25+ Ways to Multiply
This is a summary of the different ways to multiply, with links to descriptions. There are also several descriptions on Pat Ballew’s blog.
Procedures:
- Repeated Addition [listed as #24]
- On the fly shortcuts to Addition [listed as #25]
- Doubling and Halving (Egyptian, Ethiopian?, Russian?) [described here and also listed as #1]
- Doubling and Adding or Duplation (Egyptian) [described as #2]
- Shift and Add [listed as #19]
- Grid or Lattice Multiplication (Arabic, Indian?) [described as #5]
- A variation on Grid Multiplication (medieval Italian or earlier) [described as #5½]
- “Traditional” or Long Multiplication (medieval Italian or earlier) [described as #6]
- Crocetta, or Vertically and Crosswise Multiplication (Indian) [described here, and also listed as #10]
- Digit-reverse and shift, by Pappus (Greek) [described here on Pat’sBlog, and also listed as #21]
- The Method of the Cups (Spain or the Americas?) [described here, and also listed as #13]
- per Repiego, or multiplying by factors (medieval Italy or earlier) [listed as #20]
- Drawing lines like on that YouTube video [shown here as #22]
Formulas and Tables:
- Babylonian(?) (First Formula) [described as #3]
- Babylonian(?) (Second Formula) [described as #4]
- A third difference of squares (Greek) [described as #26 but really just a variation of #4]
- Prosthaphaeresis or Trig Tables (Europe) [described here as #14]
- Logarithm Tables (Europe) [described here as #15]
Physical Objects:
- Napier’s Rods (Scotland) [described here as #8]
- Genaille-Lucas Rulers (France) [decribed here as #9]
- Gunter Scale [described here as #17 with a picture here]
- Slide Rule [decribed here as #16]
- Abacus (Chinese) [described here as #18]
- “Prosthaphaeretic” Slide Rule (based on similar triangles) [described here as #23]
Other Methods:
- Similar Triangles (Greek) [described here as #7]
- Finger Multiplication by 9 [described here as the first of three methods and listed here as #11] and other multiplication tricks for multiplying by a specific digit
- Finger Multiplication between {5, 6, 7, 8, 9, or 10} (medieval Europe?)[described here as the second of three methods and listed here as #12]
- Finger Multiplication between {10, 11, 12, 13, 14, or 15} (medieval Europe?) [described here as the third of three methods and listed here as #12]
360 
February 5, 2010 at 4:36 pm |
you dont have square way???
November 8, 2010 at 4:56 pm |
yea i know square way hahaha its easy!
February 6, 2010 at 12:58 pm |
I’m not sure of what you mean by the square way — the one that comes to mind is the Babylonian formulas. Would you be able to describe the method you’re thinking of? I’d be thrilled to add more to the list!
February 6, 2010 at 6:52 pm |
“The square way” might also be a reference to grid/lattice multiplication…
November 18, 2014 at 6:11 pm |
Square way is very hard
March 29, 2010 at 11:33 am |
[…] “lattice multiplication if you want to”, or any of the other methods. I am quite interested in what sort of drawing would explain the algorithm for long multiplication; […]
May 19, 2010 at 11:16 am |
What a neat list!
Check out Vedic multiplication. It’s similar to 5-10 finger method in nature, but uses a different notation structure.
May 20, 2010 at 11:15 am |
This is a great list… thanks for posting it!
May 22, 2010 at 6:45 pm |
[…] head in seconds. Some numbers you can multiply in seconds by criss-crossing lines. So far at least 25 different methods of multiplication have been identified. Maybe the list will be infinite – as many patterns as the mind can […]
May 23, 2010 at 7:32 am |
Maria: “Vedic multiplication” is in the first group (Procedures), under the names Crocetta and “vertically and crosswise” multiplication.
July 29, 2010 at 12:21 pm |
You don’t have this way…
http://www.facebook.com/video/video.php?v=234912670345
August 1, 2010 at 8:05 pm |
Shyam, I think that’s the same as “Drawing Lines” [the last one under Procedures]. It’d be a challenge for large digits, but I think it’s really neat for small ones.
September 26, 2010 at 1:28 pm |
Don’t know how many you are up to, but I failed to mention one that is used as a way of creating a Prime Sieve You can see an example here
September 26, 2010 at 1:29 pm |
Sorry, entered too soon… This is just a type of nomograph that uses a parabola…There are lots of them for most every mathematical operation…
April 24, 2011 at 2:44 pm |
Egyptian multiplication is slightly different from Russian Peasants’:
http://www.cut-the-knot.org/Curriculum/Algebra/EgyptianMultiplication.shtml
April 24, 2011 at 4:46 pm |
Thanks Alexander (for this and your other multiplication links today)! My impression had been that there were two Egyptian methods, one of which was the same as what is called Russian multiplication, but I haven’t looked at the original sources. [I think there’s a copy of the Ahmes/Rhind Papyrus at one of the libraries in town, so once finals are done I ought to be able to verify one way or the other so that I can at least be accurate in my future teaching.]
September 21, 2011 at 3:11 am |
how the heck did you come up with this stuff
September 21, 2011 at 3:21 am |
you do know there are really 26 since you put 5 and a half
September 21, 2011 at 4:17 am |
I think that there are probably more than this, since there are a lot of places where we don’t know how the people used to multiply. This is one of my favorite topics. 🙂
September 24, 2011 at 12:10 pm |
I would like to add one more way to multiply: See here:@ http://wpgaurav.wordpress.com/2011/08/31/do-you-multiply-this-way/
September 24, 2011 at 11:49 pm |
I forgot to mention the Schönhage-Strassen algorithm for multiplying extremely large numbers using the Discrete Fourier Transform (DFT). Once a mathematical curiosity, it is now in widespread use on computers as part of the Great Internet Mersenne Prime search, which tests numbers of a specific form with millions of digits to see whether they are primes. It has found 13 Mersenne primes in 15 years.
There are numerous other digital techniques for integers of 32, 64, or some other modest numbers of bits, and for floating point numbers in various sizes, including many implemented on integrated circuits in signal processors and microprocessors. Some can be found in various computer design textbooks, such as Digital Systems: Hardware Organization and Design, by Frederick J. Hill and Gerald R. Peterson.
Representing integers in forms other than decimal notation permits many other methods. For example, integers represented by their prime factors can be multiplied just by counting up how many of each prime factor there are in each number to be multiplied. As a trivial example, 32 is 2⁵, and 64 is 2⁶, so their product is 2¹¹. This representation makes addition much more difficult, however.
Several methods, such as Egyptian and Russian peasant multiplication, are essentially equivalent to various forms of multiplication for binary numbers.
September 25, 2011 at 7:49 am |
Gaurav, I think that’s the same as Crocetta, which is already in the list — I liked how you color coded the lines in a single pattern, though.
mokurai, I’ve wondered about whether or not to include multiplication in different bases as each being different. It is in a way, though, and I’ll look at the book you mentioned — thanks!
August 8, 2012 at 3:04 am |
Thanks a lot for this quite a long list of very useful links! They really helped me today in my work.
January 7, 2013 at 1:26 pm |
I suggest you do a Google search on “geometric compass Galileo”
I think it overlaps here with a couple methods above but I think it should be mentioned somewhere in the list.
January 7, 2013 at 1:57 pm |
I just wanted to add that the geometric compass of Galileo is also known under English Sector or French Sector like this:
http://sliderulemuseum.com/Rarities.htm
March 27, 2013 at 3:46 am |
very nice
April 3, 2013 at 10:24 pm |
I am trying to find the name and book of a Naval architect who was throw in concentration camp in WW 2 who devised and later on wrote the book on alternate method of calculations the technique could beat a calculator. Can someone help.
July 27, 2014 at 8:15 am |
I read about this method (and the person who came up with it) in the 1950s while in a barbershop waiting for my haircut. Have thought about it frequently in recent years as the math-teaching discussion continues.
March 8, 2014 at 10:24 am |
Thanks for posting it was really helpful but do you have the chinise method
March 8, 2014 at 12:44 pm |
I don’t think so, boy, unless you mean on an abacus. What method are you thinking of?
August 21, 2014 at 6:24 am |
Well I m really confused with all those methods , what I m looking for , if some one can help , is ..another 5 methods NOT given in the 25 list….. Anyone please ????? Thanks
September 20, 2015 at 8:37 am |
[…] https://threesixty360.wordpress.com/25-ways-to-multiply/ […]
February 22, 2021 at 7:34 am |
[…] In #9, Heather Lewis began to explore The First Bunch of Ways to Multiply. [You can find the whole list at 25+ Ways to Multiply.] […]