Receiving Helpdesk

can the sum of 2 irrational numbers be rational

by Mr. Randi Harvey Sr. Published 3 years ago Updated 3 years ago

The sum of two irrational numbers can be rational and it can be irrational.Apr 20, 2017

Is the product of two irrational numbers always an irrational?

The product of two irrational numbers, in some cases, will be irrational. However, it is possible that some irrational numbers may multiply to form a rational product. Is it true that the product of two irrational numbers is always rational? Irrational numbers are the numbers that cannot be represented as a simple fraction.

How do you identify rational and irrational numbers?

You know, that demonstrably irrational person ... medical equipment and can identify covid variant strains rapidly, covering every sample unlike most of Canada. I expect Alberta will handily top that 10,000 number in two weeks time, because Denmark is ...

How do you calculate an irrational number?

e (Euler's Number)

  • Calculating. There are many ways of calculating the value of e, but none of them ever give a totally exact answer, because e is irrational and its digits go on ...
  • Another Calculation. ...
  • Remembering. ...
  • Growth. ...
  • Area
  • An Interesting Property. ...
  • 100 Decimal Digits
  • Advanced: Use of e in Compound Interest. ...
  • Euler's Formula for Complex Numbers

What is the quotient of two irrational numbers?

Is the quotient of two rational numbers always irrational number explain? Answer: Yes , the quotient of two rational number always a rational number . with X is any integers Y is not being Zero , then X/ Y will always be Rational number . Is an irrational number divided by an irrational number always irrational? Never.

Is 0 irrational or rational?

Any number which doesn't fulfill the above conditions is irrational. What about zero? It can be represented as a ratio of two integers as well as ratio of itself and an irrational number such that zero is not dividend in any case. People say that 0 is rational because it is an integer.

Is the sum of two irrational numbers rational?

The sum of two irrational numbers, in some cases, will be irrational. However, if the irrational parts of the numbers have a zero sum (cancel each other out), the sum will be rational. Click to see full answer.

image
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 1 2 3 4 5 6 7 8 9
8.3.21PHP Version2.79sRequest Duration2MBMemory UsageGET {post}Route
  • warninglog[20:16:21] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[20:16:21] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[20:16:21] LOG.warning: Callables of the form ["Swift_SmtpTransport", "Swift_Transport_EsmtpTranspor...
  • warninglog[20:16:21] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[20:16:21] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[20:16:21] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[20:16:21] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[20:16:21] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[20:16:21] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[20:16:21] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[20:16:21] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • Booting (10.83ms)
  • Application (2.78s)
  • 1 x Application (99.6%)
    2.78s
    1 x Booting (0.39%)
    10.83ms
    7 templates were rendered
    • themes.DevBlog.content.post (resources/views/themes/DevBlog/content/post.blade.php)34blade
      Params
      0
      post
      1
      postContent
      2
      author
      3
      updated_at
      4
      bing_rich_snippet_text
      5
      bing_rich_snippet_link
      6
      bing_related_keywords
      7
      google_related_keywords
      8
      bing_news_title
      9
      bing_news_description
      10
      bing_videos
      11
      bing_images
      12
      bing_search_result_title
      13
      bing_search_result_description
      14
      bing_search_result_url
      15
      bing_paa_questions
      16
      bing_paa_answers
      17
      bing_slider_faq_questions
      18
      bing_slider_faq_answers
      19
      bing_pop_faq_questions
      20
      bing_pop_faq_answers
      21
      bing_tab_faq_questions
      22
      bing_tab_faq_answers
      23
      google_faq_questions
      24
      google_faq_answers
      25
      google_rich_snippet
      26
      google_search_result
      27
      indexedArray
      28
      total_images
      29
      total_videos
      30
      settings
      31
      url_current
      32
      menus
      33
      sidebar
    • themes.DevBlog.layouts.master (resources/views/themes/DevBlog/layouts/master.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.head (resources/views/themes/DevBlog/panels/head.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.header (resources/views/themes/DevBlog/panels/header.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.navbar (resources/views/themes/DevBlog/panels/navbar.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.footer (resources/views/themes/DevBlog/panels/footer.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.scripts (resources/views/themes/DevBlog/panels/scripts.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    uri
    GET {post}
    middleware
    web, checkdate
    as
    post.show
    controller
    App\Http\Controllers\Frontend\json_data\PostController@show
    namespace
    where
    file
    app/Http/Controllers/Frontend/json_data/PostController.php:18-166
    7 statements were executed2.77s
    • select * from `posts` where `published_at` <= '2025-06-08 20:16:21' and `slug` = 'can-the-sum-of-2-irrational-numbers-be-rational' and `posts`.`deleted_at` is null limit 1
      2.94ms/app/Providers/RouteServiceProvider.php:54receivinghelpdeskask
      Metadata
      Bindings
      • 0. 2025-06-08 20:16:21
      • 1. can-the-sum-of-2-irrational-numbers-be-rational
      Backtrace
      • 15. /app/Providers/RouteServiceProvider.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/Router.php:842
      • 19. Route binding:39
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/framework/src/Illuminate/Foundation/Http/Middleware/VerifyCsrfToken.php:78
    • select * from `json_post_contents` where `json_post_contents`.`post_id` = 164671 and `json_post_contents`.`post_id` is not null and `rewrite_id` = 0
      12.57msmiddleware::checkdate:30receivinghelpdeskask
      Metadata
      Bindings
      • 0. 164671
      • 1. 0
      Backtrace
      • 19. middleware::checkdate:30
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/jetstream/src/Http/Middleware/ShareInertiaData.php:61
      • 22. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/Middleware/SubstituteBindings.php:50
    • select * from `nova_menu_menus` where `slug` = 'header' limit 1
      560μs/vendor/outl1ne/nova-menu-builder/src/helpers.php:32receivinghelpdeskask
      Metadata
      Bindings
      • 0. header
      Backtrace
      • 15. /vendor/outl1ne/nova-menu-builder/src/helpers.php:32
      • 17. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 19. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
      • 20. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`menu_id` = 1 and `nova_menu_menu_items`.`menu_id` is not null and `parent_id` is null order by `parent_id` asc, `order` asc, `name` asc
      550μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Bindings
      • 0. 1
      Backtrace
      • 19. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 20. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 22. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 24. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`parent_id` in (1) order by `order` asc
      270μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Backtrace
      • 24. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 25. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 27. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 28. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 29. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select `id`, `post_title`, `slug` from `posts` where `status` = 'publish' and `posts`.`deleted_at` is null order by RAND() limit 10
      2.75s/app/View/Composers/SidebarView.php:22receivinghelpdeskask
      Metadata
      Bindings
      • 0. publish
      Backtrace
      • 14. /app/View/Composers/SidebarView.php:22
      • 15. /app/View/Composers/SidebarView.php:12
      • 16. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:124
      • 17. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:162
      • 20. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:177
    • select * from `fake_users` where `fake_users`.`id` = 21612 limit 1
      700μsview::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15receivinghelpdeskask
      Metadata
      Bindings
      • 0. 21612
      Backtrace
      • 21. view::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15
      • 23. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:108
      • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 25. /vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
      • 26. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:61
    App\Models\FakeUser
    1
    Outl1ne\MenuBuilder\Models\MenuItem
    1
    Outl1ne\MenuBuilder\Models\Menu
    1
    App\Models\JsonPostContent
    1
    App\Models\Post
    11
        _token
        bixjHt7lwgQ9w7FpcvhYZcageL9fJ3QcdkLDLFwe
        _previous
        array:1 [ "url" => "https://receivinghelpdesk.com/ask/can-the-sum-of-2-irrational-numbers-be-ratio...
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /can-the-sum-of-2-irrational-numbers-be-rational
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:24 [ "cf-ipcountry" => array:1 [ 0 => "US" ] "cf-connecting-ip" => array:1 [ 0 => "216.73.216.225" ] "cdn-loop" => array:1 [ 0 => "cloudflare; loops=1" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "cf-ray" => array:1 [ 0 => "94c92da06802eb03-ORD" ] "accept-encoding" => array:1 [ 0 => "gzip, br" ] "priority" => array:1 [ 0 => "u=0, i" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "cf-visitor" => array:1 [ 0 => "{"scheme":"https"}" ] "connection" => array:1 [ 0 => "close" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "216.73.216.225, 172.71.254.170" ] "x-server-addr" => array:1 [ 0 => "154.12.239.204" ] "host" => array:1 [ 0 => "receivinghelpdesk.com" ] ]
        request_server
        0 of 0
        array:55 [ "USER" => "runcloud" "HOME" => "/home/runcloud" "SCRIPT_NAME" => "/ask/index.php" "REQUEST_URI" => "/ask/can-the-sum-of-2-irrational-numbers-be-rational" "QUERY_STRING" => "" "REQUEST_METHOD" => "GET" "SERVER_PROTOCOL" => "HTTP/1.0" "GATEWAY_INTERFACE" => "CGI/1.1" "REDIRECT_URL" => "/ask/can-the-sum-of-2-irrational-numbers-be-rational" "REMOTE_PORT" => "35156" "SCRIPT_FILENAME" => "/home/runcloud/webapps/ReceivingHelpDesk/ask/index.php" "SERVER_ADMIN" => "you@example.com" "CONTEXT_DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "CONTEXT_PREFIX" => "" "REQUEST_SCHEME" => "http" "DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "REMOTE_ADDR" => "172.71.254.170" "SERVER_PORT" => "80" "SERVER_ADDR" => "127.0.0.1" "SERVER_NAME" => "receivinghelpdesk.com" "SERVER_SOFTWARE" => "Apache/2.4.63 (Unix) OpenSSL/1.1.1f" "SERVER_SIGNATURE" => "" "LD_LIBRARY_PATH" => "/RunCloud/Packages/apache2-rc/lib" "PATH" => "/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin" "HTTP_CF_IPCOUNTRY" => "US" "HTTP_CF_CONNECTING_IP" => "216.73.216.225" "HTTP_CDN_LOOP" => "cloudflare; loops=1" "HTTP_SEC_FETCH_SITE" => "none" "HTTP_ACCEPT" => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" "HTTP_USER_AGENT" => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" "HTTP_UPGRADE_INSECURE_REQUESTS" => "1" "HTTP_SEC_CH_UA_PLATFORM" => ""Windows"" "HTTP_SEC_CH_UA_MOBILE" => "?0" "HTTP_SEC_CH_UA" => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" "HTTP_CACHE_CONTROL" => "no-cache" "HTTP_PRAGMA" => "no-cache" "HTTP_SEC_FETCH_DEST" => "document" "HTTP_CF_RAY" => "94c92da06802eb03-ORD" "HTTP_ACCEPT_ENCODING" => "gzip, br" "HTTP_PRIORITY" => "u=0, i" "HTTP_SEC_FETCH_USER" => "?1" "HTTP_SEC_FETCH_MODE" => "navigate" "HTTP_CF_VISITOR" => "{"scheme":"https"}" "HTTP_CONNECTION" => "close" "HTTP_X_FORWARDED_PROTO" => "https" "HTTP_X_FORWARDED_FOR" => "216.73.216.225, 172.71.254.170" "HTTP_X_SERVER_ADDR" => "154.12.239.204" "HTTP_HOST" => "receivinghelpdesk.com" "HTTPS" => "on" "REDIRECT_STATUS" => "200" "REDIRECT_HTTPS" => "on" "FCGI_ROLE" => "RESPONDER" "PHP_SELF" => "/ask/index.php" "REQUEST_TIME_FLOAT" => 1749393981.6014 "REQUEST_TIME" => 1749393981 ]
        request_cookies
        []
        
        response_headers
        0 of 0
        array:7 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "private, must-revalidate" ] "date" => array:1 [ 0 => "Sun, 08 Jun 2025 14:46:21 GMT" ] "pragma" => array:1 [ 0 => "no-cache" ] "expires" => array:1 [ 0 => -1 ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IitqUldmU0Q5OXh5OFk4bEZ6aEhGbmc9PSIsInZhbHVlIjoia1AvaHlmV1g2TFZOclcycnpHV1BPZitjTDVUb3MrckxtbWt0T2lBTHVRNmpPdTgycFkvSmpzRVdibnBIU1BiQUtLTCswaVlHTFNyNWpGL09Mcy9CNGNMTWZSSkRxM1k2RFJSSlo5ZkVxbkJSOHhqV0pZN3kwYlNNdnlEYzN3MGQiLCJtYWMiOiJkOTEyY2M2M2VlZGU5ZjViNDE1MjE4Zjk1NTY3MmZjM2E2YzU2Zjk1YmM4OTBkMTFjYmQwOTg3Yzk0MTQyOWUwIiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 16:46:24 GMT; Max-Age=7200; path=/; samesite=laxXSRF-TOKEN=eyJpdiI6IitqUldmU0Q5OXh5OFk4bEZ6aEhGbmc9PSIsInZhbHVlIjoia1AvaHlmV1g2TFZOclcycnpHV1BPZitjTDVUb3MrckxtbWt0T2lBTHVRNmpPdTgycFkvSmpzRVdibnBIU1BiQUtLTCswa" 1 => "askhelpdesk_session=eyJpdiI6Iko0WnMxSTQ2MkpPOTNtL1J2dlQ4SXc9PSIsInZhbHVlIjoidUhENkducUZEQ0pRY1lpL2t1MkMySTR3bTd1TGxTY2tGTzRVRDFSSDJOM052UkRLa1IvQXF5bEI1dkp3VVBQWXVqZXROZFVUdUJwQ2JYVFhkWnNqbG15REFvaHdYeEw5UUNXcWtqWnc1ZlZKVXI2R1ZyVjhJUHgvWjIxNmZMU2oiLCJtYWMiOiIxYTAxNzU0MDY2MzcwOGI3ZTYyZGY1MWE2MGNiMjAwMDJlMWRkZDljODdmYTYwNWFiZWIyMWNlN2M4NGRkZDM2IiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 16:46:24 GMT; Max-Age=7200; path=/; httponly; samesite=laxaskhelpdesk_session=eyJpdiI6Iko0WnMxSTQ2MkpPOTNtL1J2dlQ4SXc9PSIsInZhbHVlIjoidUhENkducUZEQ0pRY1lpL2t1MkMySTR3bTd1TGxTY2tGTzRVRDFSSDJOM052UkRLa1IvQXF5bEI1dkp3VVBQ" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IitqUldmU0Q5OXh5OFk4bEZ6aEhGbmc9PSIsInZhbHVlIjoia1AvaHlmV1g2TFZOclcycnpHV1BPZitjTDVUb3MrckxtbWt0T2lBTHVRNmpPdTgycFkvSmpzRVdibnBIU1BiQUtLTCswaVlHTFNyNWpGL09Mcy9CNGNMTWZSSkRxM1k2RFJSSlo5ZkVxbkJSOHhqV0pZN3kwYlNNdnlEYzN3MGQiLCJtYWMiOiJkOTEyY2M2M2VlZGU5ZjViNDE1MjE4Zjk1NTY3MmZjM2E2YzU2Zjk1YmM4OTBkMTFjYmQwOTg3Yzk0MTQyOWUwIiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 16:46:24 GMT; path=/XSRF-TOKEN=eyJpdiI6IitqUldmU0Q5OXh5OFk4bEZ6aEhGbmc9PSIsInZhbHVlIjoia1AvaHlmV1g2TFZOclcycnpHV1BPZitjTDVUb3MrckxtbWt0T2lBTHVRNmpPdTgycFkvSmpzRVdibnBIU1BiQUtLTCswa" 1 => "askhelpdesk_session=eyJpdiI6Iko0WnMxSTQ2MkpPOTNtL1J2dlQ4SXc9PSIsInZhbHVlIjoidUhENkducUZEQ0pRY1lpL2t1MkMySTR3bTd1TGxTY2tGTzRVRDFSSDJOM052UkRLa1IvQXF5bEI1dkp3VVBQWXVqZXROZFVUdUJwQ2JYVFhkWnNqbG15REFvaHdYeEw5UUNXcWtqWnc1ZlZKVXI2R1ZyVjhJUHgvWjIxNmZMU2oiLCJtYWMiOiIxYTAxNzU0MDY2MzcwOGI3ZTYyZGY1MWE2MGNiMjAwMDJlMWRkZDljODdmYTYwNWFiZWIyMWNlN2M4NGRkZDM2IiwidGFnIjoiIn0%3D; expires=Sun, 08-Jun-2025 16:46:24 GMT; path=/; httponlyaskhelpdesk_session=eyJpdiI6Iko0WnMxSTQ2MkpPOTNtL1J2dlQ4SXc9PSIsInZhbHVlIjoidUhENkducUZEQ0pRY1lpL2t1MkMySTR3bTd1TGxTY2tGTzRVRDFSSDJOM052UkRLa1IvQXF5bEI1dkp3VVBQ" ] ]
        session_attributes
        0 of 0
        array:4 [ "_token" => "bixjHt7lwgQ9w7FpcvhYZcageL9fJ3QcdkLDLFwe" "_previous" => array:1 [ "url" => "https://receivinghelpdesk.com/ask/can-the-sum-of-2-irrational-numbers-be-rational" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]