# Talk:Mental calculation

## Complete Rewrite Might Be Needed

It seems to me that his page needs to be completely rewritten to bring it up to the standards of the vast majority of the Wikipedia pages. It seems to me that the appropriate level of material is found in Arthur Benjamin's book "Mental Math Made Easy." I would very much appreciate an infomred discussion of this possibility from the Wikipedia math editors. Below is a summary of the considerations that lead me to make this suggestion.

This page looks to me like it was written by mathematcians who have used algebra to invent a pot-pourri of methods that might be suitable for mental calculation, but, it looks like they were not aware of the history of the subject, and it looks to me like the methods they invented are far below the quality of the methods actually used by people who are known as "lightening calculators."

It appears to me that the methods currently presented on this page are slower and more cumbersome than the methods used by the majority of "child prodigies" and uneducated "lightening calculators" who have sporadically appeared in various cultures ovedr the centuries.

The methods for "mental calculation" presented in Benjamin's book are not the "be all and the end all" by any means, but, they are a good step in the right direction. For example, Benjamin does not address the well know methods of determining divisibility by 7 and 13 that are based on the fact that 7*11*13 = 1001. This fact, by the way, also is the basis for a quick method of mentally computing 1/7 and 1/13. For example, 11*13 = 143 = 1001 tells you that the estimate 1/7 ~ .143 is awfully close, but that it is a little high, so the first three digits in the decimal expansion of 1/7 are 1,4,2. The next three are nine's complements of the first three, namely, 8,5,7. General principles tell you that 1/7 repeats after six digits and you are done. For 1/13, 7*11 = 77, so, 1/13 ~ .077. The repetition [period is also 6, so, 1/13 = .076923..... There are similar tricks for 17 and 19.

Two good examples of methonds missing from this page and found in Benjamin's book are "going up and going down" and "criss-cross multiplication." True, Benjamin discovered some of the methonds independently when he was a child, before he knew algebra even existed, but, more importantly, Benjamin documents the use of at least some of the methods in his book by historic "lightening calculators."

One good thing about this page is that the introduction shows that the authors are thinking in the right direction. The unfortunate thing is that they do not have (or did not have) the knowledge to carry their program out in an effective manner.

Even the introduction does not quite go far enough. The point is not made that "mental calculations" are a sport. In other words, "mental math" is an area of mathematics that is different than other areas of mathematics. The essense of the techniques are the arrangements of the calculations in a wayt that is convenient for mental calculations. For this purpose, traditional mathematical equations not not convey enough information to explain a method of mental calculation. For example, it is often necessary to explain how intermediate results are to be remembered as the calculations are being made. This is the reason that equations are only included after the explanation of the methods in Benjamin's book. Equations are used to explain "why the method works." Equations are not sufficient to explain the method itself.

Unfortunately, I am not an expert in this area. I was hoping to learn more than I already know, but was disappointed.

Specifically, I just discovered a connection between "pronic numbers" [numbers of the form n(n+1)] and mental math. I was trying to find quick ways of mentally multiplying 14 by 17 and it occurred to me that that I should be able to take advantage of the fact that they are reasonably close together. Unfortunately, their average is 15.5 so Benjamin's "going up and going down" involves carrying fractions. [You go up by 1.5 (from 15.5) to 17 and down by 1.5 to 14, so, 14 x 17 = 15.5**2 - 1.5**2 {reason why: (n-d)(n+d)=n**2 - d**2.} ] But the same method works using pronic numbers instead of squares: 14 x 17 = 15 x 16 - 1 x 2 = 240 - 2 = 238. Pronic numbers are easily calculated by n(n+1) = n**2 + n, and they are the doubles of the triangular numbers, and they are the truncated average of the square above them and the square below them, e.g., (36 + 49)/2 truncates to (36 + 48)/2 = 84/2 = 42 = 6*7. DeaconJohnFairfax (talk) 19:13, 3 September 2008 (UTC)

I agree. And a rewrite should include a section of historic "math wizards" who dazzled people with their mental calculations. Bringing in the human element would really round this out and bring in more helpful links to related articles. --Ibinthinkin (talk) 20:19, 5 April 2009 (UTC)

## Wikipedia needs "Math tricks and curiosities" page with "Math tricks" and "Math Curiosities" pseudonyms

The search "math trick" redirects here. Honestly, I was looking for information on things along the line of:

First, write the number 1089 on a piece of paper, fold it, and hand it to a friend for safekeeping. What you wrote down is not to be read until you have completed your amazing mental feat.
Next, ask your friend to write down any three-digit number, emphasizing that the first and last digits must differ by at least two. Close your eyes or turn your back while this is being done. Better still, have someone blindfold you.
After your friend has written down the three-digit number, ask him to reverse it, then subtract the smaller from the larger.
Example: 654 - 456 = 198.
Once this is done, tell your friend to reverse the new number.
Example: 198 becomes 891.
Example: 198 + 891 = 1089.
The number you wrote down at the start -- 1089 -- will always be the same as the end result of this mathematical trick.

Wouldn't the term "math trick" more correctly refer to something like the above, rather than to mental calculation? I don't believe something like the above trick belongs in this article, for sure. Applejuicefool 20:56, 21 March 2007 (UTC)

AppleJuice, I agree with you completely. In the following four paragraphs, I expand on your idea. DeaconJohnFairfax (talk) 16:33, 3 September 2008 (UTC)
We absolutely need a page called "math tricks." Maybe better would be "Math tricks and curiosities" because every math trick is in fact an example of a "math curiosity" and most "math curiosities" can be cast in the form of math tricks. We could then create pseudo-pages (really pseudonyms) called "math curiosities" and "math tricks" and redirect them to to "math tricks and curiosities." Dr. John Gayle Aiken IV, (PhD in math with 50 years experience in math, physics, engineering, and computer science). DeaconJohnFairfax (talk) 16:00, 3 September 2008 (UTC)
To add another example to your "Math Tricks" page (that I suggest be called "Math tricks and curiosities), 1492**n - 1700**n + 1860**n - yyy**n is always divisible by 1946 for any natural number "n." I learned about this from Pickover's "Wonders of Numbers." Pickover says that this was published in the January 1947 issue of the American Mathmatical Monthly. The point is that 1492, 1700, 1860, and **** are all memorable dates in the history of the US. 1700 is the date of the Boston massacre. 1860 is the date of the Gettysburg address. I've forgotten what the yyyy**n is, but it a significant 20th century date.DeaconJohnFairfax (talk) 16:33, 3 September 2008 (UTC) (this is just off the top of my head. please feel free to update.)
There is a Wikipedia page called "Mathmatical coincidences" already. However, the spirit of that page does not address the kind of "tricks and curiosities" that you and I have in mind. Also, none of this kind of material is appropriate for the "mental math" page. MAYBE if there were only as small amount of it. But, the amount (in principle) is not small. DeaconJohnFairfax (talk) 16:33, 3 September 2008 (UTC)
There is a danger here that we do not want to encourage the "occult" numerologists to contribute. There are simply too many people among them are coming from either an extreme "crackpot" mentality (do a goggle search on "the covedant" and 2050) or from a deeply psychotic mentality (like Jophn Nash was during his "irrational years). We can avoid this by demanding references to standard and generally respected references. DeaconJohnFairfax (talk) 16:33, 3 September 2008 (UTC)

## An Unsigned Comment on The Logs Section and the Introduction

Many practitioners believe it develops and trains the mind, improves visualization and intelligence.

Just a note, i added the common logs section, and if anyone understands what i was saying and wants to reword some of it so it flows better that would be great cause i dont think i wrote it as effectively as i could have.

—The preceding unsigned comment was added by Jondel (talk • contribs) .


## "Only the brain" or brain and hands (and toes)

If this article is about "doing mathematical calculations using only the human brain", shouldn't the info about doing it with the hands be removed? Alternatively, the intro should be changed from saying "only the human brain". Nurg 02:42, 17 December 2005 (UTC)

I agree in principle, the article is about mental calculation, if we include the use of hands we might as well include the use of paper/calculator/South-African monkey. However, I don't think mental calculation' competitions prohibit the use of digits, and many people seem to instinctively use digits when doing mental calculations anyway. Discussion? //Biggoggs 04:30, 5 June 2006 (UTC)
Well, there is also using a real abacus and creating a mental image of an abacus . I think if the person mentally creates the digits and hands it would be ok. However if the physical hands/digits were used then I think this should not be included.--Jondel 05:36, 5 June 2006 (UTC)

## Citation needed

"Mental calculation is said to improve mental capability, speed of response, memory power and concentration.? Said by who? I'm going to add a [citation needed] to this, it's a bit steep without a reference... //Biggoggs 04:30, 5 June 2006 (UTC)

Sorry Biggoggs, this is the pertinent reference source concerning mental capability , etc. : http://www.scciob.edu.sg/index.cfm?GPID=24 .--Jondel 05:11, 5 June 2006 (UTC)

This is another source :http://www.ucmasusa.com/MA/PressRelease.htm concentration

I hope the citation is ok now(?).--Jondel 05:39, 5 June 2006 (UTC)

I don't think those sources are acceptably pertinent', I can't see anything relevant on the ucmasusa.com site, and the scciob site basically says;
This skill has proven to be beneficial in improving one's mental capability, increasing one's speed of response, memory power and concentration power. This course has become a compulsory curriculum in many countries such as Taiwan, China and Japan, and countries such as India, Malaysia and Singapore have introduced the curriculum to schools.
Until a respectable source' can be found, I think it's best left as cite-needed or removed altogether... Biggoggs 06:58, 5 June 2006 (UTC)

Like from a psychologist, doctor,etc ? OK then, I'll restore the cite request .In the meantime I'll still search the web for such a source. --Jondel 07:21, 5 June 2006 (UTC)

Yea, something like a research paper, or at least an article with resources would be better. Biggoggs 07:51, 7 June 2006 (UTC)
Ok , that is why below , remove. But you remove, not some ip I don't know. I've put too much work into this and I believe it does improve mental capabilites not only in calculation. When I do find authoritive sources, I will reinstall and place the sources at the bottom. If we can comprimise, leave as cite needed? --Jondel 07:29, 8 June 2006 (UTC)

I personally experience that it improves concentration and it seems obvious. Can we at least say 'Proponents advocate that it improves concentration, memory power, etc?'.--Jondel 08:30, 7 June 2006 (UTC)

That would essentially be saying the same thing. I think a claim like that needs to be referenced from an academic source, not 'person opinion' or 'a friend thinks [x]'; it doesn't seem obvious to me. =Biggoggs 07:10, 8 June 2006 (UTC)=

If needed pls remove the part about concentration et al. However, I've found a few statements from doctors not too mention personal experience of others. I need time.--Jondel 00:00, 8 June 2006 (UTC)

I hope these sources are conclusive and reliable enough?--Jondel 04:33, 19 June 2006 (UTC)

## Multiplying by 2

The article claimed: "Still, the product must be calculated from right to left" I think it's not necessarily true. I've seen a method to do that from left to right: http://shifengshou.com/english/htm/shifenngshou_what1.htm

Samic 22:34, 10 August 2006 (UTC)


## Desparately needed copy-edits

This article incredibly thoroughly defies Wikipedia:Manual of Style (mathematics). I've done some copy-edits; lots more are needed. At one point someone actually used an asterisk for ordinary multiplication within TeX; how crude and vulgar can you get? Some other places still have

5*3

5 × 3

and centered TeX instead of TeX indented by a colon, or things like

x-5

x − 5

and so on. Michael Hardy (talk) 22:48, 13 September 2008 (UTC)

## Calculating square root

In the 'Finding roots' -> 'Approximating square roots' the formula

${\displaystyle {\text{root }}\simeq {\text{ known square root}}-{\frac {{\text{known square}}-{\text{unknown square}}}{2\times {\text{known square root}}}}\,}$

is used. This could very easily be transformed into

${\displaystyle {\text{root }}\simeq {\frac {{\text{known square}}+{\text{unknown square}}}{2\times {\text{known square root}}}}\,}$

yielding 20% better memorability and easier calculation.
Can I update the section? —Preceding unsigned comment added by Thomasda (talkcontribs) 23:33, 1 December 2009 (UTC)

## Multiplying 2 and 3 digit numbers in the head

When doing multiplication in the head, it is easier to visualize and remember the numbers if they are written like this:

Example:

49
73


or in general

AB
CD
`

The steps are as follows:

Multiply B with D and make note of the last digit of the result 27 - 7 (M)

Then multiply the numbers accross like this: A x D plus B x C plus the first member of the remainder M(2) (doing it in the head will be 77, the last digit of which will be noted (7) by placing it in front of the remainder (77).

Now multiply A with C and add the remainder (7), which gives you 35, and append that in front of the existing remainder to yield 3577.

Multiplying abc with def is done similarly. I guess you can find out yourself.

Genezistan (talk) 11:33, 24 December 2009 (UTC)

## Removal of "How to" template

As per advice given on Wikipedia:Teahouse/Questions#How_do_I_have_article_shortcomings_tags_removed? I am removing the {{Howto|date=November 2009}} template. I will watch this page for at least a month, so feel free to disagree.--guyvan52 (talk) 15:32, 2 February 2014 (UTC)

## Mental arithmetic as a psychological skill?

This whole section is about the interaction of cognitive and physiological performance, and not "mental arithmetic as a psychological skill." I don't know what the section title is meant to describe, but what follows is a discussion of physical exercise and cognitive task performance, which is not specific to mental calculation. I'm not sure it belongs in this article. It's saying that if you're doing mental tasks, your physical performance will be compromised, that while you are doing specific physical exercises your cognitive performance can be impaired, and that after physical exercise, cognitive performance is improved. It's really an expansion of the Yerkes–Dodson law. Ride your bicycle to the math competition but don't run in place while mentally calculating cube roots? Don't do mental calculation during a bicycle race? At minimum the title needs to be changed, but I really question whether this content fits this article. Dcs002 (talk) 17:58, 29 September 2014 (UTC)

## Multiply a number by 10, 11, ..., 19

I added a similar piece of information on the page about 11, anyway it was rejected. Since i don't want to discuss too much, i leave another piece of information here, then the community of active editors in wikipedia can choose if it is worth the article.

Given a number ${\displaystyle A}$ in base 10 with ${\displaystyle n}$ digits, we can represent it as follows:
${\displaystyle A=d_{n}\cdot 10^{n-1}+d_{n-1}\cdot 10^{n-2}+\ldots +d_{2}\cdot 10^{1}+d_{1}\cdot 10^{0}}$.
When multiplied by a number in the form ${\displaystyle (10+x),x\in [0,9]\subset \mathbb {N} \mathbb {N} }$ we have
${\displaystyle A\cdot (10+x)=(d_{n}\cdot 10^{n-1}+d_{n-1}\cdot 10^{n-2}+\ldots +d_{2}\cdot 10^{1}+d_{1}\cdot 10^{0})\cdot (10+x)}$
Therefore rewriting it as
${\displaystyle (d_{n}\cdot 10^{n-1}+d_{n-1}\cdot 10^{n-2}+\ldots +d_{2}\cdot 10^{1}+d_{1}\cdot 10^{0})\cdot 10+(d_{n}\cdot 10^{n-1}+d_{n-1}\cdot 10^{n-2}+\ldots +d_{2}\cdot 10^{1}+d_{1}\cdot 10^{0})\cdot x}$
grouping by the same power we get the method for computing
${\displaystyle d_{n}\cdot 10^{n}+(x\cdot d_{n}+d_{n-1})\cdot 10^{n-1}+(x\cdot d_{n-1}+d_{n-2})\cdot 10^{n-2}+\ldots +(x\cdot d_{3}+d_{2})\cdot 10^{2}+(x\cdot d_{2}+d_{1})\cdot 10^{1}+x\cdot d_{1}\cdot 10^{0}}$

An example could be ${\displaystyle 8745783\cdot 18}$.
We can se it as:
${\displaystyle 8\cdot 10^{6}+(8\cdot 7+4)\cdot 10^{5}+(8\cdot 4+5)\cdot 10^{4}+(8\cdot 5+7)\cdot 10^{3}+(8\cdot 7+8)\cdot 10^{2}+(8\cdot 8+3)\cdot 10^{1}+8\cdot 3\cdot 10^{0}=157424094}$

Pier4r (talk) 11:34, 8 January 2015 (UTC)

## Calculating differences: a − b / Indirect calculation too complicated

The section as the following example for indirect calculation: "For example, to calculate 8192 − 732, we can add 8 to 732 (resulting in 740), then add 60 (to get 800), then 200 (for 1000). Next, add 192 to arrive at 1192, and, finally, add 7000 to get 8192. Our final answer is 7460."

This seems overly complicated as it would require addition of 8, 60, 200, 192 and 7000. The steps also feel unnecessary, as you go from 732 to 740 to 800 to 1000 to 1192 to 7000 to 8192.

There is an easier way to do this:

- We start with 732 and want to go to 8192. The last digit is already correct, so we just don't touch it at all. - We add 60 to get to 792, so now the last two digits are correct. - We add 400 to get to 1192, so now the last three digits are correct. - We add 7000 to get to 8192, so now all four digits are correct.

Adding 60+400+7000 to 7460 is then very easy. Plus, you don't really "add" the numbers anyway; it's more like building the result from the last digit to the first. (We did not need to change the last digit, so the last digit of the result is 0, then we added 60, so the second digit from the right is 6, then we added a 4 and then a 7, giving us 7460.)

This is way easier that doing an extra steps and then having to add 8+60+1192+7000. Plus, with this method, you really need to perform the addition. It does not "build" the result from right to left like the method I outlined. Plus, why the steps in the article modify the last digit twice, although it is already correct in the beginning and does not need to be touched, I don't know. --2A02:8108:A80:7BC:888F:9674:2949:78C7 (talk) 12:14, 3 September 2017 (UTC)

## Vandalism

The tables that show approximations of powers of e is nonsense.