
How Japanese Kids Learn To Multiply - TomAnthony
http://www.magicalmaths.org/how-japanese-kids-learn-to-multiply/
======
patio11
This is a method that is occasionally taught in Japanese classrooms, but one
could also say that of chunking or the traditional long multiplication
algorithm. Want to know the _real_ super-secret-special Asian magic math
sauce? Practice. (But if calling it kungfu math lets you teach it is a US
classroom and get precious instructional hours by calling math cultural
studies then by all means Orientalize away.)

[Edit: I have a pair of chips on my shoulder about this, one on my right
shoulder and one on my left, as it were, and that's why I sort of blew up
here. I really don't think I'm wrong re: the irritant, though. The SMSC
referenced, for example, is Spiritual Moral Social Cultural (skills), and it
is pretty much exactly as froufrou as you are guessing. You'll also note that
every sentence in the post mentions "Japanese" and it's at the most rage-
inducing Stuff White People Like level of surface-learning one could imagine,
too.)

[Edit: Please up vote tokenadult's comment before this one, because while
multiplication is taught by drill2kill here, that comment is better than mine
re: math pedagogy generally.]

~~~
graeme
I tutored a student who counted on his fingers. He was 21.

Within 6 lessons, using Khan Academy, I had him adding four digit numbers in
his head, multiplying large numbers, and doing algebra.

He would learn a technique from me or the video, then apply it in practice
drills, then review at the next lesson.

Total instruction time was 12-20 hours. I think he did a bit on his own.

I concluded that no one had actually checked to see if he could understand
math. North American students hardly try calculations now; they're given
calculators at a very early age.

~~~
JPKab
Not trying to disagree with you about the crappiness of this student's
education, or how sad it is that he counted with his fingers, but computation
!= mathematics.

I HATE the fact that American schools force students to do rote computation
over and over again, with very little focus on the concepts behind the
computations. I hated "math" until my 6th grade homeroom (not math) teacher
took the initiative of writing high school level algebra word problems on the
board, and not telling us how to solve them until after we attempted them. "A
train going from St. Louis is headed toward L.A., and another train from L.A.
is headed to St. Louis. they are going 40 mph, when will they pass each other"
etc.

Suddenly, when the computation becomes a tool to satisfy an end, rather than
the end itself, math becomes interesting.

~~~
numeromancer
_... computation != mathematics._

This is exactly why giving calculators to children who have not yet mastered
any method of multiplication is bad.

------
antiterra
I'm partial to vihart's perspective on this at
<http://www.youtube.com/watch?v=a-e8fzqv3CE> It's moderately interesting, but
ultimately just a way of doing the exact same thing. Multiplication can be
reduced entirely or partially to counting, that's worth understanding. And, if
explained correctly, this can give insight into multiplying polynomials. Also,
showing that counting the intersections of 8 lines crossing 9 lines is tedious
may present times tables as convenient time savers rather than teacher
inflicted torture.

------
Someone
This is a variant of <http://en.wikipedia.org/wiki/Lattice_multiplication>
that does not make it as easy to carry tens a column to the left as that
method does.

I also think lattice multiplication makes it easier to understand for pupils
why the trick actually performs a multiplication.

So, I would teach them lattice multiplication instead.

~~~
ianb
Another approach: <http://en.wikipedia.org/wiki/Grid_method_multiplication>

Of course it's all multiplication, and all essentially the same thing. I find
the grid method the least notation-focused of at least these three, and so I
think more accessible to assimilation. Maybe I'm biased, I didn't learn
multiplication this way but found myself naturally doing it like this in my
head and then was pleasantly surprised to find it had a name. It's also
amenable to estimation.

~~~
mturmon
Thanks for the link. The grid is also helpful as a way to visualize things
like:

    
    
      (1 + a)^2 = 1 + 2a + a^2
    

There is a square with side 1+a, and the grid consists of four pieces, sizes
1, a, a, and a^2.

Thinking about approximations when a << 1, you can motivate why

    
    
      (1 + a)^2 ~= 1 + 2a 
    

really easily by imagining the grid you refer to.

In other words, keeping this grid picture in mind can be helpful to more
mature mathematicians/physicists/engineers...probably more useful than the
standard multiplication algorithm is.

------
hemancuso
This does not teach you multiplication as much as it teaches you a trick to
get the result of multiplication. I doubt anyone is transferring this
abstraction that results in the answer into something they can do in their
head or extend on paper to larger numbers.

Teach kids to open the calculator app on their phone rather than to do this.

No fast way to learn multiplication other than to practice it.

~~~
talmand
I agree, I don't see how this actually teaches how multiplication actually
works. It's a clever method to avoid solving the problem the classic way of
using a memorized multiplication table and breaking down the problem into
smaller segments. I don't even see how this saves you time once you get up to
speed on either method. The only difference is that with my old school way I
can explain to you why I get the answer I calculated while this method doesn't
seem to offer that ability.

~~~
derleth
> I don't see how this actually teaches how multiplication actually works.

[snip]

> It's a clever method to avoid solving the problem the classic way of using a
> memorized multiplication table

Do you think multiplication _actually_ works by means of a multiplication
table?

~~~
talmand
No, if that's how I came across then that's not what I intended. Although, I
feel you're a bit selective in your snips in an effort to make some kind of
point. I know how multiplication works, but the memorization of the table
speeds up the process. I was simply trying to say that this method was not
much different, nor more superior, than table memorization.

My comment about not teaching how multiplication works is aimed at the
statement made that this method is how Japanese students learn to multiply, as
in the title of this thread.

------
symmetricsaurus
The videos only show the technique using small digits. For the larger digits
this method quickly becomes cumbersome.

Latttice multiplication is much better for large numbers and deals with a lot
of the carrying of overflowing digits which occurs in both normal and
"Japanese" multiplication.

<http://www.youtube.com/watch?v=cijuPxHgZAA>

edit: spelling

~~~
boredguy8
yes, there are no '57 * 86' examples, which would be counting intersections
for quite some time.

------
tokenadult
The more important difference between mathematics education in Japan and
mathematics education in (say) the United States is how hard the problems are
and the encouragement to pupils in Japan to try to figure things out for
themselves.

I put instructional methodologies to the test by teaching supplemental
mathematics courses to elementary-age pupils willing to take on a prealgebra-
level course at that age. My pupils' families come from multiple countries in
Asia, Europe, Africa, and the Caribbean Islands. (Oh, families from all over
the United States also enroll in my classes. See my user profile for more
specifics.) Simply by benefit of a better-designed set of instructional
materials (formerly English translations of Russian textbooks, with reference
to the Singapore textbooks, and now the Prealgebra textbook from the Art of
Problem Solving),

[http://www.artofproblemsolving.com/Store/viewitem.php?item=p...](http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra)

the pupils in my classes can make big jumps in mathematics level (as verified
by various standardized tests they take in their schools of regular
enrollment, and by their participation in the AMC mathematics tests) and gains
in confidence and delight in solving unfamiliar problems. More schools in the
United States could do this, if only they would. The experience of Singapore
shows that a rethinking of the entire national education system is desirable
for best results,

<http://www.merga.net.au/documents/RP182006.pdf>

but an immediate implementation of the best English-language textbooks, rarely
used in United States schools, would be one helpful way to start improving
mathematics instruction in the United States. There are accessible
descriptions for teachers in the United States of the system in Singapore, for
example, "Beyond Singapore's Mathematics Textbooks: Focused and Flexible
Supports for Teaching and Learning" American Educator, Winter 2009-2010 pages
28-38.

[http://www.aft.org/pdfs/americaneducator/winter2009/wang-
ive...](http://www.aft.org/pdfs/americaneducator/winter2009/wang-iverson.pdf)

A critique of United States schools by Alex Reinhart that was posted here on
HN soon after it was published

<http://www.refsmmat.com/articles/unreasonable-math.html>

begins with the statement "In American schools, mathematics is taught as a
dark art. Learn these sacred methods and you will become master of the ancient
symbols. You must memorize the techniques to our satisfaction or your
performance on the state standardized exams will be so poor that they will be
forced to lower the passing grades." This implicitly mentions a key difference
between United States schools and schools in countries with better
performance: American teachers show a method and then expect students to
repeat applying the method to very similar exercises, while teachers in high-
performing countries show an open-ended problem first, and have the students
grapple with how to solve it and what method would be useful in related but
not identical problems. From The Teaching Gap: Best Ideas from the World's
Teachers for Improving Education in the Classroom (1999):

"Readers who are parents will know that there are differences among American
teachers; they might even have fought to move their child from one teacher's
class into another teacher's class. Our point is that these differences, which
appear so large within our culture, are dwarfed by the gap in general methods
of teaching that exist across cultures. We are not talking about gaps in
teachers' competence but about a gap in teaching methods." p. x

"When we watched a lesson from another country, we suddenly saw something
different. Now we were struck by the similarity among the U.S. lessons and by
how different they were from the other country's lesson. When we watched a
Japanese lesson, for example, we noticed that the teacher presents a problem
to the students without first demonstrating how to solve the problem. We
realized that U.S. teachers almost never do this, and now we saw that a
feature we hardly noticed before is perhaps one of the most important features
of U.S. lessons--that the teacher almost always demonstrates a procedure for
solving problems before assigning them to students. This is the value of
cross-cultural comparisons. They allow us to detect the underlying
commonalities that define particular systems of teaching, commonalities that
otherwise hide in the background." p. 77

A great video on the differences in teaching approaches can be found at "What
if Khan Academy was made in Japan?"

<http://www.youtube.com/watch?v=CHoXRvGTtAQ>

with actual video clips from the TIMSS study of classroom practices in various
countries.

~~~
SiVal
Tokenadult, I always enjoy your well-informed comments on topics such as this.
I think you may have misinterpreted Stigler's 1999 "The Teaching Gap", though,
as many of us did. In that book, he reports on a study of math teaching in the
US, Japan, and Germany, and finds Japan's results to be far superior to the
others and their teaching methods very different, and different in exactly the
way you describe.

But he did a followup study involving more countries to see if most or all
high-performing countries used the Japanese approach. It turned out that they
did not. Some were more like the US than the US.

Here
([http://timssvideo.com/sites/default/files/Closing%20the%20Te...](http://timssvideo.com/sites/default/files/Closing%20the%20Teaching%20Gap.pdf))
is one place where Stigler reports his updated findings and attempts to debunk
your (and my) initial conclusion---a conclusion that seemed strongly justified
by his 1999 book---that, as you posted above, "The more important difference
between mathematics education in Japan...and the US is...the encouragement to
pupils in Japan to try to figure things out for themselves."

He points out that another high-performing country in his second study, Hong
Kong, was more US-like and less Japan-like on this spectrum than the US
itself. On the dimension you're calling "more important", the low-performing
US is between the high-performing Hong Kong and the high-performing Japan.

His conclusion was that the main factor was not having kids figure things out
for themselves but having teachers carefully teach kids the relationships
among things. It didn't matter if the US kids spent time practicing
procedures. The Chinese kids spent MORE time practicing procedures and did
better, but then the Chinese teachers spent time directly pointing out
important relationships, which the US teachers didn't do much of. The Japanese
kids had to spend a lot of time figuring things out for themselves, but then
the teachers would gather them together and carefully lead them to see
relationships that they hadn't seen when working by themselves. The US
teachers would tell kids to figure things out for themselves and basically
leave their learning to whatever they managed to figure out.

Given equal IQ, time on task, etc., it's the effectiveness with which
mathematical relationships are made clear to the students (part of which
requires significant procedural drill, which Japanese kids do after school)
that matters most. A lot of time is wasted in the US doing procedural drill
with no conceptual understanding, with even more wasted on constructivist
"discovery" methods whereby kids are supposed to somehow teach themselves and
each other the mathematical relationships, and all of this led by teachers who
aren't required by their union to even _know_ anything about mathematical
relationships much less teach them.

~~~
tokenadult
Thank you, SiVal. I didn't see contact information in your user profile (and
indeed the contact information in my user profile is rather subtle until I do
a personal website update). So here I will say thanks for your comment. I'll
be revising some FAQs based on what you wrote. Feel free to contact me off-
site if you'd like to discuss these issues more. (Much of today I am updating
my personal website on its seventeenth birthday, and then I'll have to finish
a revised FAQ promised to another participant here a few days ago, a response
to a link that shows up too often in discussions on the topic of international
educational comparisons.)

------
badatmath
32 year old Japanese here. I can unequivocally say I've never seen anything
like this in my life. Possibly the younger generations, but I somehow doubt
it, as people are pretty slow to adopt completely new teaching methods. Pretty
cool though...

~~~
keithpeter
I have seen students from India use this method for small numbers and a
partial sum method for larger numbers.

[http://wsgfl2.westsussex.gov.uk/Aplaws/maths/multicultural/V...](http://wsgfl2.westsussex.gov.uk/Aplaws/maths/multicultural/VedicMultIntro.htm)

Google 'Vedic multiplication'. Its all good because it is unusual here in the
UK and it looks interesting.

~~~
badatmath
Yes, now that I recall, I believe this was a very popular topic in Japan a few
years ago (I wasn't in Japan to see it first hand). It was referred to as
"India-style multiplication," and some guy(s) sold millions of instructional
books. <http://math.hoge2.info/kakezan/pr04.html>

However, don't think it involves the drawing of lines like in the post.

------
cincinnatus
This is cool but not a part of any standard curriculum in Japan despite the
title.

~~~
gklitt
Agreed. I went to Japanese elementary school (in college now, so within the
last 15 years) and didn't learn this lines method ever.

What I found greatest about the Japanese method of learning multiplication was
actually the method for learning single-digit products. The Japanese use a
system called "kuku" (translated, "9 by 9") which involves memorizing a
rhythmic chant that goes through the entire multiplication table, with each
product being concisely expressed in a few syllables. This is made possible by
the various ways in which a number can be pronounced in Japanese.

I think the method is made necessary because to say the full products takes a
lot of syllables in Japanese (e.g. 7x7=49 would be "nana kakeru nana wa
yonjyuu-kyuu"). So perhaps it doesn't differ too much from the way you learn
multiplication tables at an English-speaking school, but a cool method
nonetheless.

Video here: <http://www.youtube.com/watch?v=1hLyzXM53IE>

~~~
talmand
It isn't that different, in my American elementary school I had to memorize
the multiplication table up to 12x12. But I don't recall anything about the
English language being a hindrance so your example is interesting. There
doesn't seem to be that much fewer syllables in 7x7=49 in English but I'm
assuming that as the numbers increase then the syllables get worse in Japanese
as opposed to English? For instance, 148x385=56980?

~~~
solox3
Using your example 7 x 7 = 49

seven [2] times [1] seven [2] equals [1] forty-nine [3] = 9

七[1] 乘[1] 七[1] 如[1] 四十九 [3] = 7 (without using primary school mnemonics)

七[1] 七[1] 四十九 [3] = 5 (using primary school mnemonics)

Even at that primitive level, a change of language can give you a 23% and 45%
speed-up of basic arithmetic operations, respectively.

~~~
shabble
A similar rote recitation exercise I recall from (a british) school was of the
form '[Seven] [Seven][s] [is|are] [Forty-Nine]', of which only the 'is' is
redundant, and is a single short syllable.

------
ricardobeat
Still quicker to do the math in your head:

    
    
        123 * 321
    
        321 * 100 = 32100
        321 * 20  =  6420
        321 * 3   =   963
                    -----
                    39483
    

You just need to know the times table and keep a little stack in your head :)

~~~
tsahyt
Keeping the stack has always been the problem for me. At least for larger
numbers. Luckily, there's RealCalc on Android which is by far the best
calculator app I've ever seen.

~~~
jff
I'm a fan of Droid48. It emulates a HP-48, so it displays the stack on-screen
and has advanced capabilities like graphing if you need them.

------
rossjudson
Interesting in that it shows how geometry can related to mathematics, but very
laborious when applied to larger numbers. Amounts to adding up the contents of
each place in the numeric result one-by-one. The work involved is the sum of
the output digits.

~~~
vog
_> how geometry can related to mathematics_

Geometry is _part of_ mathematics!

More generally, it's sad to see this attitude all over the place, as if
geometry and drawing were somehow "less mathematical" than writing and
symbols. Both, algebra/analysis as well as geometry, have their place in
mathematics, and complement each other very well.

The same holds for creativity versus stringency, by the way. A theorem has to
be proved (stringency), but how did find that proof? By aternating of
stringency and creativity back and forth! And why did you ask the question
answered by the theorem in the first place? Creativity and real-world problems
(e.g. physics or economics)!

Note that the distinction between geometry/algebra is totally unrelated to the
distinction between creativity/stringency. However, more often that not media
associate math only with the algebra+stringency part, glossing over the other
equally important aspects. This leads to a totally distorted image of the
wonderful field of mathematics.

------
esalazar
I think that it is important to note that without good instruction no method
is "good". It is really really important for anybody learning anything to
understand fundamentally what is going on. This is the problem with most
people that have math phobia's (I don't believe there are people that are not
good at math). They are taught to memorize their times tables with only a
fleeting mention that this really a summation. On this weak foundation people
add more concepts, and BOOM they are math phobic. The thing I do like about
the "line method" is it is a very visual way to show children what is going
on. The chunks, if you will, are literally the place of the digit.

------
dweekly
I wouldn't learn how to multiply from the Japanese. They are clearly pretty
bad at it.
[http://en.wikipedia.org/wiki/File:Bdrates_of_Japan_since_195...](http://en.wikipedia.org/wiki/File:Bdrates_of_Japan_since_1950.svg)

------
sean-duffy
This is quite a nice mathematical trick, but nothing more. It may allow
children to be able to perform large multiplication at a young age, but it is
also setting them up to be confused later on by providing them with a method
which abstracts from the actual arithmetic. What on earth is going to happen
when they run into algebraic multiplication? I think it is much more useful to
teach them to split up the parts of the numbers and multiply those, if they
are too difficult to be multiplied mentally. For example instead of 11x22 do
10x22 + 20 + 2.

------
prawks
_A good way to introduce this starter is to put up a map of the world and get
learners to point out Japan. As a teacher you can then move into how Japanese
Pupils learn to multiply._

Shouldn't the focus be on teaching kids to multiply, rather than teaching them
that "all Japanese people do it this way, look how strange"? How do the
Germans do math? What about Indians?

It really frustrated me that the author chose to illustrate a potentially very
helpful teaching aide by focusing so much of the apparent strangeness of
Japanese culture. Not exactly setting a very good example.

------
missing_cipher
Every time this comes up it's shot down. It isn't part of their curriculum.

------
chubbard
This method is also referred to as Mayan Multiplication which isn't
necessarily exclusive to Japan or a technique the Japanese invented. It's not
Mayan either, but just a neat technique to make multiplying numbers easier.
Although it might be hard to perform in your head this way.

The babylonians, egyptians, and romans had their own techniques because
multiplication by hand is not easy, and it's time consuming. Techniques like
these enabled those civilizations to build wonderful engineering marvels. The
point is these techniques don't violate math or subvert teaching math because
it's not the same thing.

This subject was originally called arithmetic when I was in school which was
different than math. Arithmetic was the rudimentary techniques for how numbers
were added/subtracted/multiplied/divided where math was word problems that
required you know arithmetic to solve but also required logic and abstraction.
So yes arithmetic != math, but that doesn't mean teaching arithmetic is NOT
useful. They are different concepts that are related, but not the same.

I think this is perfectly fine to teach someone this. The whole computation
isn't math argument is a misplaced. We're talking about 2nd or 3rd grade
students. They'll learn math at the higher levels again.

------
frouaix
By the way, if you or you children need a quick and easy way to practice
mental calculations, find an implementation of the game
<http://en.wikipedia.org/wiki/Des_chiffres_et_des_lettres> on your phone.

~~~
zwieback
Thanks for bringing back memories of ski vacations in the French Alps. We
couldn't understand most of the stuff on TV but loved this show.

------
mumrah
I never understood why this is easier than just manually doing the FOIL. 13
_12 = (10 + 3)_ (10 + 2) = 10 _10 + 10_ 2 + 3 _10 + 3_ 2\. Drawing all those
lines and counting intersections seems tedious.

------
aghull
How convenient that all the examples use digits in the 1-3 range. This looks
painful for normal numbers with a mix of large and small digits.

This method always made more sense to me:
[http://www.ehow.com/video_12244670_solve-multiplication-
prob...](http://www.ehow.com/video_12244670_solve-multiplication-problem-
using-diagonal-lines.html) as it combines the idea of the diagonal lines in
the "japanese" method with the mnemonics of the [1-9]x[1-9] multiplication
tables that everyone should also learn.

------
agumonkey
Nice trick, but it would be better if explained in correlation to the
traditional algorithm, showing you can separate work by units and that they're
really the same at the core.

------
muloka
What's more impressive is flash anzan. The ability to visualize an abacus and
then with it do additions of large sums rapidly and accurately.

~~~
partomniscient
Indeed. This short video shows the above in action, nothing short of amazing.

<https://www.youtube.com/watch?v=6m6s-ulE6LY>

------
sinker
It's a neat trick, but the method doesn't do much to reveal why multiplication
works as it does. Most students will just sit there drawing lines and counting
intersections without understanding why it gives them the right answer.
Memorizing a routine without understanding why it works is just one step above
memorizing a table.

------
niels_olson
My sister-in-law is a Japanese school teacher. I showed this too her and she
said she does not recognize it at all. My mom has a master's in math education
and teaches at the local junior college. I'll let you know if she has any
comment on this.

------
nikolakirev
This is indeed very interesting technique, but I don't think it is very
practical and fast.

------
jhancock
My son's home from school when I see this. We try out a few examples, starting
with those in the videos. Then we do a few more...and then we do: 13 x 23. I'm
obviously missing something hidden in this method, as our result, 416...fail
;(

------
shn
it's hardly practical. Yes, interesting, but I would not use it at all. The
way I know how to do multiplication "manually" is superior than this. You line
up numbers on top of each other. Then you start multiplying first digit of the
bottom number with all the digits of above number, from right to left and
adding any carry over. You start writing the result on the same column you are
multiplying. Then you add them all.

    
    
        123
        123 
      x______
        369
       246
      123

+________

    
    
      15129
    

That's what I learned in elementary school and have been using since then.
It's interesting to know how others to do it.

~~~
signa11
> That's what I learned in elementary school and have been using since then.
> It's interesting to know how others to do it.

well, i generally do it using a 'covolution approach' like this:

    
    
         45
       x 86
       ------
           0 (5*6 == 30) -> take the units place, and put '3' as carry over
          4  (6*4+8*5+3 == 47) -> take the units place, and put '4' as carry over
        36   (8*4+4 == 36)

\-------------

    
    
       3640
    

\-------------

with sufficient practice, you can multiply arbitrary 3 digits in approx 10-15
seconds or so ;)

[edit-1]: my formatting _sucks_ , cannot seem to align stuff nicely at all.

------
wilhow
This method does not convey any understanding of the problem. I'm not sure
it's a good idea in the long run. It'll create a class of people who know how
to solve a particular mathematical problem, but don't understand why.

------
Jabbles
Great. Now do it for 79*86.

~~~
Ntrails
realistically all this or any other process is doing is expansion and taking
advantage of the commutative nature of multiplication.

(70 + 9)x(80 + 6)

70x80 + 9x80 + 6x70 + 6x9

56(00) + 72(0) + 42(0) + 54

Guess what - long story short you need to know your times tables. I don't see
how the lines method is any easier than long multiplication, which in itself
is not hard with a little time and explanation. If you can use the lines, or
any other method, as a way of conveying what I did above - then you've
succeeded.

Edit: Stars go to italics and I don't know the escape character

~~~
pbhjpbhj
* It's not backslash!

I think you can only do * with an unpaired * terminator * ?

I think you can only do * with an unpaired * terminator ?

I think you can only do x _x with an unpaired x_ x terminator x _x ?

[that line is this with no spaces around _ :I think you can only do x * x with
an unpaired x * x terminator x * x ?

Nope, looks like as long as it's spaced from the prior/next character then it
doesn't cause italicisation.

Help says: " _Text surrounded by asterisks is italicized, if the character
after the first asterisk isn't whitespace._ "

------
tuananh
Interesting indeed. Reminds me of this [Chinese
suanpan](<http://en.wikipedia.org/wiki/Suanpan>).

------
schultz9999
Cool but impractical - hard to do in your head and cumbersome when drawing
something more than 3 lines. But indeed magical, no argument here :)

------
acomjean
I see the video, but how does this work?

on unix/linux I just use bc -l

~~~
Aardwolf
bc is a pretty misleading calculator, without specifying rounding it thinks
2^(40/1.5) is 67108864.

It's not even close.

If you trust your finances to that, you could get in trouble.

~~~
keithpeter
The man speaks the truth, but I do get a warning

    
    
       2.0^(40.0/1.5)
       Runtime warning (func=(main), adr=18): non-zero scale in exponent
       67108864.00000000000000000000
    

Apparently the ^ function only works with integer powers. The exponential and
natural log functions work as expected

<http://www.johndcook.com/blog/2010/07/14/bc-math-library/>

    
    
       keith@xeon4:~$ bc -l
       e(l(2)*(40.0/1.5))
       106528681.30999083085437360438
    

nice example.

------
notemouse
Don't try to use this method to do mind calculation, it may cause senile
dementia.

------
est
nice trick for numbers with each digit smaller than 5

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Aardwolf
I think only smaller than 4...

I wonder if this works in binary :D

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3chelon
That was my first thought, so I checked, and yes, it does. You just change the
carrying rules. But by the time you've accounted for the number of digits
you'd need to do anything useful, it's fairly useless except from a
theoretical computational perspective - even when converted to logic circuits,
I'm guessing it's not the most efficient solution out there.

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tuxdev
Actually, this method is one of the most efficient ways to multiply in
software. It's called the "Comba" method in that context, and is efficient for
multiplying 32x32 words or smaller, due to cache effects and function overhead
required by more complicated algorithms. Over about 32 words on many
architectures, the algorithmic advantage of the Karatsuba method wins out.

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Aardwolf
It doesn't work for 25x25, so what's the point?

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leddt
It does work for 25x25.

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Aardwolf
You're right, I see it now, it represents 400 + 20*10 + 25 which is indeed
625. They just avoided those cases in the article and only showed those where
the number of crossing neatly formed the digits.

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tylermauthe
Now, using this new technique, attempt to solve this problem

4x + 2x^2 + 5 = 100

:)

