Improving Your Mental Math Skills?

Infrared-Archer asks: "I want to learn how to do most math calculations in my head. That way I won't have to reach for the calculator for problems I should be able to do mentally. Of course there are various websites (beat the calculator) that show many tricks, but I am looking for a comprehensive solution (books, websites) that shows how to solve of wide range of math problems mentally. Any suggestions?"

Comments

No substitute for hard work

[Anonymous Coward]

All the tricks are fine, but there is no way around it, you have to practice and keep your skills up. Start adding things up when shopping, calculate tips and sales taxes, etc. When ever you rach for the calculator, see if you can't do it in your head first, at least for a quick estimate.

Re:No substitute for hard work

[pla]

All the tricks are fine, but there is no way around it, you have to practice and keep your skills up

True, but the tricks do help quite a lot, in some cases.

For example, I expect most geeks can add, subtract, and multiply arbitrarily long numbers in their sleep. Division, however, (at least for me) has always proved somewhat tricky when the numbers grow beyond two or three digits.

My solution? Look up "duplation" on Google. The Egyptians used to use it to multiply numbers, basically in what amounts to a bitwise manner (though understanding binary helps to speed up the process, you can do it with nothing more complicated than "multiply by two" and "greater than").

However, as I said, doing multiplication doesn't present much of a problem. But you can also do division by using the inverse of duplation! You basically can break an arbitrary largeish division problem into a set of "divide by 2, compare" operations. Basically just long division in binary, but it requires a shorter mental stack (which seems like the key to all the tricks I've seen - ways to reduce the number of items on the brain's stack during the calculation).


So, I'll agree that nothing can beat plain ol' practice for improving one's math skills. But the tricks can make some operations go from "annoyingly hard" to the almost mindlessly easy "step a, step b, step c, repeat 5 times, get an answer".

Best way

[Arngautr]

The best way is to simply limit your calculator usage. I like to show off with the folks I tutor by doing their calculations in my head before they can type them into calculators. A strong basis in algebra can help you beak apart calculations into managable chunks, the trick is remembering how to put those chuncks back together. For instance (contrived example so not great but...): 95*23=100*25-100*2-5*23=2500-200-115=2185

Re:Best way

[jonjohnson]

And, my favorite trick is to multiply any number by 5, divide it by two, move the decimal place over (multiply by 10). It makes it much easer to grok that in my head, at least. So, 5*1024 is the same as 1024/2 * 10 = 512 * 10 = 5120.

Work backwards for dividing by 5.

Re:Best way

[rixstep]

Yeah, I basically do it the same way, except I usually use logarithms and double precision floating point, then I just round off (ceiling or floor) to the nearest 128-bit integer.

Takes a bit of practice, but once you get the hang of it, it's a piece of cake.

First get your arithmetic up to scratch

[twem2]

A good way to practice is every time you get a bus or train (or get any sort of ticket with numbers on) add up the digits on it in your head. After doing this for a while you'll get quicker and more accurate.
For added challenge translate every letter on there into a number using its place in the alphabet (or even its ascii number) and add them on.

You can then make up your own versions using other arithmetical operators and fractions.

After your arithmetic is up to scratch other areas of maths will be easier to do in your head (although beyond anything simple it is still best to write it down)

Vedic Mathematics

[manjunaths]

Try vedic mathematics. There are several books out there, you can try amazon.com. Where I am from (Bangalore, India) we get these books for 1-2 dollars a piece and they come in several volumes. But I saw that they are fairly expensive on amazon.com. If you know someone from India you can ask then to get it for you, it may work out cheaper.

You could also try a google search I found some interesting websites

http://www.vedicmaths.com
http://www1.ics.uci.edu /~rgupta/vedic.html
http://vedmaths.tripod.com

Hope this helps.

Re:Vedic Mathematics

[russellh]

My mathematician wife, by the way, pictures numbers as colors and can somehow do back-of-the-envelope calculations that way. I'm not entirely sure that's a sign of a healthy mind, but it seems to work for her.

I do. Not really numbers, but letters. It has deteriorated over the years for me. Apparently, it is called synesthesia

Re:Vedic Mathematics

[arvindn]

No!! Vedic mathematics is a scam.

The guy who wrote it, Tirthaji, was a fraud. Every word and every claim in the book reg. the history is fabrication. The math is also pure junk and utterly useless.

Seriously. I did a term paper on it last year. You don't have to take my word, of course: read this article by Prof. S. G. Dani, School of Mathematics, Tata Inst. of Fundamental research (the premier research inst. in India.) There's also a much more detailed version.

Unfortuntely, the book fits the political ideology of the current Hindu-fascist government in power in India, and so they've been promoting it big time.

blind leading the blind

[MatrixBandit]

Awhile ago I realized that since highschool my own math skills had deteriorated beyond belief. The breaking point was when I was going to buy a 21" monitor and I wanted to figure out what the height and width of the screen would be so I could actually get a feel for what it was I was paying $400 for. It took me about 4 hours of racking my brain trying to remember old algerbra rules to transform the pythagorean theorem to use the diagonal (20" viewable) and a generic aspect ratio 1.333 to derive the height / width.

My point is that if you want to get quicker with your mental math skills or keep your current pace, you have to keep using it or else it will atrophy like everything else. Translation: college math courses or at home math excercises, but either way don't expect to be able to ever be "done" with it.

Good luck with that by the way, you're a better man than I.

Just do it!

[Captain Kirk]

Research proves there is no trick or secret. People who rely on calculators are poor at mental math because of lack of practice. While some people do have innate skills in maths, everyone has the ability to train the brain to to basic math. Take a look at this study
Memory, mental arithmetic and mathematics

Logarithm tricks: Rule of 72

[ubiquitin]

I like estimating tricks.

The rule of 72 helps to figure out how long it takes for something to double or halve. Divide 72 by the percentage rate of growth or decrease and you'll get the number of time periods in which something will double or halve. For example, let's assume Moore's law says double CPU speeds every 18 months. 72/18=4. So CPU speeds increase by 4% every month. Or another example: your phat mutual fund gets 12% per year, so 72/12=6. So your money will double in 6 years.

This trick is so simple that even the finance guys always know it. :) Anyone else have logarithm tricks to share?

Feynman

[xenephon]

There's an amusing story about Feynman and an abacus salesman in Brazil (found in Surely you're Joking, Mr. Feynman). Feynman was eating in a cafe where he often went, and an abacus salesman came in, trying to sell to the staff. He challenged them to some math problems, and (apparently by chance) they suggested he compete with Feynman instead. They started with an addition problem, and the abacus guy won by quite a bit. They moved on to multiplication, and the abacus won again, but not by very much. Sensing a challenge, the abacus salesman suggests they do cube roots. Quoting now:

"Cube roots! He wants to do cube roots by arithmetic! It's hard to find a more difficult fundamental problem in arithmetic. It must have been his topnotch exercize in abacus-land.

"He writes a number on some paper--any old number--and I still remember it: 1729.03. He starts working on it, mumbling and grumbling: "Mmmmmmmmagmmmmbrrr"--he's working like a demon! He's poring away, doing this cube root.

Meanwhile I'm just sitting there.

One of the waiters says, "What are you doing?"

I point to my head. "Thinking!" I say. I write down 12 on the paper. After a little while I've got 12.002.

The man with the abacus wipes the sweat off his forehead: "Twelve!" he says.

"Oh, no!" I say. "More digits! More digits!" I know that in taking a cube root by arithmetic, each new digit is even more work than before. It's a hard job."

Feynman goes on to explain the approximate method he used to get the result, and then gives his analysis:

"I realized something: he doesn't know numbers. With the abacus, you don't have to memorize a lot of arithmetic combinations; all you have to do is learn how to push the little beads up and down. You don't have to memorize 9 + 7 = 16; you just know that when you add 9 you push a ten's bead up and pull a one's bead down. So we're slower at basic arithmetic, but we know numbers.

Furthermore, the whole idea of an approximate method was beyond him, even though a cube root often cannot be computed exactly by any method. So I never could teach him how I did cube roots or explain ho lucky I was that he happened to choose 1729.03."

The rest of that chapter (entitled "Lucky Numbers") talks about his experiences in trying to improve his mental math skills. Definitely worth a read.

You want Trachtenberg Speed-Math.

[Anonymous Coward]

Run a google-search on "trachtenberg math".

You're looking for sites like Trachtenberg Speed System or Trachtenberg Math (Multiplication).

Professor Jakow Trachtenberg was a brilliant mathematician. Imprisoned by the nazis during WWII, he kept his mind busy to survive by applying advanced mathematical techniques to numeric computation. Eventually developing a number of techniques that provide for rapid mental computation without massive rote memorization.

For example:

0 Zero times any number at all is zero.

1 Copy down the multiplicand unchanged.

2 Double each digit of the multiplicand.

3 First step: subtract from 10 and double, and add 5 if the number is odd.
. Middle steps: subtract from 9 and double, and add half the neighbor, plus 5 if the number is odd.
. Last step: take half the lefthand digit of the multiplicand and reduce by 2.

4 First step: subtract from 10, and add 5 if the number is odd.
. Middle steps: subtract from 9 and add half the neighbor, plus 5 if the number is odd.
. Last step: take half the lefthand digit of the multiplicand and reduce by 1.

5 Use half the neighbor, plus 5 if the number is odd.

6 Use the number plus half the neighbor, plus five if the number is odd.

7 Use double the number plus half the neighbor, plus five if the number is odd.

8 First step: subtract from 10 and double.
. Middle steps: subtract from 9, double, and add the neighbor.
. Last step: Reduce the lefthand digit of the multiplicand by 2.

9 First step: subtract from 10.
. Middle steps: subtract from 9 and add the neighbor.
. Last step: reduce the lefthand digit of the multiplicand by 1.

10 Use the neighbor.

11 Add the neighbor to the number.

12 Double the number and add the neighbor.

Re:Try an abacus.

[mzs]

Here is a more complete excerpt. This is how he explained how he was able to approximate the root so quickly:

The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03 is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root's excess is one-third of the number's excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.

Visualize

[DaoudaW]

When I was a kid I found this already old book called (?) "The Art of Ciphering". That's a guess since I haven't seen the book in probably 35 years. But I remember some of the techniques in it. I was a farm kid at the time so while doing field work I'd have long blocks of time (as much as 10-12 hours a day) without much to occupy my mind. So I filled the time doing math in my head. I got pretty good at multiplying 4-digit x 4-digit, 5-digit x 5-digit, etc. in my head. Also extracting square roots, doing Roman fractions, and other stuff.

As I did these arithmetic problems, I found that my mind developed a kind of blackboard. I could visualize the problem and effectively "write" the answer without worrying about keeping track of everything as separate digits.

My advice: Find a good algorithm, practice a lot (yep, hours and hours), draw a picture in your mind.

The bonus of doing this is that later when I started studying math, the visualization I'd developed helped lots in advanced courses. I could "see" solutions almost instantly that would take others awhile to derive and even then they wouldn't really understand the relationships which led to the solution.