I love it when people find their Aha! moments, overcome longstanding fears about learning, and are inspired to create their own explanations. Thanks for the support!

I just wanted to give you a shout to let you know your book has been an absolute godsend for me! Im a music technologist so Im usually doing coding to create something musical or interactive like that but I've always struggled with the basic maths needed for DSP and other coding (even with an MSc in Music Tech that covered this stuff!). After just a few hours reading I feel like I've gained a hugely improved level of understanding and can finally start to have an idea of whats going on in algorithms instead of relying on copy and paste for known working methods!

Thats pretty much all, just a thanks for helping me out! Good to see people making the effort to engage students in an understanding of the subject in that way!

-Tim F.

Math, Better Explained: the title says it all

If you are like me you always had this dismal feeling about concepts like imaginary numbers, the true nature of the Euler constant, the meaning of the result of the dot product of two vectors, and the list goes on. Confronted with questions about those fundamental topics and dismay about not understanding them thoroughly my teacher would give answers like "Get over it, just learn to use it." or "Those are just concepts you use to get results."

Now the times of vagueness are over. Young Kalid Azad from Princeton University also was driven by a deeply felt curiosity about the nature of mathematical concepts, and he had the resources available and the brains to work it out. And here we go with a book about all those concepts we've learned to apply, explaining them in a new way, showing new approaches and new angles, and suddenly things become clear.

What really sets him apart is his joy in discovering, his lucid language, and his knack for explaining complex things in a clear, enjoyable and catchy way. This is the book to give to your kids, to your old math teacher back at school, or to yourself and find joy in math again.

-Amazon Review

I have been thinking about writing a textbook (and now thinking about a textbook along with a website) to explain mathematical topics in a way that is extremely intuitive (a concept you're familiar with).  I just looked at your website a couple days ago and have browsed around a little and what you've done is quite inspiring.

Your approach is QUITE different to what I have in mind; like today I looked at your "Surprising Uses of the Pythagorean Theorem" page, and although I planned on using that same similar triangles proof to explain the theorem, I'm relieved that my explanation will be differ significantly.

As of now, you'll be my "measure" for my work, i.e. when coming up with explanations for topics I'll think about your website as the level of understandability that I want to achieve in my work.

-Derek M

Hi Mr. Azad, my name is Joshua and I'm a high school student currently in AP Calculus and I just wanted to give you a HUGE thanks for helping me actually understand calculus and giving me an actual chance at appreciating it or even mastering it. I'm still learning and am still a little shaky on some concepts but I would have been far worse if I hadn't taken your free course on the internet. Once again, I just wanted to say thank you for all the knowledge and insight your course has provided me with.

  • Joshua H

Just wanted to say that I really appreciate all that you're doing. I've never learned calculus in the intuitive way that you're teaching it before and I feel like I never completely understand the basics until now. I'm a little embarrassed because I actually have a Master's Degree from Stanford in engineering and have been taking and using calculus all my life without intuitively understanding how I was using it.

Just wanted to thank you personally and to tell you that you're doing a great service and I hope your method is used to teach future generations of students. Never realized until I found your website how badly math education needs to be updated.

  • Curtis

You may or may not remember the email I sent a number of months ago, when I was talking about some of the more general benefits of various areas of math study. You responded and reminded me that even as adults, we seldom are motivated by knowledge of some future benefit. Instead, we are motivated by the joy and beauty in the activity itself, and then enjoy the future benefits in addition to the original benefit.

I've thought a lot about your comments since then, and I've applied them in my tutoring work. One young man in particular comes to mind. He took AP Calc this past year, and had not been doing his homework. Somehow I was able to help him to see the emotional satisfaction that he could get from throwing his heart into it. Throughout the year, he has enjoyed our sessions, especially when I could help him to find the beauty in what he was doing. While he will probably never pursue theoretical mathematics as a career, I believe that I was able to make this year less painful than it would have been.

Thank you for helping me to grow as a teacher. There have been a number of mathematical concepts that I understand better because of your explanations, but more importantly to me, I have found out a lot about my position as an educator because of your influence.

Along with people like Vi Hart, Paul Lockhart, Randy Palisoc, Jo Boaler, Arthur Benjamin and others, you have had and continue to have a significant impact on my educational philosophy.

  • Peter A

Hey Kalid, I just wanted I am absolutely so thankful for this site.    Over the course of my four years in college, the math courses have always been so mechanical. As much as I enjoyed the topics and insight learned, it was always the redundant 'follow these steps' and apply it on the exams.  At best, I only grasped a basic and algorithmic understanding of the math.  But reading over the various topics of this site, I gained a greater intuition and strong understanding of these topics and the underlying intuition behind them.  I'm slowly making much more sense of the math that learned all those years ago.


I have been reading your site for the past days as a preparation for my upcoming university session in computer engineering in 4 weeks and I have to say that more 'aaaahhhhh!' moments have emerged using your site than when using most online math resources in the past months (my second favorite being Khan Academy for Ipad). Truly a beautiful feeling, thanks for having the initiative of putting this site together.

At age 24, it was difficult leaving my medical residency (facing the chronic pain of unsatisfactory life choices to fit other's ideals of a career) and re-establishing enough order in my life to choose a new path out of mental clarity.

Any little thing that helps me along this process truly makes me happy.

Have a good day, Anthony

Thank you, thank you, thank you, Kalid!!!

I've been pursuing Bayes' theorem for 40 years but never got a truly intuitive sense of what it meant.  But today I came across your posting on Understanding Bayes Theorem With Ratios and now I get it!! What I find amazing is that of all the books and papers I've read, it took your unassuming, unpretentious approach to enable me to find the way forward.

You have a really special talent for explaining obscure concepts.  Thank you for sharing it with the rest of us.

Hi Kalid,

After going through your intuitive approach to TRIGONOMETRY , I think I have become your fan! In all my 50 + years of teaching Trigonometry, I haven't found your method anywhere. It is fantastic! I have designed a lot of intuitive ways of teaching math at the school level, through hands on activities and games.

I recently offered to come out of semi-retirement and teach high school English at a failing school in the local district in Miami. Even though I have a PhD in English and have taught for over 30 years, I still had to take a test of General Knowledge, which included a Math section. It's been 40 years since I was last in a math class, and it was the only subject in which I never had success. The sight of a quadratic equation actually can make me physically uncomfortable, even today. It's the same syndrome I recognize in kids who read very poorly or ESL students who just can't seem to "get" English.

Anyway, in my preparation for the test (which I passed--hooray!), I found your website and for the first time in my 57 years, I learned to handle combinations and permutations painlessly. Your explanations are not only generous and kind, but also crystal clear. After a few hours with your examples, I really did understand the principles rather than just memorizing a formula.

I wish I could repay your kindness and patience personally. Of course, if there is anything I can ever do for you or a friend or family member about editing or explaining arcane points of grammar, please feel free to dump it in my lap.

But in case the need never arises, I still promise to "pay it forward" with every student in my classes. It may be me explaining why there are really twelve vowels in English, not 5, but it will be you behind the effort.

Thank you so much,

Greg B

Kalid, Man, I'm writing to thank you for this awesome site, for your time and efforts, for sharing your knowledge and willing to help. Absolutely new to maths, and 35, I sometimes find it difficult to understand even the simplest things. I've been gazing a little on your explanations and I must say they are knowledge at its finest.- Say, thanks, indeed. Hope you get back exponentialized what you give =D


PS. Excuse my English, it's not my native language.


Hello Kalid! I recently stumbled upon your video while trying to wrap my head around the concept of imaginary numbers. After watching just one of your videos, everything became clear which is amazing since my calculus prof a terrible teacher. I'm not too big on reading since I fall asleep 90% of the time, and was wondering why you don't make any more videos. I'm in first year engineering and your explanations are saving me from failing! Please make more videos! Thank you for everything you do! You're an amazing teacher.

Best Regards, Alan

I failed algebra one year in high school.

I thought I was absolutely the worst person in the world at math. Turns out, crappy teachers and a teaching methodology that is diametrically opposed to one's optimal learning style count for a lot in school.

However what was amazing to me was how I went from zero confidence in my math skills to actually being excited about math when I took geometry. I never studied once in that class--just absorbed and immediately internalized what the teacher said. I could look at a proof and it just sort of visually made sense to me and clicked and I could step through it because of the pattern recognition. To this date I've never experienced anything intuitive in that fundamentally primal manner.

What has been great lately is Khan Academy and a growing interest in teaching myself software development has rekindled my interest in math. I also owe a huge debt of gratitude to Kalid @ BetterExplained.com (he frequents HN) for getting me past my fear that higher-level maths were beyond my capabilities. They weren't--I just needed to find a way of applying them in an intuitive manner that I could easily internalize vs. staring at equations and their definitions.

I wouldn't be surprised if I would have ended up as an engineer if I hadn't had such a poor experience with math when I was younger. I am pretty resentful of it. Fortunately I can take steps to change that, and I am.

Hi Kalid!

I recently stumbled upon your website and I subscribed for the weekly emails out of sheer curiosity. I wasn't expecting any kind of a mind blowing article or anything of that sort. Oh well, I am thoroughly surprised.

This very first email gives me chills. Although, I have come across "intuitive thinking in math" as a concept before, never before have I encountered such a miscellany of explanations for just one math equation. This is just pure knowledge. This kind of an insight is what I missed during my school days.

Thank you so much for teaching everyone that there is not just one way of solving a math problem but many. I just purchased your book: "Math, Better Explained" and I am currently reading it.

I would like to read more on this topic. If you have any suggestions, I would be very happy to hear from you. I am yet to finish reading your book. I am so thrilled right now.

I can't stress this enough but thank you so much for your work!

Best regards,

Sowmya H.

I am certain you hear this often and I sincerely mean it.  Thank you.  Not only is your insight most welcome, but your style and delivery have allowed my children (and I) to circumvent the often  seemingly insurmountable mountain of mathematics.  I have struggled with some (Okay, most) of these ideas for years, now I see the light.


Thank you, thank you, thank you. I dropped out of maths at high school aged about 15 because I didn't "get" calculus etc - it was too abstract, I didn't understand how it related to anything "real". I enjoyed stuff like chemistry & biology, but dropped them eventually because you couldn't do them at university level without maths. As an adult I had a go at learning maths a few times - I kept an interest in science & got into computers in the 80s. I dated a mathematician for a while & got him to try teaching me. Bad idea!! Ha ha, he had no idea how little I knew. I tried self-teaching from books, but still no joy. I started to think maybe I had a kind of dyslexia but with numbers.

Five minutes on BetterExplained and I think maybe this is it. At the age of 63 I might finally learn some maths. The analogy approach is like a magic trick! You have showed me the light switch and I could not be more grateful. Brilliant.

I just finished your first book and I'm writing to thank you. Thank you for writing it and helping us understand better. I have a BSC in engineering but I struggled with math concepts throughout my whole education. If this book had existed then, I would have enjoyed my studies much better. :) I'm certainly going to reread it at some point. I'm not gonna be a scholar, but I like to understand the world a bit better. And your book does that. Can't wait to dive into your Calculus course!

I just finished "Math, Better Explained" and am actually excited to start reading your treatment of Calculus (something I never expected to be 'excited' about); so I had to say thank you. It has been 15 years since I was in school and I remember saying at the time, "I like math, but I can't stand math class because nobody ever bothers to explain the why to me". Maybe my college professors tried and failed, maybe they didn't understand it themselves, but I wanted to thank you for being the first person to manage to teach me math in a way that helped me understand the why. Your intuitive approach is just what I needed, so keep up the excellent work. I am enjoying math again for the first time in a long time; thank you.

  • Jason

Kalid, Hi! I just wanted to thank you for this wonderful site. It has the best explanations I've seen on the internet, and I feel like I'm truly gaining an intuitive grasp of what math is all about from reading through these articles. They're also fun to read! Just wanted to show my appreciation. Keep up all the good work, and serious props/thanks for what you've done so far!

  • Alexander

Hi, I don't have any feedback to provide specific to your website, I'm just contacting you to give my sincere appreciation for everything you do. A little bit of backstory about me, I'm someone who dropped out of high school at 16 years old (I'm 20 now) and someone who always struggled with math and thought of it as a useless subject. My typical routine would be to take a course, fail, take remedial summer school, and repeat until I dropped out for the last time at 16. I worked minimum wage jobs from the ages of 16 - 19 until one day I realized that I had no skills, and no prospects for anything. I decided this wouldn't be my future. I decided that I wanted to try this math thing again, but not out of any interest, simply because I had never done well in anything math related and simply was not a "math person". First thing I do is google "how to learn math" which led me to your website, and explained how neat math could be, and how it isn't just formulas and equations, but possibly our attempt to understand the world. I began working to catch up with math through khan academy, and textbooks I got from the library. I finished per-algebra, algebra 1, algebra 2 and trigonometry and am going to start learning pre-calculus. In September, I will be starting a Bachelor of Science in Physics at my local university which is something I would have laughed at if you suggested it to me merely three years ago. If you stuck through reading this until now, I just wanted to say thank you so much for inspiring me to start studying math and showing me the true joy that comes from solving problems, and overcoming your own obstacles. In turn, I hope to inspire this passion in somebody else one day. I recommend your site to everybody I know whenever the subject of math comes up. Once again, thank you very much for what you do. Sincerely, A humble reader

  • Sam

I found this today and this is EXCELLENT! I'm a 45 year old women with a degree is Accounting. I'm a CPA and I have an MBA from Washington University. I took Calc I, II and III in undergrad years ago. Calculus has always mystified and eluded me!! I'd ask someone to explain to me the concept so I could better understand it. All I could ever get from them was more odd mystifying formulas and talk about limits which helped me none. My son is a freshman at Washington University and taking calculus. He has rendered his teacher nearly useles. He goes to class, but says the professor cannot/does not teach; he just does proofs. He works in study groups and works with tutors. Our Calc tutor, in fact, is $80 an hour. I searched Google to see if I could find something that would help me, so I could maybe better help him understand. I came across your site and I find it AMAZING! So, amazing that I am really, really, really intrigued. I even thought about going back to school to study it. It has actually excited me. I shared it with my son. This is great. Really, really great. I had to let you know. Thanks,

  • Denise