenderComponent com_content (1.58MB) (2.57%)	18.14ms
1 x Before Access::getAssetRules (id:1 name:root.1) (1.38MB) (1.95%)	13.77ms
1 x afterDispatch (634.77KB) (0.77%)	5.46ms
1 x After Access::preloadPermissions (com_content) (1.85MB) (0.57%)	4.01ms
1 x Before Access::preloadComponents (all components) (466.48KB) (0.39%)	2.76ms
1 x afterRenderModule mod_custom (statcounter) (44.45KB) (0.16%)	1.16ms
1 x afterRenderRawModule mod_custom (statcounter) (40.59KB) (0.13%)	952μs
1 x After Access::preloadComponents (all components) (155.77KB) (0.1%)	701μs
1 x beforeRenderComponent com_content (34.4KB) (0.06%)	400μs
1 x After Access::getAssetRules (id:8651 name:com_content.article.5593) (27.55KB) (0.04%)	256μs
1 x Before Access::getAssetRules (id:8651 name:com_content.article.5593) (130.8KB) (0.02%)	117μs
1 x After Access::getAssetRules (id:1 name:root.1) (6.95KB) (0%)	20μs
1 x Before Access::preloadPermissions (com_content) (4.35KB) (0%)	14μs
1 x beforeRenderModule mod_custom (statcounter) (720B) (0%)	7μs

47 statements were executed, 17 of which were duplicates, 30 unique14.31ms230.73KB

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SELECT `data` FROM `w7ut0_session` WHERE `session_id` = ?94μs1.68KBParams/libraries/vendor/joomla/session/src/Handler/DatabaseHandler.php:261Copy
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SELECT `extension_id` AS `id`,`element` AS `option`,`params`,`enabled` FROM `w7ut0_extensions` WHERE `type` = 'component' AND `state` = 0 AND `enabled` = 1221μs9.9KB/libraries/src/Component/ComponentHelper.php:399Copy
SELECT `id`,`rules` FROM `w7ut0_viewlevels`70μs17.01KB/libraries/src/Access/Access.php:955Copy
SELECT `b`.`id` FROM `w7ut0_usergroups` AS `a` LEFT JOIN `w7ut0_usergroups` AS `b` ON `b`.`lft` <= `a`.`lft` AND `b`.`rgt` >= `a`.`rgt` WHERE `a`.`id` = :guest97μs1.68KBParams/libraries/src/Access/Access.php:868Copy
SELECT `folder` AS `type`,`element` AS `name`,`params` AS `params`,`extension_id` AS `id` FROM `w7ut0_extensions` WHERE `enabled` = 1 AND `type` = 'plugin' AND `state` IN (0,1) AND `access` IN (:preparedArray1,:preparedArray2) ORDER BY `ordering`441μs5.09KBParams/libraries/src/Plugin/PluginHelper.php:294Copy
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SELECT template FROM w7ut0_template_styles as s WHERE s.client_id = 0 AND s.home = 178μs1.05KB/libraries/rokcommon/RokCommon/PlatformInfo/Joomla.php:31Copy
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SHOW FULL COLUMNS FROM `w7ut0_content`333μs2.61KB/libraries/vendor/joomla/database/src/Mysqli/MysqliDriver.php:625Copy
UPDATE `w7ut0_content` SET `hits` = (`hits` + 1) WHERE `id` = '5593'228μs0B/libraries/src/Table/Table.php:1325Copy
SELECT `a`.`id`,`a`.`asset_id`,`a`.`title`,`a`.`alias`,`a`.`introtext`,`a`.`fulltext`,`a`.`state`,`a`.`catid`,`a`.`created`,`a`.`created_by`,`a`.`created_by_alias`,`a`.`modified`,`a`.`modified_by`,`a`.`checked_out`,`a`.`checked_out_time`,`a`.`publish_up`,`a`.`publish_down`,`a`.`images`,`a`.`urls`,`a`.`attribs`,`a`.`version`,`a`.`ordering`,`a`.`metakey`,`a`.`metadesc`,`a`.`access`,`a`.`hits`,`a`.`metadata`,`a`.`featured`,`a`.`language`,`fp`.`featured_up`,`fp`.`featured_down`,`c`.`title` AS `category_title`,`c`.`alias` AS `category_alias`,`c`.`access` AS `category_access`,`c`.`language` AS `category_language`,`fp`.`ordering`,`u`.`name` AS `author`,`parent`.`title` AS `parent_title`,`parent`.`id` AS `parent_id`,`parent`.`path` AS `parent_route`,`parent`.`alias` AS `parent_alias`,`parent`.`language` AS `parent_language`,ROUND(`v`.`rating_sum` / `v`.`rating_count`, 1) AS `rating`,`v`.`rating_count` AS `rating_count` FROM `w7ut0_content` AS `a` INNER JOIN `w7ut0_categories` AS `c` ON `c`.`id` = `a`.`catid` LEFT JOIN `w7ut0_content_frontpage` AS `fp` ON `fp`.`content_id` = `a`.`id` LEFT JOIN `w7ut0_users` AS `u` ON `u`.`id` = `a`.`created_by` LEFT JOIN `w7ut0_categories` AS `parent` ON `parent`.`id` = `c`.`parent_id` LEFT JOIN `w7ut0_content_rating` AS `v` ON `a`.`id` = `v`.`content_id` WHERE ( (`a`.`id` = :pk AND `c`.`published` > 0) AND (`a`.`publish_up` IS NULL OR `a`.`publish_up` <= :publishUp)) AND (`a`.`publish_down` IS NULL OR `a`.`publish_down` >= :publishDown) AND `a`.`state` IN (:preparedArray1,:preparedArray2)264μs20.74KBParams/components/com_content/src/Model/ArticleModel.php:215Copy
SELECT `c`.`id`,`c`.`asset_id`,`c`.`access`,`c`.`alias`,`c`.`checked_out`,`c`.`checked_out_time`,`c`.`created_time`,`c`.`created_user_id`,`c`.`description`,`c`.`extension`,`c`.`hits`,`c`.`language`,`c`.`level`,`c`.`lft`,`c`.`metadata`,`c`.`metadesc`,`c`.`metakey`,`c`.`modified_time`,`c`.`note`,`c`.`params`,`c`.`parent_id`,`c`.`path`,`c`.`published`,`c`.`rgt`,`c`.`title`,`c`.`modified_user_id`,`c`.`version`, CASE WHEN CHAR_LENGTH(`c`.`alias`) != 0 THEN CONCAT_WS(':', `c`.`id`, `c`.`alias`) ELSE `c`.`id` END as `slug` FROM `w7ut0_categories` AS `s` INNER JOIN `w7ut0_categories` AS `c` ON (`s`.`lft` <= `c`.`lft` AND `c`.`lft` < `s`.`rgt`) OR (`c`.`lft` < `s`.`lft` AND `s`.`rgt` < `c`.`rgt`) WHERE (`c`.`extension` = :extension OR `c`.`extension` = 'system') AND `c`.`access` IN (:preparedArray1,:preparedArray2) AND `c`.`published` = 1 AND `s`.`id` = :id ORDER BY `c`.`lft`468μs7.23KBParams/libraries/src/Categories/Categories.php:375Copy
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SELECT LOWER(REPLACE(u.title, " ", "")) FROM w7ut0_usergroups as u WHERE u.id IN ('1','9')79μs1.23KB/plugins/system/tabsaccordions/src/Sets.php:413Copy
SELECT LOWER(REPLACE(a.title, " ", "")) FROM w7ut0_viewlevels as a WHERE a.id IN ('1','5')63μs1.23KB/plugins/system/tabsaccordions/src/Sets.php:148Copy
SELECT LOWER(REPLACE(u.title, " ", "")) FROM w7ut0_usergroups as u WHERE u.id IN ('1','9')58μs1.23KB/plugins/system/tabsaccordions/src/Sets.php:413Copy
SELECT `c`.`id`,`c`.`asset_id`,`c`.`access`,`c`.`alias`,`c`.`checked_out`,`c`.`checked_out_time`,`c`.`created_time`,`c`.`created_user_id`,`c`.`description`,`c`.`extension`,`c`.`hits`,`c`.`language`,`c`.`level`,`c`.`lft`,`c`.`metadata`,`c`.`metadesc`,`c`.`metakey`,`c`.`modified_time`,`c`.`note`,`c`.`params`,`c`.`parent_id`,`c`.`path`,`c`.`published`,`c`.`rgt`,`c`.`title`,`c`.`modified_user_id`,`c`.`version`, CASE WHEN CHAR_LENGTH(`c`.`alias`) != 0 THEN CONCAT_WS(':', `c`.`id`, `c`.`alias`) ELSE `c`.`id` END as `slug` FROM `w7ut0_categories` AS `s` INNER JOIN `w7ut0_categories` AS `c` ON (`s`.`lft` <= `c`.`lft` AND `c`.`lft` < `s`.`rgt`) OR (`c`.`lft` < `s`.`lft` AND `s`.`rgt` < `c`.`rgt`) WHERE (`c`.`extension` = :extension OR `c`.`extension` = 'system') AND `c`.`access` IN (:preparedArray1,:preparedArray2) AND `c`.`published` = 1 AND `s`.`id` = :id ORDER BY `c`.`lft`563μs7.23KBParams/libraries/src/Categories/Categories.php:375Copy
SELECT DISTINCT a.id, a.title, a.name, a.checked_out, a.checked_out_time, a.note, a.state, a.access, a.created_time, a.created_user_id, a.ordering, a.language, a.fieldparams, a.params, a.type, a.default_value, a.context, a.group_id, a.label, a.description, a.required, a.only_use_in_subform,l.title AS language_title, l.image AS language_image,uc.name AS editor,ag.title AS access_level,ua.name AS author_name,g.title AS group_title, g.access as group_access, g.state AS group_state, g.note as group_note FROM w7ut0_fields AS a LEFT JOIN `w7ut0_languages` AS l ON l.lang_code = a.language LEFT JOIN w7ut0_users AS uc ON uc.id=a.checked_out LEFT JOIN w7ut0_viewlevels AS ag ON ag.id = a.access LEFT JOIN w7ut0_users AS ua ON ua.id = a.created_user_id LEFT JOIN w7ut0_fields_groups AS g ON g.id = a.group_id LEFT JOIN `w7ut0_fields_categories` AS fc ON fc.field_id = a.id WHERE ( (`a`.`context` = :context AND (`fc`.`category_id` IS NULL OR `fc`.`category_id` IN (:preparedArray1,:preparedArray2,:preparedArray3,:preparedArray4)) AND `a`.`access` IN (:preparedArray5,:preparedArray6)) AND (`a`.`group_id` = 0 OR `g`.`access` IN (:preparedArray7,:preparedArray8)) AND `a`.`state` = :state) AND (`a`.`group_id` = 0 OR `g`.`state` = :gstate) AND `a`.`only_use_in_subform` = :only_use_in_subform ORDER BY a.ordering ASC513μs8.09KBParams/libraries/src/MVC/Model/BaseDatabaseModel.php:166Copy
SELECT `key` FROM w7ut0_os_gallery_api WHERE `service`='os_gallery_activation'55μs912B/components/com_osgallery/helpers/osGalleryHelperSite.php:1997Copy
SELECT `a`.`id`,`a`.`title`,`a`.`catid`,`a`.`language`, CASE WHEN CHAR_LENGTH(`a`.`alias`) != 0 THEN CONCAT_WS(':', `a`.`id`, `a`.`alias`) ELSE a.id END AS `slug`, CASE WHEN CHAR_LENGTH(`cc`.`alias`) != 0 THEN CONCAT_WS(':', `cc`.`id`, `cc`.`alias`) ELSE cc.id END AS `catslug` FROM `w7ut0_content` AS `a` LEFT JOIN `w7ut0_categories` AS `cc` ON `cc`.`id` = `a`.`catid` WHERE `a`.`catid` = :catid AND `a`.`state` = :state AND `a`.`access` IN (:preparedArray1,:preparedArray2) AND (`publish_up` IS NULL OR `publish_up` <= :nowDate1) AND (`publish_down` IS NULL OR `publish_down` >= :nowDate2) ORDER BY `a`.`title`615μs16.62KBParams/plugins/content/pagenavigation/src/Extension/PageNavigation.php:188Copy
SELECT `c`.`id`,`c`.`asset_id`,`c`.`access`,`c`.`alias`,`c`.`checked_out`,`c`.`checked_out_time`,`c`.`created_time`,`c`.`created_user_id`,`c`.`description`,`c`.`extension`,`c`.`hits`,`c`.`language`,`c`.`level`,`c`.`lft`,`c`.`metadata`,`c`.`metadesc`,`c`.`metakey`,`c`.`modified_time`,`c`.`note`,`c`.`params`,`c`.`parent_id`,`c`.`path`,`c`.`published`,`c`.`rgt`,`c`.`title`,`c`.`modified_user_id`,`c`.`version`, CASE WHEN CHAR_LENGTH(`c`.`alias`) != 0 THEN CONCAT_WS(':', `c`.`id`, `c`.`alias`) ELSE `c`.`id` END as `slug` FROM `w7ut0_categories` AS `s` INNER JOIN `w7ut0_categories` AS `c` ON (`s`.`lft` <= `c`.`lft` AND `c`.`lft` < `s`.`rgt`) OR (`c`.`lft` < `s`.`lft` AND `s`.`rgt` < `c`.`rgt`) WHERE (`c`.`extension` = :extension OR `c`.`extension` = 'system') AND `c`.`access` IN (:preparedArray1,:preparedArray2) AND `c`.`published` = 1 AND `s`.`id` = :id ORDER BY `c`.`lft`430μs7.23KBParams/libraries/src/Categories/Categories.php:375Copy
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SELECT LOWER(REPLACE(u.title, " ", "")) FROM w7ut0_usergroups as u WHERE u.id IN ('1','9')61μs1.23KB/plugins/system/tabsaccordions/src/Sets.php:413Copy
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SELECT m.id, m.alias, m.path AS route, m.level, m.parent_id FROM w7ut0_menu AS m WHERE m.menutype = 'mainmenu' AND m.parent_id > 0 AND m.client_id = 0 AND m.published >= 0 ORDER BY m.lft1.2ms6.62KB/libraries/gantry5/src/classes/Gantry/Framework/Menu.php:791Copy
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SELECT LOWER(REPLACE(a.title, " ", "")) FROM w7ut0_viewlevels as a WHERE a.id IN ('1','5')98μs1.23KB/plugins/system/tabsaccordions/src/Sets.php:148Copy
SELECT LOWER(REPLACE(u.title, " ", "")) FROM w7ut0_usergroups as u WHERE u.id IN ('1','9')64μs1.23KB/plugins/system/tabsaccordions/src/Sets.php:413Copy
SELECT `key` FROM w7ut0_os_gallery_api WHERE `service`='os_gallery_activation'69μs912B/components/com_osgallery/helpers/osGalleryHelperSite.php:1997Copy
SELECT `session_id` FROM `w7ut0_session` WHERE `session_id` = ?144μs1.9KBParams/libraries/vendor/joomla/session/src/Handler/DatabaseHandler.php:291Copy
UPDATE `w7ut0_session` SET `data` = ? , `time` = ? WHERE `session_id` = ?301μs840BParams/libraries/vendor/joomla/session/src/Handler/DatabaseHandler.php:318Copy

01.21 Ισοδύναμα Κλάσματα

Τι είναι τα ισοδύναμα κλάσματα ;

Πώς δημιουργώ ισοδύναμα κλάσματα ;

Πώς ελέγχουμε αν δυο κλάσματα είναι ισοδύναμα ;

Αν πολλαπλασιάσουμε «χιαστί» τους όρους 2 ισοδύναμων κλασμάτων, τα δύο γινόμενα που προκύπτουν είναι ίσα μεταξύ τους.

Τι είναι απλοποίηση ενός κλάσματος ; Ποιο κλάσμα λέμε ανάγωγο ;