Reddit reviews Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
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Secrets of Mental Math The Mathemagician s Guide to Lightning Calculation and Amazing Math Tricks
He's my major advisor, and he loves occasionally showing off (who wouldn't?). I find it very entertaining. As far as I can tell, it's just a lot of practice plus some pattern recognition. For multiplying large numbers he just uses the distributive property combined with a certain method of remembering numbers in his head he uses.
I also read his book Secrets of Mental Math back in high school. He outlines some of the techniques there although its more basic.
Stay away from Numberphile. Numberphile oversimplifies mathematical concepts to the point where they will give you misconceptions about common mathematical notions that will greatly impact your learning later on. I'm noticing this happening a lot with the "1+2+... = -1/12" video because it doesn't explain that they aren't using the standard partial sum definition of series convergence.
Not sure how "mathematical" it is, but Secrets of Mental Math is a great, useful book that will help you do really fast calculations in your head.
I'll share my reading list for the next 12 months as it's how I plan to become a better learner:
 
Learning
Improving Maths
Improving Reading Speed
Algorithmic thinking
Understanding the Mind
Productivity
It's an ambitious reading list for the next 12 months as there is some heavy reading in there but hopefully you can get one or two useful suggestions from it!
$9.36 and free shipping.
Honestly. You'll be improving yourself while being able to amaze others at your "magic".
I read a good portion of this book: http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?ie=UTF8&qid=1341547865&sr=8-1&keywords=secrets+of+mental+math
It has many tricks like that. I enjoyed it.
memory, just pick one book the basics are the same: A Sheep Falls Out of the Tree, Quantum Memory Power, not just memory techniques but with a section on Improve your intelligence
math: secrets of mental math
Among many others who can be given the title of the world's most intelligent person is Marilyn vos Savant: one of her books
Good question OP! I drafted a blog article on this topic a while back but haven't published it yet. An excerpt is below.
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With equations, I sometimes just visualize what I'd usually do on paper. For arithmetic, there are actually a lot of computational methods that are better suited to mental computation than the standard pencil-and-paper algorithms.
In fact, mathematician Arthur Benjamin has written a book about this called Secrets of Mental Math.
There are tons of different options, often for the same problem. The main thing is to understand some general principles, such as breaking a problem down into easier sub-problems, and exploiting special features of a particular problem.
Below are some basic methods to give you an idea. (These may not all be entirely different from the pencil-and-paper methods, but at the very least, the format is modified to make them easier to do mentally.)
ADDITION
(1) Separate into place values: 27+39= (20+30)+(7+9)=50+16=66
We've reduced the problem into two easier sub-problems, and combining the sub-problems in the last step is easy, because there is no need to carry as in the standard written algorithm.
(2) Exploit special features: 298+327 = 300 + 327 -2 = 625
We could have used the place value method, but since 298 is close to 300, which is easy to work with, we can take advantage of that by thinking of 298 as 300 - 2.
SUBTRACTION
(1) Number-line method: To find 71-24, you move forward 6 units on the number line to get to 30, then 41 more units to get to 71, for a total of 47 units along the number line.
(2) There are other methods, but I'll omit these, since the number-line method is a good starting point.
MULTIPLICATION
(1) Separate into place values: 18*22 = 18*(20+2)=360+36=396.
(2) Special features: 18*22=(20-2)*(20+2)=400-4=396
Here, instead of using place values, we use the feature that 18*22 can be written in the form (a-b)*(a+b) to obtain a difference of squares.
(3) Factoring method: 14*28=14*7*4=98*4=(100-2)*4=400-8=392
Here, we've turned a product of two 2-digit numbers into simpler sub-problems, each involving multiplication by a single-digit number (first we multiply by 7, then by 4).
(4) Multiplying by 11: 11*52= 572 (add the two digits of 52 to get 5+2=7, then stick 7 in between 5 and 2 to get 572).
This can be done almost instantaneously; try using the place-value method to see why this method works. Also, it can be modified slightly to work when the sum of the digits is a two digit number.
DIVISION
(1) Educated guess plus error correction: 129/7 = ? Note that 7*20=140, and we're over by 11. We need to take away two sevens to get back under, which takes us to 126, so the answer is 18 with a remainder of 3.
(2) Reduce first, using divisibility rules. Some neat rules include the rules for 3, 9, and 11.
The rules for 3 and 9 are probably more well known: a number is divisible by 3 if and only if the sum of its digits is divisible by 3 (replace 3 with 9 and the same rule holds).
For example, 5654 is not divisible by 9, since 5+6+5+4=20, which is not divisible by 9.
The rule for 11 is the same, but it's the alternating sum of the digits that we care about.
Using the same number as before, we get that 5654 is divisible by 11, since 5-6+5-4=0, and 0 is divisible by 11.
PRACTICE
I think it's kind of fun to get good at finding novel methods that are more efficient than the usual methods, and even if it's not that fun, it's at least useful to learn the basics.
If you want to practice these skills outside of the computations that you normally do, there's a nice online arithmetic game I found that's simple and flexible enough for you to practice any of the four operations above, and you can set the parameters to work on numbers of varying sizes.
Happy calculating!
Greg at Higher Math Help
Edit: formatting
First off, I'd recommend looking into a book like this.
Second, when doing something like multiplication, it always helps to break a problem down into easier steps. Typically, you want to be working with multiples of 10/100/1000s etc.
For multiplying 32 by 32, I would break it into two problems: (32 x 30) + (32 x 2). With a moderate amount of practice, you should quickly be able to see that the first term is 960, and the second is 64. Adding them together gives the answer: 1024. It can be tricky to keep all these numbers in your head at once, but that honestly just comes down to practice.
Also, that same question can be expressed as 32^2 . These types of problems have a whole bunch of neat tricks. One that I recall from the book I linked above has to do with squaring any number ending in a 5, like 15 or 145. First, the number will always end in 25. For the leading digits, take the last 5 off the number, and multiply the remaining digits by their value +1. So, for 15 we just have 1x2=2. For 145, we have 14x15=210. Finally, tack 25 on the end of that, so you have 15^2 = (1x2)25 = 225, and 145^2 = (14x15)25 = 21025. Boom! Now you can square any number ending in 5 really quick.
Edit: Wanted to add some additional comments that have helped me out through the years. First, realize that
(1) Addition is easier than subtraction,
(2) Addition and subtraction are easier than multiplication,
(3) Multiplication is easier than division.
Let's go through these one by one. For (1), try to rewrite a subtraction problem as addition. Say you're given 31 - 14; then rephrase the question as, what plus 14 equals 31? You can immediately see that the ones digit is 7, since 4+7 = 11. We have to remember that we are carrying the ten over to the next digit, and solve 1 + (1 carried over) + what = 3. Obviously the tens digit for our answer is 1, and the answer is 17. I hope I didn't explain that too poorly.
For (2), that's pretty much what I was originally explaining at the start. Try to break a multiplication problem down to a problem of simple multiplication plus addition or subtraction. One more example: 37 x 40. Here, 40 looks nice and simple to work with; 37 is also pretty close to it, so let's add 3 to it and just make sure to subtract it later. That way, you end up with 40 x 40 - (3 x 40) = 1600 - 120 = 1480.
I don't really have any hints with division, unfortunately. I don't really run into it too often, and when I do, I just resort to some mental long division.
To me its about what you can do in your head. Get a book for example, BOOK is good.
Also, subscribe to /r/math. Finally, ANYTIME you see a number do something with it. Factor it, think of a historical significance, etc.
I have a book on mental math, and this is essentially the technique that the author uses to square numbers mentally really quickly.
In other words,
x^2 = (x+k)(x-k) + k^2
where you substitute x's into the equation you gave.
This is the book.
Check this book out!
It absolutely changed my mental math ability. Arthur Benjamin also has videos all over the Internet with some quick mental math tricks.
http://www.amazon.com/gp/product/0307338401/ref=oss_product
"Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks"
I found this at my brother's house and got it for myself. It's a fun book :)
I read this book a few years ago, and it is pretty much the way I do any basic arithmetic in my head now. http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?ie=UTF8&qid=1333153637&sr=8-1
The book The Secrets of Mental Math has some great tricks in it to help you along.
If you want to learn to calculate quickly in your head, probably the most fruitful thing is to pick up a bunch of tricks for mental math. One good video course for this is Secrets of Mental Math put out by The Great Courses. The same lecturer published out a very good book on the subject as well.
Of course, if you want to go old school, then it's hard to beat the utility of memorizing logarithm tables...
I have the same problem. Its a lot about efficiency. Ive been reading secrets to mental math and that's helpful.
https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
Check out Secrets of Mental Math by Arthur Benjamin. Benjamin is amazing, I've seen him at MAA meetings. He does lightning fast calculations in his head, and his book shows you how to do it. Your students may or may not think this is cool, but I do :) And the bonus is that they will never learn this kind of thing in school at any grade, so you won't be stepping on anybody's toes by teaching it to them now.
Also, the "third grade team" sucks. Screw those guys.
I'm only a high school sophomore, so I can't really help you with most of your questions, but if you want to improve your mental math, buy "Secrets of Mental Math" by Arthur Benjamin.
It's written in a way that makes sitting in your room doing mental calculations seem fun and it is very accessible. I have only gotten through 3 chapters (the addition/subtraction/multiplication chapters) and I can confidently add and subtract 3-digit numbers in seconds. I can even mentally cube two-digit numbers in a few minutes.
[Anyway, here's a link to the book] (http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?s=books&ie=UTF8&qid=1381633585&sr=1-1&keywords=mental+math)
[If you don't want to buy it, you can use this PDF version of the book] (http://www.uowm.gr/mathslife/images/fbfiles/files/Secrets_of_Mental_Math___Michael_Shermer___Arthur_Benjamin.pdf)
[And here is the author, Arthur Benjamin, performing what he likes to call "Mathemagics"] (http://www.youtube.com/watch?v=e4PTvXtz4GM)
I hope this has been helpful and you succeed in whatever uni you go to :)
I began doing it in my head the same way. For clarity, my thought processes were based on the idea of "don't do something hard, do something easier instead and then fix it up afterwards", roughly:
(I stopped there because I just wasn't looking forward to adding 973 to 243250, but was pretty sure I could slog my way through it if I actually had to.)
But there are lots of tricks you can do to make mental math easier. I don't know them, but like the above, I know that I could* go and learn them. For example, here is a book by one of the world's best people at mental arithmetic, Arthur Benjamin; the book is filled with techniques you can use to make mental arithmetic easier. See him on TED here.
Secrets of Mental Math May be helpful for filling in some gaps. Also A Mind for Numbers gives helpful meta learning info: how to study, etc.
Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
A lot of these tricks are very easy. He explains them all in his book Secrets of Mental Math
When you say everyday calculations I'm assuming you're talking about arithmetic, and if that's the case you're probably just better off using you're phone if it's too complex to do in you're head, though you may be interested in this book by Arthur Benjamin.
I'm majoring in math and electrical engineering so the math classes I take do help with my "everyday" calculations, but have never really helped me with anything non-technical. That said, the more math you know the more you can find it just about everywhere. I mean, you don't have to work at NASA to see the technical results of math, speech recognition applications like Siri or Ok Google on you're phone are insanely complex and far from a "solved" problem.
Definitely a ton of math in the medical field. MRIs and CT scanners use a lot of physics in combination with computational algorithms to create images, both of which require some pretty high level math. There's actually an example in one of my probability books that shows how important statistics can be in testing patients. It turns out that even if a test has a really high accuracy, if the condition is extremely rare there is a very high probability that a positive result for the test is a false positive. The book states that ~80% of doctors who were presented this question answered incorrectly.
Secrets of Mental Math
The following easy to read book teaches kids (and adults) you how to do it. Its actually really easy:
Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
My favorite book that has a ton of these is this book. I remember seeing the author do all kinds of math tricks on talk shows. My favorite was determining what day of the week any date in history was (or at least, after the start of the Gregorian calendar)
This is a thing. I read a decent book with a lot of cool math tricks
https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
I'm premed, the most I know is just 2 semesters of calculus. However I am reading [Mental Math tricks] (http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401). I don't know what good that'll do me other than make look more of a nerd than I do now. I am also learning how to program and work with computer. I'm starting small with PyScript and trying to get A+ certified.
Get this. Greatly improved my calculation time.
This is one of the methods suggested in this book: https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
It’s a really nice read for doing mental math. The author, Arthur Benjamin, has some really impressive videos on YouTube IIRC
Little mental trick you can do to show off to some people:
any number * 11 is easy. Even in the 2 digits.
Let's do 32 again.
*32 11
Separate 32 into two digits, add them, and then put that number between those two digits. For example:
3 + 2 = 5
place between the two original digits:
352
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This works with three digits as well (but I have to go figure out how to do that one again). There is a book on the Apple Store that is an awesome read if you're into it. All of the things I am showing you are possible to do mentally. I can currently square 4 digit numbers in my head sorta reliably, and can square 3 and 2 digit numbers without fail. It is really fun and I enjoy doing it.
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EDIT:
PLEASE PLEASE PLEASE support this guy and do not download a pdf of the book. He is absolutely incredible with what he can do and is sharing it with people so they can do it too. Give him credit!
Book on Amazon
Book on iBooks
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Youtube video of this guy
Here's another one that's pretty good
Read the book by Arthur Benjamin. He's one of my role models. :D The book has the most amazing mental math tricks ever, and I can square 2, 3, and even 4 digit numbers in my head. Getting to 5 digits soon. There are a lot of other cool tricks in there as well.
Asked myself the same question this morning. I found this book is supposed to be a good start.
http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401
I read this book in high school when it was originally published as "Mathemagics." https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=pd_lpo_sbs_14_t_2?_encoding=UTF8&psc=1&refRID=WQYSFNW9WRJY77M30PZG
Its a collection of tips and shortcuts to make mental math easier. I really enjoyed it and found it very useful.
Benjamin Arthur is great at this. He wrote a book that may interest you.
I would enumerate on the various techniques I've used over the years, which drove my early math teachers somewhat mad, but, well, those little tricks and more are readily available in the book The Secrets Of Mental Math. I never finished the book, but it's got quite a few very useful tips, just in the opening couple of chapters, and it builds on them to add other neat things.
I used to be just like you, then really became fascinated by physics, which was very difficult given my deficiencies in math. I figured I would start with flash cards and what not, so I started browsing amazon and came across this. This guy is a genius, and teaches you a lot of tricks to do math quickly in your head. The next thing I did was checked out Khan Academy. I can not over-exaggerate how utterly fucking awesome this site is. Not only does he have like 2300+ videos on every topic, but he has something like 125 math modules that allow you to practice. It's completely free and all you need is a facebook or gmail account to log in...
Aside from Khan, The Secrets of Mental Math was extremely helpful in this endeavor.
Honestly, I highly recommend this book, and pretty much anything else by Arthur Benjamin. He's the real deal when it comes to mental math. Take it seriously, and do tons of practice problems. Feel free to go "fast" through the book the first time through, but go super slow the second time through and get everything super solid.
After completing the book you'll be able to do squares, multiplication, division, addition, subtraction pretty damn fast up to around 3-4 digits. With more practice you can eventually get as good as Prof. Benjamin (he doesn't leave anything out! Tells you the entire technique). By more, I mean years more, but hey, at least it's possible