Frustrated with rote definitions? I was too -- until I started looking for the intuition behind each concept.
Math, BetterExplained condenses a dozen math concepts into clear, easy-to-read tutorials. I want you to enjoy those aha! moments when you aren't going through the motions and an idea really clicks. Check out the samples below and see how thousands have been able to learn the math concepts that have frustrated them.
Every topic can be intuitive if explained correctly: grow your math understanding!
What's inside?
- 12 chapters drawn from the blog articles, essential to any student:
- Contents
- Introduction
- 1. Developing Math Intuition (Chapter preview)
- 2. The Pythagorean Theorem
- 3. Pythagorean Distance
- 4. Radians and Degrees
- 5. Imaginary Numbers
- 6. Complex Arithmetic
- 7. Exponential Functions & e
- 8. The Natural Logarithm (ln)
- 9. Interest Rates
- 10. Understanding Exponents
- 11. Euler’s Formula
- 12. Introduction To Calculus
- Afterword
- Print-quality design made for easy onscreen reading & printing
- PowerPoint slides and images for all diagrams used in the chapters (deluxe version)
- 12 video guides that explain the analogies in the chapters - each averages a succinct 10 minutes.
- Sample Video (Chapter 1: Developing Your Intuition)
- 3-part (1-hour total) interview with Scott H Young. Scott has written extensively about study techniques and we discuss ways to generalize the approach of building your intuition.
- Concept worksheet discussing common pitfalls/additional ideas, drawn from the most common questions asked about each topic
Who's it for?
- Students: Save hours of frustration -- get things as I wish they were explained to me.
- Teachers/Home Schoolers: Get high-quality educational materials & ideas for your lesson plans.
- Self-learners: Go back and understand concepts at a level rarely discussed in textbooks or classrooms.
Testimonials
Doug Hogg, Former PrincipalI have several books on calculus (Calculus for Dummys, Math for the Millions, etc. etc. - never was able to read them) but your explanation is what I have needed all these years.
Ferose Khan JI am an engineer by profession... but never had that a-ha moment. I had it when I read your article. I don't have words to explain my happiness, you have unwound a knot that was tied 12 years before...
C. DhavejiThis is a great explanation! I am 49 years old and have never known what e is all about. It is thanks to your article that I get it and now can explain it to my son who is 13 years old...
Daniel EzellI've been following you for nearly two years...I find the intuitive approach to the subject and lucid writing unparalleled.
@mugunthkumarOne of the best e-books I've ever read in my life...
@anoemiJust started reading the ebook Math Better Explained by Kalid Azad, loving it so far
Explanations That Work
The explanations inside have been developed after years of study, and are consistently among the most popular in their field. I use analogies and diagrams to make concepts click: we're visual creatures, let's use that part of our brain!
For example: Concepts like imaginary numbers just pop into place when you see them as numbers in two dimensions:
We're used to the number line stretching left-to-right; when it spreads vertically, new numbers emerge. This simple insight saves hours of frustration when visualizing imaginary numbers.
Written For Human Beings
Rigid, stiff writing is uncomfortable for us both. I explain the way I wish I was taught: clearly, concisely, and with a sense of humor. Let's learn with conversations, not lectures.
For Your Computer, Phone or eReader
I don't like DRM restrictions any more than you. The ebook is an unrestricted PDF and can be enjoyed anywhere.
Quality Design
Some ebooks look like they were copied out of Word -- not this. Book-quality layout and typography mean it's a joy to print and read at your leisure. Readability is king: wide margins give plenty of room to write notes, and whitespace helps the words flow.
About Kalid Azad
I became fascinated with intuitive learning after a tough math class at Princeton University. After cramming for days, the formulas finally clicked -- "Argh! Why couldn't they explain it like that in the first place?" I had to share my insights to save other people my frustration.
That "aha!" moment spawned BetterExplained.com, whose articles have been read by millions worldwide. It's regarded as one of the best places to really understand math. Memorization is fragile, but insights stick.
I've been writing technically for a decade, from chapters in programming textbooks to whitepapers for Microsoft.
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The order goes through Paypal, but an account isn't required.
Satisfaction Guaranteed
I deeply believe Math, BetterExplained will dramatically improve your understanding of math. But if you aren't satisfied, for any reason, just contact me for a refund. I'm happy to say over 99% of customers are happy with their purchase!
97 thoughts on “Learn To Unlock Your Math Intuition”
I’ve been following your website for months now; this is excellent news! And since this is the first time I’m posting I just wanted to say thank you Kalid, this site has gotten me excited about math again.
I’ve been following you for nearly two years. I have several sources for my math explorations (I put them all on the collaborative blog). Still, I find the intuitive approach to the subject and lucid writing unparalleled. What really sets you apart is your excitement about discovery.
I work through your articles with my tutoring students who want to work outside their math curriculum. I eagerly open any new Better Explained article that hits my newsreader. Thank you, Kalid.
Awesome news!! Love your stuff. By chance, will there be a hard copy of the book too?
@Sebastian: Thank you for the kind words! You are more than welcome, and I’m really happy you were able to rekindle your interest
.
@Daniel: Thanks so much, it really means a lot to me when teachers are able to help other students learn. I get really excited when I finally get those breakthroughs and am thrilled they’re coming in useful — thanks again!
@Karmen: Thanks! I’d love to produce a hard copy of the ebook, I’m taking this as the first step to get experience with publishing
. If and when the hard copy comes out, I’ll do my best to make sure everyone who got the electronic copy gets a discount on the hardcopy as well.
I agree with your comments. I think this is a great idea and would like to see a hard copy one day
I myself am a victim of being a critic to the theories behind math, but this step up approach to understanding it from “outside the box” of number crunching is essential. Thanks Kalid!
I’ve been following your website for months.I think it’s great!
i just bought your ebook and as i was paying i got the idea that this could be a great concept for an iphone app or interactive ebook. i think its something you might want to consider depending on the book’s success.
Yes, this is great news. I’ve followed you for quite a while now and was recently thinking to myself that I wish you had a book that would incorporate website topics as well as others! Bought my copy today
Thanks for all your hard work.
@Valerie: You’re more than welcome! I’d love to see a hard copy too
.
@Meng: Thanks for the support!
@Hamad: That’s a great idea! I had thought about just reading the PDF, but there are so many more things you can do. Now you’ve got me thinking…
@ktr: You’re more than welcome, thanks for the kind words and support!
That’s great news Kalid!! I really like your blog and your approach towards life & maths
Your ebook will be a cherished collection in my library
Keep up the good work! All the best!
Regards,
Prateek.
@Prateek: Thanks for the note — I really appreciate your encouragement!
What do you use for typesetting ? (Just curious).
@uwe: The ebook was created using Adobe InDesign. I hired a designer, James Wondrack, to make the template.
this will be money well spent.
Hi,
You seem to link to the articles on the website from the book. It would be a good idea to to have those articles included in the book because its very annoying to click those links.
I’ve been reading this site for some time now. Keep up the good work!
@Maulin: Thanks for the feedback! Yes, I’ll be changing the links to be to the other chapters — I appreciate the note!
when can we expect a full version to be released?
@John: I’ve gotten some great feedback and am going to be doing some work to make the ebook very printer-friendly. I think there’ll be another beta coming out in early October, and assuming all goes well, the final version should be released near the end of that month or in early November.
Hi Khalid,
I’ve just come across your site by recommendation through reddit. I really appreciate the work you’re putting in. I know my maths as much as I need to, but I’ve just bought your ebook because you explain concepts better than I do to my students.
I hope to send you tips and feedback for the final version!
Thanks,
Afsar
Khalid,
thank you for releasing this – i have found this site endlessly useful. You put things in a way that just fits!
Its good to buy an ebook to take away with me and have things put into a more structured and familiar form (take as a compliment). Worth ever £ and i will be recommending to many!!
Kind Regards,
Doug
@Afsar: Thanks for the comment! I’m happy if anything can be used to help your students. Feedback is definitely welcome!
@Doug: You’re welcome, thanks for the comment and support! Yes, I want the ebook to be like a portable version of the site, I have some changes to the beta coming up that will help with just that
.
I have some decent intuition (beyond the proofs
) of the Chain and Power rules for derivatives. Just thought I’d share, whether or not you already know.
Chain: This “rule” is really the simple concept of “correcting” for different variables when you’re looking at relationships. It says that d/dx f(g(x)) = f’(g(x)) times g’(x)
In other words, figure out the direct relationship between f and g, not thinking about x just yet, then multiply by the relationship between g and x.
Example: Let’s say Al takes 3 steps whenever Bill takes 1 step. That would mean the ratio of Al to Bill is 3:1, and if Al were the y-axis and Bill were x, then there would be a line of slope 3/1. Now let’s say that _Bill_ takes 2 steps whenever Carrie takes 1 step.
The question is, how many steps does _Al_ take when Carrie takes 1 step?
Not hard. When Carrie takes 1 step, Bill takes 2 steps. And when Bill takes 2 steps, Al will have to take 6 steps!
Rate of A wrt B = 3
Rate of B wrt C = 2
Rate of A wrt C = (3)(2) = 6
Maybe it’s already obvious to you and everyone else, but judging by the misleading use of the word “rule” to describe this simple concept, I can’t be too sure. Tell me if I haven’t been clear and detailed enough.
Power: Geez, I wish I could draw a picture. This is a new insight, and I have only a visual, rather than arithmetical, intuition. It says that d/dx x^n = nx^(n-1)
Looks really weird, and the best proof I’ve seen uses the binomial theorem, but hopefully it’ll make more sense now, if it doesn’t already. (You know so much!)
Imagine a square, x^2, to start. Imagine the side of the square grow a tiny bit, which will be “dx”. How much does the area of the square increase? Well, there are two parts to this; two rectangles are formed, each with a long side of x (the side of the square) and a short side of dx, along with a super small square of side dx. Think (a+b)^2=a^2+2ab+b^2. The key point here is that dx^2 so small, you don’t even have to think about it. So the square grows by two thin rectangles with total area 2xdx. But the derivative is a ratio, which means when you compare the amount the area grows, 2xdx, to the amount the side grows, dx, you get 2x. Imagine two _lines_ bulging out of the sides of the square! 2x!
Look at a cube, x^3. Using the same logic, imagine three _squares_ bulging out of the faces of the cube! 3x^2!
This is the pattern: If x^n is your n-dimensional object, and x is the length of a side, then the area/volume of each bulging cross section is x^(n-1)dx. Actually, this has the same number of dimensions, except one of the sides is now the growth in x. How many bulges will we need? One for each dimension, n (visualize the cube).
You might have figured this out yourself by now, if this blog is any indication. Just wanted to share anyway. I’m a 16 yr old homeschooler, and after I first met you via your post on e a while back, I really got inspired to truly become comfortable with math, instead of just preparing for the ap tests! I was sort of already trying to understand, but you boosted and energized my efforts. Now I’m past the stage where you think math is a mess of logic, formulas, and conventions, and I still have my whole life ahead of me! Thanks.
Also, a typo I caught in your ebook:
Pg. 1: “Afterward” might be “afterwOrd”
@Kent: Thanks for the note! That’s really, really awesome that you’re starting to see math in this deeper way, I wish I had realized the importance when I was your age. I think you’ll find it helps you in a ton of subjects, not just math — looking for the deeper connections just helps everything click for me.
Chain rule: Yep, that’s a great way to look at it. Another way I’ve found is to think of it as a series of “wiggles”. That is, A moves 3x whenever you “wiggle” B, and B moves 2x whenever you wiggle C. So when you wiggle C just a bit, you get the multiplied effect you mention.
Power rule: That’s exactly how I think about it! For each dimension you have, you gain a “slab” on each side. So for x = 2 (square), you indeed gain 2 slabs, and so on.
It’s really great you’re finding these insights so early
[Thanks for the typo, it's fixed now].
great ebook, i have purchase it hope to see more good sharing in the near future….keep it up!
@See Zeng Hoe: Thanks for the support!
when the final version will up?
@See Zeng Hoe: I’m pulling in the final round of feedback now, and aiming for late October for the final version.
UPDATE: Hi everyone, the final version is now available! If you have purchased a beta copy, you should receive the final version in your email. Please contact me if you didn’t receive it or are having trouble.
hi khalid,
Do you plan to publish a new ebook related to math?
@See Zeng Hoe: Yes, I’d love to publish another one — I’m thinking about what to include, and writing a few more posts
.
Hi Khalid, I love your website and wanted to buy the Beta version but due to some problems (i.e. my tendency to procrastinate and my absent-mindedness), I didn’t. And now it’s gone up to a price I don’t have a budget for at the moment.. yikes! I’ll definitely buy it coz yr explanations on math concepts simply rock! *is still saving money*
I like a hard copy better tho, so if it’s out, pls announce it here, thanks!
@Ami: Hi, thanks for the comment & feedback! I’m thinking about lowering the price which should hopefully be in the range of more people
. And yep, I’ll definitely be announcing the hard copy as soon as it’s ready.
How much of this is articles from the website, other than the extra chapter?
@Andy: The extra chapter, foreward, and afterward are unique to the book (the premium version also includes the original powerpoint slides used to make the diagrams) . The other chapters are based on the articles, with changes/updates to make them flow nicely in a book format. The primary benefit of the book is having the content in a portable, DRM-free format for comfortable printing and reading. Hope this helps!
Cool, I’ll probably pick it up, I’d like to support the site anyway. By the way, you should make a Facebook fan page, I’m sure your readers would be glad to give you a little extra publicity.
@Andy: Thanks for the support, and great idea on the facebook fan page. I’m traveling now but will probably set one up when I return
.
I’m sad because I don’t have a credit card just cash. I think that if I had an organized ebook ot would be easier to understand because I would read in the right order. I can afford the ebook but don’t have a credit card
@Seamus: I’ll see what I can do
.
@Kent hey man that’s amazing that you’re getting such “deep” insights already. Man seriously, this stuff is what should be taught in schools. These insights are the real deal of what math’s about.
I never got till today any intuitive understanding of the chain or power rule. I only knew the algebraic proofs… I didn’t even think an intuitive understanding was possible.
put a link to this page on your main navigation, maybe you will get more sales
@Anon: Good idea, just added!
Thank you Kalid, very insightful information.
@Thaer: You’re more than welcome!
I recently was searching something about calculus and when i see this article of introduction to calculus. Its just amazing. I knew that all this Math had some thing basic like this which we were not told in our classes. But ur description is just awesome.Keep up the good work.
Actually i was searching for you and found u ……….so lucky
@Anirudh: Thanks for the encouragement! I’m happy you found the article helpful
.
can u make it a bit cheaper for asian friends
It’d be nice if this were a bit cheaper. A price like $9 (or $9.99) would attract more buyers. And of course, if it were $4.99, then it’d make it affordable to many more people (not necessarily increase the total volume of sales). There ought to be a betterexplained article on the psychology of pricing and the ways to make people buy instead of pirating.
@Cheapo: Thanks for the feedback — I may consider a coupon or discount for educational users. The psychology of pricing is very interesting indeed
. I wanted a price which was comparable to a real book, and much, much cheaper than even an hour of tutoring.
Greetings from Malaysia!
I just bought your ebook after browsing your explanation on “e” – which i simply loved….
thanx so much for your efforts and look forward to more postings…I’d love to see you take on and simplify various concepts in statistics as well….
@randhill: Thanks for the support, glad you are enjoying it! Yes, I’d love to write more on stats, thanks for the suggestion.
Yes. I Found it. Great Book…
Dear sir,
I read an article on this ebook , can i get hard copy on this book
Dear sir,
Can we purchase this book through credit card , how we will get the e book and how will i be able to read this on my PC. what is procedure to laod it on my PC , and what happens if i had to format my PC.
@Ashok: Hi, the book is offered as a PDF, which you should be able to read on any PC with Adobe Acrobat Reader. There isn’t a hardcopy version yet, but you can print out the PDF if you like (see the sample chapter for an example). If you have to format your PC, you should take a backup of the file (and all your other data!).
You can purchase through credit card, Paypal, or Google checkout. Hope this helps!
Dear Kalid,
Outstanding, Amazing, Defying the Fear of Maths….Invaluable
I love your book and your site (I’ll ensure my daughter reads through the book and regularly visits the site when she grows up)I’ll continue keep doing so:)
Just a point you may like to rectify http://betterexplained.com/articles/math-betterexplained-ebook-available/ the Book is priced at $19 ($24 for Slides too) while on http://instacalc.com/ (the bottom part of page it’s advertised at $14 (although it takes to the previous page)
Thanks & Regards,
Jai
@Jai: Thanks for the comment and support! I’ve fixed the outdated ad, thanks.
how can i have the ebook by using account in BDO(philippines)
@Darel: Hi, you can purchase the book for any country if you have a Paypal or Google Checkout account.
Hi Kalid.
I’d like to purchase your book. But the problem is I live in Iran — the country is sanctioned and people don’t use international bank accounts or major credit cards.
Any thoughts? please help!
Regards.
excellcnt book.please include the image and expla in the topic means very quicly understand.
@Mehran: Thanks for the comment — let me email you privately to work it out.
@ruckmani: Thank you — yep, the images and slides are included in the premium version.
Good stuff, but I’m still a little hazy as to how the value of e is determined. I can readily the realtionship between c and D to reach pi, but the e value eludes me.
@Patrick: e appears as we compound 100% interest more and more — check out http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ for a table showing how we get the actual value.
Kalid,
I’ve been to your website numerous times, and was delighted to see you’ve finally created an ebook. I bought it immediately. Thank you so much for your insights; you’ve shown how beautiful math is.
@Marty: Thank you for the kind words and the support!
Want to warn people who want to buy the book for it’s “calculus” content. There is not much in it about calculus per se. There is more of a consideration that calculus is taught in a technical way rather than using intuition. I’ve bought this book for calculus only and was a bit dissapointed as I already know the other things.
Hi Olya, I’m sorry about the confusion, I should make the table of contents more clear (there is only 1 chapter on calculus in this ebook, I plan on making another dedicated to it). I’ve issued a refund, let me know if you have any issues!
i just came across you site accidently. but this a woderful site. really i enjoyed the complex no explanation. the only constraint to buy this book is the cost. but never mine i would buy it
@manim: Thank you for the note! Glad you enjoyed the complex number article, that’s one of my favorites
.
Does the e-book have more content than the blog posts on this website?
@Karina: The standard ebook has the content from the blog posts (along with a foreward/afterward), organized and interlinked for a more “book” feel (i.e., organized to reference other chapters, linked table of contents, etc). The premium version includes the original PowerPoint slides & images. Hope this helps!
Sir,
I want to purchase ebook , but there are three version , i am confused which to buy , is there hard copy too of this book Math, BetterExplained
@Ashok: Hi! I don’t have a hardcopy of the ebook yet, but I would like to in the near future. The standard ebook is good for most people; the premium is nice if you are a teacher or parent and want to make diagrams of your own, based on the ones in the book. The screencast version has video walkthroughs, and can help reinforce the concepts, especially if you prefer audio/visual in addition to reading. Hope this helps!
please provide the book free
its good but prices to high
@tanmoy, abhinav: Thanks for the feedback — there are many free articles on the website, but a *lot* of effort went into professionally producing the ebook
. I am working on a Kindle-only version to be released very soon, which will be less expensive. Everyone who buys or has bought the PDF version will be emailed a free copy of the Kindle one too.
Hi Kalid,
As a fan of your articles where I learnt simple maths, I want to buy the ebook. Just send me your actual e-mail so that I can buy it. Thanks a lot.
Hi Mario, thanks for the note — you can reach me at kalid@instacalc.com if you have any questions (note, you don’t need to email to purchase though!).
I appreciate your contribution towards making the learning of
mathematics easy. Do you have materials that is beneficial for children age-
3-12 years.
@Egbujuonuma: Thanks for the comment! I don’t have an ebook for that age range yet, but these articles might be good for 10-12 year olds:
http://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/
http://betterexplained.com/articles/surprising-patterns-in-the-square-numbers-1-4-9-16/
http://betterexplained.com/articles/how-to-analyze-data-using-the-average/
Hope that helps!
thank you kalid, you are a sample of the person who really helps students to understand mathematics. Math. is the only subject that prevents me from getting the first place in my class. it’s so because i feel extremely scared before any math quiz.Can u help me know the cause of this hyper-test anxiety?(P.s.:i used to have A’s in math 2 years ago, but after my teacher became a very old teacher, this problem affected me very much).
thank u sooo much, maria.
@maria: Thanks for the kind words. Sorry to hear about the test anxiety! My friend Scott Young (http://www.scotthyoung.com/) has a lot more on study tips that you might find useful.
I think part of getting over the anxiety may be realizing a quiz is a way to show what you know (it’s like a checkup to find problems, vs. a “punishment”). I.e., see the quiz as a chance to fill in gaps in whatever you know — if you miss some questions, it’s something you know you can fix later on. I realize this can be tough though — as you practice more math, it should get easier
.
Hi Kalid
Great website. Your approach has really helped me with intuition — I thought I already knew this math, but your insights have really made me feel I own it. Thx.
Today, I decided to buy the book etc to support your work. Are you planning on a new rev of the book in the near future?
Hi Neil, thanks for the support. I love that feeling of “owning” it, that’s exactly it — we want the ideas to be our own, not something a person told us and we dutifully remember.
I’d like to do periodic revs on the book, I don’t have anything planned immediately, but have been thinking about special events / webinars for book readers.
I am currently editing a multiple-authored technical book, and just received a chapter on an advanced statistical technique. Unfortunately, I found the chapter totally incomprehensible, and I had even hand-picked this author in the hopes that he would finally be the one to explain it clearly.
In my note back asking for a rewrite, I said, “I’ve always felt that anything worth knowing should be understandable to anyone who is scientifically literate. I always try to put myself in the mind of my typical master’s student, who has a BS in biology and solid but not spectacular GREs.”
Worried that this was maybe too radical or unrealistic of a view, I stumbled onto this site. I look forward to reading more in the days ahead.
Hi Frank, thanks for the note! I totally agree — things should be written so a reasonably curious student should be able to make sense of it. It’s possible!
. Hope you enjoy the site.
sir, iam a student of class 10 and aspiring for iitjee 2015, till today i thought that ncert is the best explainer for basics, and a self satisfying book that doesnt need any tutor, but after reading the Calculus: Building Intuition for the Derivative article from your website, i felt as my brain is being rewired and right wires going to write place and circuit is working more successfully producing more energy in a cheaper and easier manner, i in reality felt it much better than ncert +1 book. thanks for all…………
kalid azad sir, you are great than even the greatest.
and also sir a very very happy new year
Great great book. Reads perfectly on a Nook. Looking forward to volume 2 (if there is one) or just poking around on the rest of the site.
Thanks Lovedeep, I really appreciate the note! I love that feeling over rewiring our understanding, untangling what as previously confusion. Really appreciate it.
Mr. Omlette, glad it’s working! I hope to do some follow-ups on specific subjects (Calculus, Statistics, etc.) as time goes on
.
hi Kalid,
will your book work with students taking Intermediate Algebra and other remedial math courses?
Hi George, I think the high-level analogies would work for most readers. That said, the actual details would be best appreciated with an understanding of Algebra.
excellent man.what a job man.these intuitive explanations have helped me so much.
finally there is someone who understands my frustration. Today’s teachers and the education system is are/is not set up to tell you how and why maths work they simply tell you things and state things. Its absolutely horrible, they have taken out all the proof and do not teach it. Even the simplest of things seems so hard to grasp in maths. Its not that i can’t do maths well but my style of learning is different. I love and need proof and step by step clear instructions regarding the why and how of getting to the answers. Today’s school books are also atrocious in this sense.
Thanks for the comment Tarun — I agree, we have to focus on the intuition, not the rote details.