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

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

[Otter]

Some of the older folks may remember Chisenbop or Korean finger math. They used to advertise some instructional program for it all the time during cartoons. One kid in my school learned it but I've never heard anything about it since the 70s. ("Korean" was a lot more exotic then than it is now.)

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.

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

Visualisation?

[jago25_98]

Some links (click the 1's). Some are for dylexics but still relevent for all since pretty much all of us are capable of visual thought...:

1 1 1 1 1 1 1 1 & similar 1 1 1

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?

Re:blind leading the blind

[nefertari]

You forgot to take the square of 4/3

This one is the right way:

(3/3a)^2 + (4/3 a)^2 = 20 ^2
since 3^2+4^2=5^2 we get
(5/3 a)^2 = 20 ^2, so
20 = 5/3 a and a=12.
So: height = 12, width = 16.

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.

One Speed Math System

[ec_hack]

Long ago in high school, I competed in what was then called "Number Sense" - doing math problems mentally, no aid of scratch paper. (Calculators were an expensive novelty - 4 functions, Nixie tube displays, plugged into the wall, had 4 functions.) The system we all worked from is now called the "Trachtenberg Speed System of Basic Mathematics", and it had lots of tricks for converting decimals to fractions and vice versa, multiplication of pairs of 4 digit numbers, etc. There are a lot of drills on visualization that helps in holding intermdiate results in the head. See http://www.speed-math.com or find the book on Amazon.

Re:Try doing square roots in your head.

[nelsonal]

It works better on craps, as the odds are tighter, in roulette, the green squares are both losers giving you about about a 47% chance of winning. The pass (or nopass, but you get dirty looks from the shooter) line is north of 49% (a bit better if you take odds). The issue is that in strings of random numbers long sequences of the same result are more common than conventional wisdom would have us believe and you only get about 9 losses before most table limits kick in (do the math on the doublings between minimums and maximums) You could increase your initial bet (which is what you win back each sequence), if you had a big enough starting stake.

For the lazy...

[sleepingsquirrel]

And for those of us who haven't mastered the art of quick mental hyperlinking, here's an actual clickable link...

http://www-gap.dcs.st-and.ac.uk/~history/HistTopic s/Mental_arithmetic.html

Memory improvement

[Vexware]

One activity you can indulge in can simultanesouly improve your memory, make you feel good and allow you to show off in front of your friends so they will think that you are a really intelligent person (which I am not saying you aren't, but people who aren't really into this kind of brainy and "geeky" activity will surely be very impressed) is to memorize 1000 digits of pi. It's funner than you may think, as it's a real challenge and over time will increase your capacity to use the full potential of your memory properly.

Re:No substitute for hard work

[Monkelectric]

You got it. Want to meet someone who can do math in their head? Talk to a casino dealer. I dated a girl who was a casino dealer and she could do just about any simple calculation in her head. And when you think about it, thats half what a casino dealer does: calculate payouts.

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".

Two books I have...

[JGski]

Two books I have related to doing math in your head:

"Consider a Spherical Cow" and there's a 2nd book "Consider a Cylindrical Cow" :-) - which is about how to do "back-of-the-envelope" estimates. How many pairs of shoes can be made from a single cow? Consider a spherical cow. :-)

And a Dover reprint: "How to Calculate Quickly" which has many of the tricks and rules of thumb people used to all know before calculators.

From my "antiquarian" collection I have a number of "arithmetic" textbooks (all pre-1930) that have lots of little rules of thumb for checking sums and products - many are familiar to accountants. Also great chapters like "Arithmetic of Thrift", "Arithmetic of Agriculture", etc. with problems like "...girls in a class in millinery need 20 yd. of ribbon..."