Neural Networks Part 8: Image Classification with Convolutional Neural Networks (CNNs)

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  • Опубликовано: 10 янв 2025

Комментарии • 784

  • @statquest
    @statquest  3 года назад +24

    Learn how to code a simple Convolutional Neural Network with this fully annotated Jupyter Notebook: lightning.ai/lightning-ai/studios/build-train-and-use-a-convolutional-neural-network
    The full Neural Networks playlist, from the basics to deep learning, is here: ruclips.net/video/CqOfi41LfDw/видео.html
    Support StatQuest by buying my books The StatQuest Illustrated Guide to Machine Learning, The StatQuest Illustrated Guide to Neural Networks and AI, or a Study Guide or Merch!!! statquest.org/statquest-store/

    • @evelinewuytens2890
      @evelinewuytens2890 3 года назад +3

      Great playlist! Can you also make a video on variational autoencoder networks?

    • @statquest
      @statquest  3 года назад +1

      @@evelinewuytens2890 I'll keep that in mind.

    • @bennybenbenw
      @bennybenbenw 2 года назад

      Hey josh, is that possible to make videos about rcnn, fast rcnn, faster cnn & yolo
      I watched some videos and read some paper, didnt clear explain math part(only understand basic concept)
      Especially how to caculate selective search, how to train(one image contain many classification, how to train when we have many many images)

    • @statquest
      @statquest  2 года назад +1

      @@bennybenbenw I'll keep those topics in mind.

  • @GeekyMan
    @GeekyMan 3 года назад +496

    I can't imagine how much time and effort you put for:
    1. Creating the content and simplifying it for us
    2. Create the animated ppts
    3. Explaining every step with great detail and simplicity
    I just wanna give a huge hug to you sir! You are an asset. ❤❤

    • @statquest
      @statquest  3 года назад +37

      Wow, thanks!

    • @olli3686
      @olli3686 3 года назад +19

      @@statquest for real! This is amazing! Thank you so much

    • @raunak5344
      @raunak5344 10 месяцев назад +4

      4. Writing the song lines and adding attractive music to them to add some entertainment to the whole matter

    • @samlinus836
      @samlinus836 6 месяцев назад

      Yes he is

  • @ddb276
    @ddb276 2 года назад +188

    As a Cambridge qualified PhD Mathematician, I cannot begin to describe how fantastic your series are. The way you simplify the concepts, yet keep true to the underlying Mathematics is quite amazing. Not to mention the great animations, dynamic graphs and equations, etc. Well done Josh, for making principled data science accessible to the general audience.

    • @statquest
      @statquest  2 года назад +10

      Wow! Thank you very much!

    • @birdwalkin
      @birdwalkin Год назад +8

      As a person with a PhD in Subjective Applied Mathematics from the University of American Samoa, I approve this message

    • @RomaineGangaram
      @RomaineGangaram 8 месяцев назад

      Josh needs to be on brilliant,can you guys help

  • @TuanLe-oc9te
    @TuanLe-oc9te 3 года назад +62

    You saved my life. The best CNN explanation I've ever seen

    • @statquest
      @statquest  3 года назад +6

      Hooray!

    • @danielniels22
      @danielniels22 3 года назад

      why did it save you? do u have Neural Network homework or what?

  • @wa1601
    @wa1601 Месяц назад +6

    As a phd student in business analytics in germany i want to thank you from all of my heart, you played a huge role in why im a phd student right now, thank you so much!!!!

    • @statquest
      @statquest  Месяц назад +1

      BAM! Good luck with the PhD!

  • @KoffeeKlaire
    @KoffeeKlaire 4 месяца назад +11

    As a PhD candidate in machine learning at Harvard, I cannot stress how simple and beautiful your videos make complex concepts. Well done

  • @sagnikbhattacharya7597
    @sagnikbhattacharya7597 10 месяцев назад +10

    Just wanted you to know that I'm earning my bread and butter just because of you. Thank you teacher!

    • @statquest
      @statquest  10 месяцев назад +4

      Congratulations! BAM! :)

  • @shubhamraj6170
    @shubhamraj6170 4 месяца назад +8

    Why am I not able to stop the playlist? Why am I still not bored!?
    Great effort in making things simple and fun. BAM!

  • @dmitriykalashnikov85
    @dmitriykalashnikov85 3 года назад +33

    Amazing work. I'm started learning DS and I can't imagine how I can handle all of this information without your videos. Big thanks for everything you've done, do and will do

    • @statquest
      @statquest  3 года назад +3

      Thank you very much! :)

  • @surajsamantha
    @surajsamantha 4 месяца назад +3

    The Moment I Saw The first video of this series , I immediately placed an order of the book ! Cant Describe How well u are explaining. Blessed To Have RUclipsrs Like You .

  • @grantkim9124
    @grantkim9124 2 года назад +8

    I was a undergrad poli-sci data analytics student three years ago. I couldn't imagine myself going into data science because I know I am not a STEM student nor do I have a great working brain for math. But when I watched your videos back then, I was able to get confidence that I can give myself a chance to study DS which I love. Here, three years later, I am in the MSDS program at Columbia University studying data science. This was only possible because of your ml/stats videos. I still find myself studying your videos to understand concepts, which allows me to read the text without spending countless days stuck. I sincerely thank you very much for giving me a chance to actually dive on such a complex but cool subject.

    • @statquest
      @statquest  2 года назад +2

      WOW!!! Congratulations!!! That is awesome. It's an honor to be a small part of your success and it motivates me to do more. Thank you!

  • @davidpinedo4008
    @davidpinedo4008 Год назад +8

    I can't stop thanking you for your content! I am a master in data science student and usually before engaging with the commonly unfathomable statistical learning books I come to your channel to grasp the topics.

  • @nikhilb5800
    @nikhilb5800 Год назад +2

    "I don't know how much time does Artificial Neural Networks take to train, learn the input data, But you are putting more efforts and it taking much time in your training time".Thanks to your efforts sir.
    , your videos really explains very well and it helps us in visualizing easily.

  • @alexg7082
    @alexg7082 3 месяца назад +3

    Another simple and human-readable explanation of the rather complex concepts of Neural Networks.

  • @piasophie8345
    @piasophie8345 2 года назад +2

    I've never seen such a simple yet very good explanation of a CNN. Thanks a lot! As a non-native english speaker I really love the simplicity and the written texts in your videos.

  • @rituchaudhary1832
    @rituchaudhary1832 10 месяцев назад +6

    You are a gift of God to the education of Machine Learning!!!! Thank you so much!!

  • @social.2184
    @social.2184 10 месяцев назад +1

    I don't know how can I thank you for pouring such knowledge to us for free.

  • @KhoaLe-oc6xl
    @KhoaLe-oc6xl 3 года назад +6

    Just to let you know how much I appreciate your quests! Absolutely simple but not missing any concept

  • @Techdadphd
    @Techdadphd 2 года назад +3

    Oh my goodness! this is the simplest way CNN has ever been explained while still keeping true to the Maths. Thanks so much, Josh!

    • @statquest
      @statquest  2 года назад

      Glad you liked it! :)

  • @AMVSAGOs
    @AMVSAGOs 3 года назад +49

    Eagerly waiting for LSTM and it's varients. awesom explaination ...

  • @shamshersingh9680
    @shamshersingh9680 8 месяцев назад +4

    This is the best explanation for CNN I have ever come across. I am very sure this is best I will ever see. I cannot you thank you enough. I have had explanations from my instructors who are PhD, MTechs and what not!! even they could not explain why filters are able to extract features and why we use global pooling. The answer I got was to reduce the number of inputs nodes to NN (which is partly true also) but the way you have explained the importance of pooling, I was amazed and equally happy to see. Thank You Josh Sir. I think you should be knighted for your efforts 😃👏🏻👏🏻👏🏻👏🏻👏🏻

    • @statquest
      @statquest  8 месяцев назад +2

      Thank you! :)

    • @liviumircea6905
      @liviumircea6905 8 месяцев назад +2

      Sir Josh Starmer first of his name ,the ruler of StatQuest realm 🙏

    • @statquest
      @statquest  8 месяцев назад +2

      @@liviumircea6905 Ha! you made me laugh. :)

    • @flyawayhome3
      @flyawayhome3 Месяц назад

      Hello! Please can you tell me how the filters themselves are decided? Is the filter structure another element that is estimated by the model? Or pre-determined?

  • @wolfywolfgang2498
    @wolfywolfgang2498 3 года назад +2

    After watching the unskippable lectures of my professor for hours and understanding nothing, this 15 min viode did wonders, thanks Josh! 😌

  • @pushkarkhedkar9243
    @pushkarkhedkar9243 3 года назад +1

    Never seen easier version of neural network! whole neural network series is a blessing. A BIG THANKS!

    • @statquest
      @statquest  3 года назад

      Thank you very much! :)

  • @111dogger
    @111dogger 3 года назад +4

    What a simple way to explain such a complex topic. Perfect explanation Josh.

    • @statquest
      @statquest  3 года назад

      Thank you very much! :)

    • @111dogger
      @111dogger 3 года назад +2

      @@statquest Thank you so much for making these amazing videos for us Josh. I hope in the near future we get to see how RNN's with LSTM work :)))

  • @Unbiased27
    @Unbiased27 2 года назад +1

    This is one of the best explanations of CNNs I have ever seen!! You’re a gifted teacher man! Thanks for the refresher!

  • @kimjong-un4521
    @kimjong-un4521 3 месяца назад +1

    This is the best explanation for CNN you could ever find. wow just wow

    • @statquest
      @statquest  3 месяца назад

      Thank you very much! :)

  • @TheElementFive
    @TheElementFive 3 года назад +7

    My favorite channel to do with all things data has finally done a video on my favorite data science topic... TRIPLE BAM!

  • @karthigasankarananth5520
    @karthigasankarananth5520 Год назад +2

    I can't imagine how you can explain so simply ...hats off to your work ..great and superb explanation...need lot of statistical videos like this

  • @majd2882
    @majd2882 Год назад +1

    What professors try to explain for weeks can u explain in 15 minutes.
    Thank you man

  • @keremkezer6826
    @keremkezer6826 11 месяцев назад +2

    This is the best explanation i've checked to many resources but no one simplified that much!

  • @adrianwleung
    @adrianwleung 3 года назад +2

    I literally binged your neural network videos in a day like a Netflix show and now realized that I am at the end of the series to date and I need to wait for a new episode!

  • @schaefferyang4073
    @schaefferyang4073 3 года назад +1

    Your videos have inspired me a lot when I was a master student in data science. And now when I go further as a PhD student, your video is still inspiring me!!!! Many thanks for your videos!!! pls go further and further

    • @statquest
      @statquest  3 года назад

      Thanks and will do! :)

  • @julianberger234
    @julianberger234 3 года назад +1

    Came for this video, ended watching half of the series. Just learned this last week in deep learning and wanted to repeat everything neat and nice, thank you very much!

    • @statquest
      @statquest  3 года назад +1

      Glad it was helpful!

  • @srmsagargupta
    @srmsagargupta 3 года назад +7

    Everytime I see a new video from you I feel like I got a GOLD coin for free. Thank you Sir!

  • @EyMaxl
    @EyMaxl 6 месяцев назад +2

    Ive seen multiple videos on CNN and nothing made me understand convolution, but this! Thanks statquest

  • @NicholasHeeralal
    @NicholasHeeralal 2 года назад +4

    I was really confused on this concept before I came across this video, now I feel I understand it way better. You really helped a lot! Thank you so much!

  • @abhiraj533
    @abhiraj533 3 года назад +1

    Seriously!!!!! you are the best among Tech content creators .... always love the way you explain things ....

  • @cheerioz83
    @cheerioz83 2 года назад +5

    You are an amazing teacher -- we're lucky to have you.

  • @twandekorte2077
    @twandekorte2077 3 года назад +8

    Thanks for the video! I watched (and took notes) of the whole Neural Network series :) Like others have said: you explain difficult concepts in such an elegant simple way, while staying true to the basic mechanisms of the concept.

    • @statquest
      @statquest  3 года назад +1

      Awesome, thank you! And thanks for your support!

  • @chrisvyBacon
    @chrisvyBacon 10 месяцев назад +1

    i love your little musical jingles at the beginning. thank you so much for sharing your knowledge in such a fun way. Youre the goat

    • @statquest
      @statquest  10 месяцев назад

      Thanks so much!

  • @gayathrigirishnair7405
    @gayathrigirishnair7405 3 года назад +1

    The best explanation I've ever seen of a CNN.

  • @octavia7530
    @octavia7530 2 года назад +1

    This series is the best thing that happened to me before my Deep Learning exam lol. Everything is explained in such a simple and fun matter and it made me actually enjoy learning these concepts and makes me want to learn even more about the subject.

    • @statquest
      @statquest  2 года назад +1

      Good luck on the exam! BAM! :)

    • @octavia7530
      @octavia7530 2 года назад +1

      @@statquest now that i've been blessed with the famous BAM i think i really will have good luck :D thank you!!

    • @statquest
      @statquest  2 года назад

      @@octavia7530 :)

  • @andresvodopivec5950
    @andresvodopivec5950 3 года назад +7

    Hi Josh. Thank you so much for these videos. All videos you do are fun and so easy to understand. Without any doubt when I see your explanations I can conclude that things are not difficult, they are just badly explained. Your explanations are fantastic. I decided to support you. I am sorry I cannot provide the amount you deserve for such a quality education, but I am merely a student. However, I will not forget you when my condition improves. Please do not stop helping us.

    • @statquest
      @statquest  3 года назад +1

      Wow! Thank you very much! It means a lot to me that you care enough to contribute. :)

  • @DIY_Foodie
    @DIY_Foodie Год назад +1

    This is the best neural network layers explained in the entire video community

  • @Vishal_Bania
    @Vishal_Bania 2 года назад +1

    I am watching this a day before my exam and i don't even know how to thank you ! i am a big fan of your work (double bam)!

  • @julieirwin3288
    @julieirwin3288 3 года назад +6

    Thank you Josh! It is so amazing you explained CNN clearly in just 15 minutes :)

  • @thecompanioncube4211
    @thecompanioncube4211 Месяц назад +1

    I love your content so much. It's a great stepping stone/revision material for the topics. Also the effort taken to finalize bam distribution per video is greatly appreciated

  • @reyyantemel340
    @reyyantemel340 Год назад +1

    Abicim yemin ederim hayatımda bu kadar iyi anlatım gördüğümü sanmıyorum

  • @sampathkumar3405
    @sampathkumar3405 2 года назад +1

    Hello josh, without you, my journey in data science wouldn't be this easier.

  • @sharadpkumar
    @sharadpkumar 3 месяца назад +1

    amazing.....what an effort....the great animations, dynamic graphs and equations, etc. Well done

    • @statquest
      @statquest  3 месяца назад

      Thank you so much 😀!

  • @JM-vu2tt
    @JM-vu2tt Год назад +1

    I am speechless, of your work and how you achieved your teaching intention, at least in my case, I would say that this explanation is PERFECT, I havent watched a lot of youtube friendly explaining videos on CNNs but surely this one is perfect, dont need to see another one🎉

    • @statquest
      @statquest  Год назад

      I'm glad you like this video! :)

  • @shrutik9673
    @shrutik9673 8 месяцев назад +5

    BEST CNN VIDEO IN THE INTERNET

  • @AP-ni6zh
    @AP-ni6zh 3 года назад +2

    Thankyou. Thankyou! You are amazing master of explaining things... break it down to the smallest chunk and and build it up in step by step. I very much look forward for more marching learning tutorial. There is lot of tutorial out there and I spend lot of time understanding it but nothing match the the way you explain. Thank you Tons for awesome contents you have created and the insights you provide

  • @abhishekrupakula1613
    @abhishekrupakula1613 3 года назад +2

    Hey Josh! Thank you so much for this video. This is the best CNN Explanations I have ever seen.

    • @statquest
      @statquest  3 года назад

      Thank you very much! :)

  • @205-cssaurabhmaulekhi9
    @205-cssaurabhmaulekhi9 2 года назад +1

    i'm reading François Chollet's book and whenever i need visualization or i'm stuck i come to your channel, great work.

  • @Amir_mo
    @Amir_mo Год назад +4

    Hands down to the best video and great channel. Thank you for the incredible effort and dedication!

  • @theboredtutor
    @theboredtutor 2 месяца назад +1

    Simply the best explanation of CNN out there!

  • @O5MO
    @O5MO 3 года назад +1

    Best playlist on neural networks i've ever seen. Thanks for this effort.

  • @SaltyYagi
    @SaltyYagi 11 месяцев назад +1

    GOAT video. What a great simple, relaxed way of explaining things. Cheers!

  • @AnimApiano
    @AnimApiano 3 года назад +1

    Thank you very much for the explanation! The best explanation i've ever had on CNN, made me wanted to watch the entire neural network series

    • @statquest
      @statquest  3 года назад +1

      Glad you enjoyed it!

  • @MononeRocks
    @MononeRocks 8 месяцев назад +2

    Thank you Josh!!! You truly are the best at explaining these concepts. I would love to see future videos on how to train the kernels, and more on image recognition/computer vision (clearly explained of course). I also got your book and it's really nice, maybe there can be a part 2 in the future 👀

    • @statquest
      @statquest  8 месяцев назад

      I'm writing a new book all about neural networks right now and hope to have it done soon.

  • @noahrubin375
    @noahrubin375 3 года назад +1

    I've watched a lot of statquest videos but I think this is my new favourite!

  • @vasanthc55
    @vasanthc55 Год назад +1

    wow, you're the best at explaining's things for easy understanding. "Simply Great"

  • @GriselElianaQuispeAramayo
    @GriselElianaQuispeAramayo 9 месяцев назад +2

    Thank you very much you are saving my master's degree

  • @LanguageHelp
    @LanguageHelp 3 года назад +25

    Please consider doing an NLP series from regression to bert

    • @statquest
      @statquest  3 года назад +7

      I'll keep that in mind.

    • @buithanhlam3726
      @buithanhlam3726 3 года назад +1

      I think transformer and self-attention is a very hard topic for explain, so let him some time.

    • @LanguageHelp
      @LanguageHelp 3 года назад +1

      @@buithanhlam3726 True. It was just a suggestion for the future.

    • @statquest
      @statquest  3 года назад +1

      @@xiaoqingwan1912 noted! :)

    • @최은빈-z1q
      @최은빈-z1q 3 года назад

      I third that! Would love to see an NLP series.

  • @abdulwajedfarhat969
    @abdulwajedfarhat969 Год назад +1

    Me searching any ML topic on youtube; Convolutional neural network Statquest. Thanks for the great explanation.

  • @madhujegishetti4102
    @madhujegishetti4102 Год назад +1

    BAM!!... You explained very easily and clearly.... Bam!!!

    • @statquest
      @statquest  Год назад +1

      Glad it was helpful!

    • @madhujegishetti4102
      @madhujegishetti4102 Год назад +1

      @@statquest BAM!! It's first time a big you tuber replied to comment... 🙏🙏

    • @statquest
      @statquest  Год назад

      @@madhujegishetti4102 BAM! :)

  • @kamranabbas2438
    @kamranabbas2438 3 года назад +1

    Wow, that was the best explanation of CNN. Looking forward to RNN explanation made so succinct and to the point. Thanks a lot!

    • @statquest
      @statquest  3 года назад +1

      Thank you very much! :)

  • @rominasepulveda570
    @rominasepulveda570 2 года назад +1

    This is my favorite video so far! I not so familiar with math but I want to learn all this stuff because I love science and I need this background and your videos have made my journey not just easier but possible! Thank you so much!

    • @statquest
      @statquest  2 года назад

      Glad it was helpful!

  • @konstantynstats7833
    @konstantynstats7833 2 года назад +2

    simply an amazing explanation.

  • @franssjostrom719
    @franssjostrom719 2 года назад +1

    These lessons must be really cool to do. Just seeing the model work for a shifter X must be an fantastic experience

  • @envynoir
    @envynoir Год назад +1

    You're literally a saint for this content! Thank you very much.

  • @saima6759
    @saima6759 Год назад +4

    Somehow "bam~" has become my own pet phrase in real life.

  • @grostire
    @grostire 2 года назад +1

    You explain things VERY well. Thank you very much ! The colors really help.

  • @jennifertang33
    @jennifertang33 3 года назад +1

    omg you guys are so amazing, really explain very complicated concepts in a clear, quick as well as funny way!!!

  • @caokang4957
    @caokang4957 Год назад +1

    Thanks!

    • @statquest
      @statquest  Год назад +1

      Thank you so much for supporting StatQuest!!! TRIPLE BAM!!! :)

  • @fabio336ful
    @fabio336ful 2 года назад +1

    I saw the entire NN series. Thank yoy Josh! BIG BAM!!

  • @davidsalazar154
    @davidsalazar154 6 месяцев назад +1

    This is the most beautiful video on all youtube ! Awesome !

  • @dengzhonghan5125
    @dengzhonghan5125 3 года назад +2

    The best tutorial as always. Thanks. Looking forward to seeing RNN.

    • @statquest
      @statquest  3 года назад

      Glad you enjoyed it!

  • @ceylanyozgatligil9722
    @ceylanyozgatligil9722 2 года назад +2

    All of your videos are amazing. You are very talented to explain complicated things in a simple way. I am looking forward to seeing embedding, attention and transformers videos from your point of view.

    • @statquest
      @statquest  2 года назад +1

      Awesome, thank you!

  • @mohamadmohtasham372
    @mohamadmohtasham372 2 года назад +1

    I passed my exam by your video )))) thank you ))

    • @statquest
      @statquest  2 года назад +1

      Congratulations!!! TRIPLE BAM! :)

  • @hcv_study
    @hcv_study 2 года назад +1

    Huge thanks to you! Can't believe I can learn this for free.

  • @DurgaBurande
    @DurgaBurande Год назад +1

    Hey, Thanks for the simplified explanation.... it's too good!!

  • @siddharthmodi2740
    @siddharthmodi2740 3 года назад +1

    You are unbelievable man , lots of respect for you to putting a lot of efforts to make things easy for us. Thank you so much ❤

  • @Shaywu827
    @Shaywu827 3 года назад +1

    more video for NLP please!!! cannot say how much I enjoy watching your videos!

  • @shrutik9673
    @shrutik9673 8 месяцев назад +1

    STOP LOOKING FOR 1HR LONG VIDS THIS GUY HAS DONE THE IMPOSSIBLE

  • @pfever
    @pfever 2 года назад +1

    Give this man a Nobel prize 🏆

  • @gopalsharma2053
    @gopalsharma2053 11 месяцев назад +1

    Hi Josh, you know you are awesome, you know you and I both are in this domain and I have also started learning to play guitar. I hope this channel will help me in my journey.

  • @patrikszepesi2903
    @patrikszepesi2903 2 года назад +1

    I love your videos, I have binge watched your entire machine learning series. One suggestion I might add is the following: It can be confusing to use 1s to represent black pixels and 0s to represent white pixels, because in Computer Vision a black pixel has a value of 0 and a white pixel has a value of 255. So when normalized Black = 0 and White= 1. Thank you so much for these videos btw I love them.

  • @elielberra2867
    @elielberra2867 2 года назад +2

    Amazing video! As always, I am incredibly thankful for all the time an effort you put on to these lessons :)

  • @RKN998
    @RKN998 3 года назад +1

    you are a LEGEND, this saved me probably days. thank you so much

  • @joao.vitor.franco
    @joao.vitor.franco 3 года назад +3

    Thanks for the awesome content, just loved this neural networks series!

    • @statquest
      @statquest  3 года назад

      Thank you very much! And thank you for your support!!! :)

  • @Vinicius-nd8nz
    @Vinicius-nd8nz 2 года назад +1

    BAM best explanation with amazing well thought example to follow, I can't thank you enough!

    • @statquest
      @statquest  2 года назад

      Glad it was helpful!

  • @ia0722
    @ia0722 2 года назад +1

    I love you so much, I can actually process the information while breathing normally and without crying (it’s been a tough year😅)
    I’m just gonna go ahead and put in a tiny wish for videos about graph NN and fully convolutional NN

    • @statquest
      @statquest  2 года назад

      Thanks! I'll keep those topics in mind.

  • @joyjitchoudhury
    @joyjitchoudhury 9 месяцев назад +1

    This is unbelievably good!

  • @shirinbas
    @shirinbas Год назад +1

    Excellent video, To the point and complete description. I was actually looking for a good course on NN. I will use the playlist provided in the channel

  • @ashleypeterson1105
    @ashleypeterson1105 Год назад +1

    Wow!
    What an explanation!
    Thank you so much for all these wonderful contents.
    Keep it up! 🎉

  • @nonalcoho
    @nonalcoho 3 года назад +3

    Omg!!
    Thank you so much for making CNN video. Bammmmm!

  • @Eman_sq
    @Eman_sq 2 года назад +1

    Thank you for helping me and my friends in our journey

  • @Fibo1321
    @Fibo1321 Год назад +1

    THIS IS AMAZING, Josh. Thank you so much. now I understand the concept.