# Secrets of Mental Math

#1
Tue, 03/22/2011 - 22:00

#### Secrets of Mental Math

I started reading about mental math before I had heard of memory techniques. The book I have is called *Secrets of Mental Math* by Arthur Benjamin and Michael Shermer. I've only read a couple of chapters but I'm interested in pursuing it.

Anyone else interested in mental math? Are there other books on the subject you recommend?

Yes, I'm interested. I think the type of person interested in memory techniques and memorizing cards and numbers and binaries is attracted to quick calculating techniques. Ben Pridmore is a quick calculating devotee (and competitor)

I have an iPhone/iPad app called Mathemagics which is very good. A few years ago I started to make my way through Benjamin and Shermer's book but lost steam (it was going well and I amazed myself). Benjamin is a mnemonics enthusiast. There is video on YouTube with Benjamin doing quick math for an audience. It re-sparked my interest.

Btw, Arthur Benjamin has a video Great Courses "Secret of Mental Math" course: http://www.teach12.com/tgc/courses/course_detail.aspx?cid=1406

There is a section about mental speed math in Ben Pridmore's book, but I haven't worked through it yet. I think his method is different than mentioned in Benjamin's and Shermer's book...

Trachtenberg has a good system for multiplication, division, addition and subtraction. There is an out-of-print book you can read, but the internet has a few resources--not perfect resources but good resources. I have an app called Mercury Math which is basically Trachtenberg. You can use the Major System to keep the numbers in your head. Mental math is basically the ability to visualize the numbers in your mind.

This is a book that I've gotten a great deal out of,

http://www.amazon.com/How-Calculate-Quickly-Course-Arithmetic/dp/048620295X

It's filled with tons of structured drills, the idea being that over time you'll develop "number sense," something that used to be common among bookkeepers, grocers, shop clerks, etc. before calculating machines became widely available. It is a ton of hard work, though, which will put off most people.

Looks good -- I will add it to my reading list.

Can someone tell me what the fundamental skills are. I'm guessing the ability to add single digits easily.

What else?

Thanks

Here is a page listing some elements:

https://secure.wikimedia.org/wikipedia/en/wiki/Mental_calculation

Peepz,

I developed a website where you can practice your mental math and put a ranking. So show how quick you are!

It is appreciated,

Greetz Chip

I would love to. What is the website address?

If you have strong association for digit pairs (eg. PAO, major) do you think it affects your "normal", logical mental calculations?

Does it helps with some subconsciuos connections or does the now unwanted images disturb you in some (sub)consciuos level? Or do you think it affects you in some other way?

I found it a good book. A good start. For me it wasnt difficult to follow. But.....This was my first book. Maybe you can find better books about mental math?

I don't know, but I think it could probably only help with mental math, because images are easier to remember than numbers.

For a while, I was experimenting with keeping track of my shopping cart price at the supermarket using only my mnemonic images. Each of my images has a one syllable pronunciation, so I would calculate like this:

$2.40 + $1.99 = $4.39

would be done with this pronounciation:

U-RO plus I-PUH equals A-MUH

Then I would hold "A-MUH" ($4.39 -- flag and Marshal stack) in my mind until I added the next item. Then I would see how close my final mnemonic image was to the check out price. I really should start doing that again... :)

I don't think it helps with the calculation, but it might be good practice for combining mnemonic images with mental calculation.

Forgot to post the adress: calculationrankings.com

I've just noticed the Shifengshou rapid calculation from above link. It's really impressive. i see that it is different from other mental math method: vedic math, abacus or trachtenberg. Does anyone here know about the method of Prof. Shifengshou? Here's the link for your reference: http://www.sfsrc.com/ & http://www.youtube.com/watch?v=5mEtb7dHqVE.

It's crazy why i not know about this earlier! Sadly no material in english is available now. Thanks a lot for your opinon.

Looks interesting. I found an example here::

http://web.archive.org/web/20091028191722/http://shifengshou.com/english...

... and created a new wiki page on Shi Fengshou. :)

I found an English site:

http://www.sfsrc.com/En/Index.aspx?UserInfo_ID=217687

"English translation of new textbooks are now under preparation"

:bigsmile: :love: Thanks Josh. Great searching! I am thinking of learning chinese to understand more about this ...i wish that someone here will post or guide the full details of this method :)

I've sent them an email asking about the English translations. I will post an update if they reply to me. :)

i have a copy in chinese. if anyone is interested or willing to make a translation in English plz send mail to me.

They emailed me today and wrote that they have it translated into English already, but they haven't told me how to order it yet. I'll post an update when they reply again.

Peepz,

I'm quick with easy calcs, but I´m still not so quick with decimals, questions like 2.05 x 1.29 (I put some questions on my website in the tests section on calculationrankings.com)

Anyone has an idea how to calculate it quickly?

thanks!

Sure. just play with it. Here goes.

2.05 X 1.29 is the same as 205 X 129. Just put the decimal point back in after calculation.

205 X 129 = 200 X 129 + 5 X 129.

200 X 129 = 200 X 130 - 200 X 1 = 26,000 - 200 = 25,800

5 X 129 = 1290 / 2 = 645.

25800 + 645 = 26445

Now put the decimal back in: 2.6445.

Hi Josh. It seems that they forget to reply you...:)

I guess so... :)

Maybe the products are not ready yet, and they are waiting until they have something available? I don't know...

I still following this post by analyzing the chinese book and found some ways of calculation. The most exciting point of this method is it will calculate from left to right (vs the traditional way of right to left).

For short explanation, i take the short number to be multiplied only. But it does not a matter how long you like.

Plz share with you as below:

* General rule: Before the first digit of the multiplicand add a zero

1. Multiply by 2

Ex: 67394 x 2 ==> 067394 x 2

- Each figure of result = unit digit of left most figure multiple 2, plus 1 if the "neighbour" is ">" or "=" 5

+ Step 1: 0 x 2 = 0, its neighbour is 6 that bigger than 5 ==> first digit of result = 0 + 1 = 1

+ Step 2: 6 x 2 = 12, taking only its unit digits is 2. The neighbour of 6 is 7 (>5) so the next digit of result = 2 + 1 = 3

+ Step 3: 7 x 2 = 14, taking 4 only. The neighbour of 7 is 3 (<5) so will not plus 1. the next digit of result = 4

+ Step 4: 3 x 2 = 6, the neighbour of 3 is 9 (>5) ==> plus 1. The next digit of result = 6 + 1 = 7

+ Step 5: 9 x 2 =18, taking 8. The neighbour of 9 is 4 (<5), not plus any. The next digit of result = 8

+ Final step: 4 x 2 = 8. Taking 8 as the final digit of result.

==> The result is: 134788

For further, plz visit the site that Josh suggests: http://web.archive.org/web/20091028191722/http://shifengshou.com/english...

2. Multiply by 5

Ex: 475 x 5 ==> 0475 x 5

Rule:

- If the neighbour from the left is 0, 1 ==> plus nothing

- If the neighbour is 2, 3 ==> plus 1

- If the neighbour is 4, 5 ==> plus 2

.....

- If the neighbour is 8, 9 ==> plus 4

* I guess the number added here is the ten unit of 5 x the neighbour itself. Now here we go

+ Step 1: 0 x 5 = 0. The neighbour of 0 is 4 ==> plus 2. The first digit of result is: 0 + 2 = 2

+ Step 2: 4 x 5 = 20, taking its unit digit 0 only. The neighbour of 4 is 7 => plus 3: The next digit of result: 0 + 3 = 3

+ Step 3: 7 x 5 = 35, taking the unit digit 5. The neighbour of 7 is 5 => plus 2. The next digit of result: 5 + 2 = 7

+ Final step: 5 x 5 = 25. Taking the unit digit 5 only. The next digit of result is 5

==> The result is: 2375

Other cases i not know, i am still studying... and finding someone know chinese to translate but not available yet :)

Thanks!

Is there a way to scan the pages and then use OCR software?

https://www.google.com/search?q=chinese+ocr

You could then copy/paste the text into Google Translate...

You can use lots of apps for Android...beside this, I recommend LotaMath because of his design, it looks a bit like ancient C64 software :-). For your knowldedge, I recommend some books (not already given):

MatheMagicsAsisa

get brillilant@calculation

for most well known short-cuts

(other short-cut books and websites you`ll find easily)

George Lane

Mind Games

for the championship disciplines and a good overwiew

Ronald W. Doerfler

Dead Reckoning

...only for cracks, if you like to go on.

For German Readers:

Gert Mittring

Rechnen wie ein Weltmeister

for some insight

Karl Menninger

Rechenkniffe

ancient, lovely book that features a ton of tricks

You may start to enjoy looking on numbers sheets as well and tables with primes, squares etc. will start to surround you.

Have fun!

Now i fully understand how it work :) Since the calculation start from the left to right, it means that the calculator must know what's the carry-number from the next calculation. The method is that there should be a number to make a comparison with the next number to estimate the carry number!

Below is the list of number to be compared when multiplication.

@ Multiply by 2:

1/2 = 0.5 ==> compare with 5. If the right neighbor is >, = 5 then +1 and reverse.

You already know this from the previous post.

@ Multiply by 3:

1/3 = 0.3333 => compare with 34. If the right neighbor is >, = 34 ==> +1

2/3 = 0.6666 => compare with 67. If the right neighbor is >, = 67 ==> +2

Ex: 04958 x 3

- 0 x 3 = 0 and 49 > 34 ==> plus 1. We have 0 + 1 = 1. This is the first digit of the result number.

- 4 x 3 = 12. Always taking the unit only. And 95 > 67 ==> plus 2. We have 2 + 2 = 4. This is the second digit

- 9 x 3 = 27, taking 7 only. And 58 > 34 => plus 1. We have 7 + 1 = 8. This is the next digit.

- 5 x 3 = 15, taking 5. And 80 > 67 ==> Plus 2. We have 5 + 2 = 7

- 8 x 3 = 4. Taking 4 as the last digit

==> Result: 14874

@ Multiply by 4:

1/4 = 0.25 => compare with 25. If the right neighbor is >, = 25 ==> +1

2/4 = 0.50 => compare with 50. If the right neighbor is >, = 50 ==> +2

3/4 = 0.75 => compare with 75. If the right neighbor is >, = 75 ==> +3

Ex: 04958 x 4

- 0 x 4 = 0 and 49 > 25 ==> plus 1. We have 0 + 1 = 1. This is the first digit of the result number.

- 4 x 4 = 16. Always taking the unit only. And 95 > 75 ==> plus 3. We have 6 + 3 = 9. This is the second digit

- 9 x 4 = 36, taking 6 only. And 58 > 50 => plus 2. We have 6 + 2 = 8. This is the next digit.

- 5 x 4 = 20, taking 0. And 80 > 75 ==> Plus 3. We have 0 + 3 = 3

- 8 x 4 = 32. Taking 2 as the last digit

==> Result: 19832

@ Multiply by 5

1/5 = 0.2 ==> +1

2/5 = 0.4 ==> +2

3/5 = 0.6 ==> +3

4/5 = 0.8 ==> +4

The rule is same for the following number: 6, 7, 8, 9

@RetoCH: I'd love to hear some technique or system from George Lane. Is there any difference the "Mind Games" and Secrets of Mental Math by Mr Arthur Benjamin? Can you give some details about that book?