The Second Bunch of Ways to Multiply

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Napier 1Hey, let’s do some more multiplication!   (See Ways (1)(6) here.)  I promised some that wouldn’t be taught in school (at least not as a practical way to multiply), so that’s where I’ll start.

(7) Greek Multiplication.  This is really different from other kinds of multiplication because it’s based on Geometry.  The Ancient Greeks certainly had the concept of discrete quantities, but in a lot of their mathematics numbers were interpreted as lengths; to multiply a and b like an Ancient Greek, you start with the quantities a and b and also the reference point 1.

Draw two intersecting lines and, from the point of intersection mark lengths 1 and b on one line and length a on the other line, and then draw a line between 1 and a.

Greek Mult 1Starting from b, draw a line that is parallel to the line you just drew.  (I could really have said “construct” because all of this can be done with a straightedge and compass.)

Greek Mult 2

See how the length x is where that line you just drew intersects the line with a?  It turns out that x is the length of the product ab!    It’s actually pretty easy to show this:  the triangle with sides x and b is similar to the triangle with sides a and 1 [they have one angle in common, and because their third side are parallel the remaining pairs of angles are equal as well].  Since they are similar, x/b must equal a/1, and cross multiplication gives x equal to ab.

One nice feature of this is that you can actually see how if b<1 then the positions of 1 and b will be switched, and the product ab will be less than a.

(8) Napier Rods.  These were first published in 1617 in the Rabdologia by John Napier, a Scottish mathematician who is more famous for introducing the idea of logarithms.  Napier Rods, which I just realized are supposed to be called Napier‘s Rods (or Napier’s Bones) are like a portable version of Grid Multiplication, so if they were allowed on Standardized Testing it might take care of that whole time issue.  (Then again, one could ask why these would be allowed and not, say, a calculator.  So I guess this also falls under the cool just because it’s cool category of multiplication.)

Godzilla is here to show how they work.  He’s going to demonstrate how to multiply 3558  by 274.

Start by getting yourself  some Napier Rods.

Napier 1

Do you need to see those up close?  Here they are:

Napier 2

Those are still hard to see aren’t they?  That’s because I don’t know how to focus my camera.  And THAT’S why I love Wikipedia, folks.  Here’s a drawing that (someone?  I can’t tell who) posted on Wiki under GNU-FDL.

Bones_of_Napier_(board_and_rods)

So each rod (or bone) has a digit on top, and the multiples of that digit are written underneath.

Since Godzilla wants to multiply 3558 by 274, we’ll start by picking up rods for 3, 5, 5, and 8 and lining them up.  [I printed out two copies of this set of Napier rods since the digit 5 appeared more than once.]  There’s a rod with nothing at the top:  this is called the Index Rod, and it can be put at the right or left.  It just helps you keep track of the rows.

Napier 3

We’re going to multiply this number by 274, so we’ll start by looking in Row 2.

Napier 4a

See how that looks like a little tiny grid from lattice multiplication?  If you add along the diagonals, it shows that 3558×2=07116.

Now let’s multiply by the 7 of 274.  We’ll look in Row 7.

Napier 4b

This shows us (after adding along the diagonals) that 3558×7=24906.  We had to do a little carrying here.

Finally, we’ll multiply 3558 by the 4 of 274.  Look in Row 4:

Napier 4c

So 3558×4=14232.

Now that we have all the pieces we need, we add, keeping track of the place value (by staggering on the left):

Napier Sum

Yay!  We have our answer!  If you prefer to play with a java version that allows you to switch the base, you can find it here at Cut The Knot.

(9) Genaille-Lucas Rulers.  These take the basic concept of the Napier Rods and modify it using shaded triangles in a way that completely eliminates adding and carrying within each row (though you’ll still have to add to get the total at the very end).  They were invented by Frenchmen Henri Genaille and Edouaird Lucas just over a century ago.  Godzilla is using paper versions that Brian Borchers created.

Genaille 1

He’s already lined up the 3558 to do the same multiplication (3558×274) as before.  Let’s take a closer look.  Notice that with this set, the Index piece is aligned on the left.

Genaille 2

OK, we’re going to use the same basic idea as before, where we’ll look in rows 2, 7, and 4.  Let’s start with Row 2:

Genaille 2 row

This row should give us 3558×2.  To read it, start on the RIGHT with the number on the very top.  That’s a 6.  Then you read towards the LEFT, following the little gray triangles.  The final number is 7116.

Now we’ll look at 3558×7:

Genaille 7 row

Again, you start with the number in the TOP of the right-hand column (which is a 6).  Follow the little gray triangles to the next number, to get that 3558×7=24906.

Finally, here is Row 4:

Genaille 4 row

As before when we start from the top of the rightmost column and read to the left we see that 3558×4=14232.

This is a completely mindless way to multiply until you get to the end, where you align your answers (staggered on the right):

Napier Sum

And we have the same answer as before!  And right here is a picture of some of the rods from 1885 in France.  (They and Napier’s Bones were actually more like square prisms, with a different Ruler printed on each side so that each Rod could show one of four different sides depending on what digits you need.)

This post is already rather long, so I’ll end by briefly mentioning three final methods:

(10) Vertically and Crosswise Multiplication (which TwoPi wrote about just over a year ago).  TwoPi tells me that it should probably be called Trachtenberg multiplication, since it was created by the Ukrainian engineer Jakow Trachtenberg while he was imprisioned in a concentration camp in WWII.  (There are widespread claims that this technique is of Vedic origins, but there is little historical evidence to support that.)

Edited to add: Whoops, Trachtenberg might have come up with it on his own, but he wasn’t the first.  It was published in Italy over 500 years ago and was known as Crocetta.  (See Pat’s comment below.)

Edited again (7/8) to add: Looking in “Capitalism and Arithmetic”, I saw that this is attributed even further back, to the Indian Lilavati of around 1150.

(11) and (12) Speaking of multiplication we’ve already talked about, in this post from New Year’s Eve 2007 I wrote about multiplying by 9 on your fingers (I’ll call that #11) and also two Medieval ways to multiply on your fingers.  They’re pretty similar, but one works for multiplying any two of {5, 6, 7, 8, 9, 10} and the other works for multiplying any two of {10, 11, 12, 13, 14, 15}.  Together I’ll count those as #12.

That’s it for now!  I can think of three more ways offhand (abacus, slide rule, and trig functions) so I’ll write up something about those in the next couple days.  This is fun!

16 Responses to “The Second Bunch of Ways to Multiply”

  1. David Petersen Says:

    If you’re just trying to get to 25, you could include calculator/computer. I know a little bit about code, but I don’t know the algorithms that calculators use to multiply. Do they have some “times tables” stored and look up certain results? Do they just add over and over? I’ve always wondered.

  2. pat ballew Says:

    I’m pretty sure Tractenberg didn’t invent the method you mention in 9) . F. Swetz in “Captialism and Arithmetic” lists this as one of eight methods shown by Pacioli in his 1494 work. The Italian name was per Crocetta, literally, “by the cross”. Swetz mentions it as a candidate for the origin of the X for multiplication since it was common to draw the “cross” between the numbers. It was also sometimes called the Pigeonhole (I assume that was by people who used the vertical lines in the figure as well.

    By the way, if you need it to get to your 13 ways, he also mentions “per repiegio”, factor one number and multiply the object by the factors and add the results… (not too handy for two pirmes) I actually have a couple of historical methods at http://www.pballew.net/old_mult.htm

    Thanks for the interesting historical notes…

  3. Ξ Says:

    Thanks for the correction Pat — I love learning new ways, so between your suggestions and David’s maybe I’ll get to 25. 🙂

    [Oh, and before I turned on my computer this morning I suddenly thought “Did I list (9) two times?” In fact I had, so I corrected the second (9) to (10).]

  4. Jason Dyer Says:

    There’s that geometric method that’s been floating about the Internet and posted about here.

    I vaguely recall a few more but I need to rummage around my books to be sure.

  5. Jason Dyer Says:

    Multiplication by bitwise shifts.

    The idea is if you have the number 0101 then 1010 is the same number multiplied by 2. Addition can compensate when multiplying by numbers that aren’t powers of 2.

  6. Jason Dyer Says:

    While some “mental math” methods are too specific to qualify as new ways to multiply, I’d say “compensation” is general enough. That is:

    49 x 11 = (50 x 11) – (1 x 11) = 550 – 11 = 539

    34 x 27 = (30 x 27) + (4 x 27) = 810 + 108 = 918

    The second example is just standard multiplication, but the first one compensates by subtraction rather than addition to round to the nearest 10.

  7. jedward706 Says:

    seems to me that I recall playing with roman numerals with some middle schoolers and using “stones” and a board to manipulate and multiply with roman numerals — basically a bunch of quick “grouping” — you know 10 stones on this “V” line make one stone on the “L” line

    my understanding was that Roman merchants had and used these boards

  8. Jason Dyer Says:

    Ok, I guess I should have checked Wikipedia specifically on multiplication first on this. Not only does it mention the shift and add but it goes into 5 more computer algorithms. Combined with all the ones listed above (I found at least two new ones at Pat’s link) we’re up to 25.

  9. Ways to multiply « The Math Less Traveled Says:

    […] at 360 has a couple of posts (with more to come!) listing a bunch of different algorithms for performing multiplication. Did you […]

  10. pat ballew Says:

    Ok, and not to be pedantic, but your mention of Napier’s Bones suggest that he presented logarithms in the same book as the rods, Rabdologia. In fact I believe “Mirifici logarithmorum canonis descriptio”, which introduced logarithms, was published in 1614. Rabdologia was not published until 1617. By then, I believe Briggs had already set out the common logarithms after his famous meeting with Napier in 1615, where, so the story goes, “almost one quarter of an hour was spent, each beholding other with admiration, before one word was spoken. At last Mr Briggs began, -‘My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, viz. the Logarithms”
    I recently came across an article which I seem to have misplaced, that says his Rods were used into the 1960’s in many places… Now where is that thing… I know I had it… hmmmm

    oh well…

  11. Ξ Says:

    Dang, Pat, I thought I checked that. Goes to show what happens when you’re more focused on taking pictures out on the porch than on making sure the details are correct. I changed the wording a little to make it correct. [And BTW, I really appreciate the corrections — it’s like having spinach in your teeth: you’d rather you didn’t, but if you did then it’s much better if someone tells you than for you to walk around all day with it!]

    jedwards, I’ve seen pictures of those abaci but I don’t know how to use them. Or, rather, one could use them the same way as a Chinese or Japanese abacus, but I don’t know if there were difference in how the Romans did calculations.

    Jason, Yay about collectively getting to 25! So I’m guessing there will be at least two more blog posts after this one.

  12. pat ballew Says:

    Just a couple more ways to add to your list…. before there was a slide rule, the mechanical method of adding logarithms was accomplished by Edmund Gunter called Gunter’s scale… an image of the scale is here

    Another way is the use of nomographs (sometimes called nomograms) which are physical graphs accompanied by a straightedge…
    The property at work uses a graph of y=x^2 … and employs the simple technique that the line segment containing two points on y=x^2 at x=-a and x=b will have a y-intercept at (0, ab)
    The algebraic justification is within the reach of any good pre-calc student and is left (as is the mathematical tradition) as an exercise for the reader.

  13. Of Ethopian and Egyptian Multiplication « Technical Musings of a Curious Mind Says:

    […] reasoning about some of the other ways in that post and the next. Makes […]

  14. apoorva anand Says:

    if the bone have 3,4,5,6 digits or numbers how should we write.

  15. Ξ Says:

    I’m not quite sure what you mean by 3, 4, 5, 6 digits on them. In the example I was multiplying a 4-digit number by a 3-digit number: essentially you just do one digit at a time (the 4-digit number times one digit at a time) and then add them. The bones don’t need to have more than one digit on them, because you put them together [though you might need several sets if your number has several of the same digit].

  16. Michel Mouyssinat Says:

    you can visit my web site and the exibition ‘homo calculus’ (search with google) Ihe history of calculus.
    The ‘kit calculus’ is used to compute with Neper Bones and Genaille’s rulers. It is very easy to use for children.

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