Archive for the ‘Multiplication’ Category

A 26th way to multiply. Sort of.

August 2, 2009

plastic_numbersI’m moving offices:   just next door, but I have about 20% less shelf space so I figured it was a good opportunity to see if there were any books that I didn’t really use (answer:  yes).  In looking through one (Man and Number by Donald Smeltzer, which is a distracting title even though I can see that this book was published 50 years ago) I found another way to multiply!  It’s not anything dramatic, but was apparently described by Nichomachus of Alexandria over 1900 years ago in his Introduction to Arithmetic.  It uses the following fact:

x^2 - y^2 = (x+y) \cdot (x-y)

Here’s the method:  suppose first that a and b are both odd or both even.  Let x be the average of the two numbers (so x is a whole number, because their sum must be even) and let y be the positive difference between x and a or b.  The radius, as it were.  Then:

a \cdot b = (x+y) \cdot (x-y) = x^2 - y^2

The example given in the book is:

24 \cdot 14 = (19+5) \cdot (19-5) = 19^2 - 5^2 = 361-25=336

You still have to know your perfect squares, but if you happen to have a table of squares like the Babylonians, that’s no trouble at all.    (Indeed, this is basically Formula (4) here,  so I don’t know if this should count at all.  But I like having an actual citation for the method.)

But what if you have an even and an odd number?  Never fear, just ignore that pesky odd bit and add it on at the end.  For example, if you have 24·15, you know that this is 24·14 plus an additional 24.  So you find 24·14 as above, and add 24 to get 360.

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Seven More Ways to Multiply

July 8, 2009

Prosthaphaeretic_Slide_Rule2.svgThis section should be entitled “Ways Other People told me about” because it pretty much comes from other people’s recent comments, with a couple of extras thrown in.  But the best part?  We’re up to 25 ways!

In an effort to appear organized, I created a single page with links to all of the ways to multiply.  If anyone writes up any additional ways, post a link in the comments and I’ll add them.

On to the final(?) ways!

(19) Shift and Add, described by Rick Regan of Exploring Binary.  From the comments of Ethiopian Multiplication:

14 is 1110 in binary and 12 is 1100. Multiply the two in binary: 1100 x 1110 (think of 1110 as being on the bottom — I can’t draw it that way since the formatting won’t work). The partial products, in order, are 0000 (0), 11000 (24), 110000 (48), 1100000 (96). The nonzero products are copies of the top number, 12, shifted left — doubled — an appropriate number of times. Adding the partial products gives 10101000 (168).

(This is also mentioned by Jason Dyer in these comments.)

(20) per Repiego, described by Pat Ballew of Pat’sBlog in the comments here.  This is listed by Pacioli in the Trevisio, and is essentially breaking a number into factors and then mulitplying by each of those in turn.  So to multiply 14 by 12, for example, you might multiply 14 by 2 (getting 28), then multiply that product by 2 (getting 56), then multiplying THAT by 3, to get a product of 168.

(21) This method from Pappus, also described by Pat Ballew in this article.  It starts off:

I will instead use 257 x 62 to shorten the process.

Write the first number on a piece of paper, and then the second number should be written backwards on a seperate piece of paper.
Align the second sheet under the first as shown below, and multiply only the numbers aligned vertically (the 7 and 2) to get 14. Record the four, and keep the one that is to carry in the mind (this is the mental part referred to in the title):
_______257
_________26
_______ 4

The bottom number is shifted, more pairs are multiplied and added, and you can read about the rest of the process here.

(22) This method from YouTube, which Jason Dyer of The Number Warrior pointed out in this post and in the comments here.    It’s like a visual depiction of grid multiplication.  It’s listed as Mayan multiplication, but I think that’s likely to be false:  In Victor J. Katz’s A History of Mathematics he says that the Mayan documents that were not destroyed don’t show how the Mayans did calculation.  (Hmmm.  Katz also says the Babylonians used tables, and doesn’t mention those formulas I mentioned earlier.)

(23) This is described as the Prosthaphaeretic Slide Rule in this article (brought to my attention in these comments by Jason Dyer), but it really doesn’t use prosthaphaeresis at all.  Rather, it’s a physical item that creates the similar triangles that the Greeks used for multiplication.Prosthaphaeretic_Slide_Rule2.svg

(24) Repeated Addition.  Simple enough.  It only works for integers, but that’s true of several of the other methods as well.

(25) On the fly shortcuts to Repeated Addition.  For example, Jason Dyer’s example:

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

And we’re at 25 ways!   Hooray!

I will instead use 257 x 62 to shorten the process.

Write the first number on a piece of paper, and then the second number should be written backwards on a seperate piece of paper.
Align the second sheet under the first as shown below, and multiply only the numbers aligned vertically (the 7 and 2) to get 14. Record the four, and keep the one that is to carry in the mind (this is the mental part referred to in the title):
_______257
_________26
_______ 4

Multiplying on an Abacus

June 25, 2009

Mult FinishedContinuing with different ways to multiply, here’s just the one

(18) The Abacus.  To start, you need to know how to write numbers on the Chinese abacus.  Numbers are written with the 1s column on the far right, then the 10s column, then the 100s column, etc.  and you only look at which beads are touching the center bar.  Beads on the top count as 5, while beads on the bottom count as 1.  Here’s the number 358:

358

Notice that you could get by with each rod only having 1 bead on the top and 4 on the bottom.  Some have that; it’s called a 1/4 abacus, and I believe it’s been popular in Japan for about 60 or 70 years.  It’s called a soroban.  The 2/5 abacus (which is what I have) is used in China, and it’s called a Saun-pan.  It just makes certain carrying easier, plus it’s what I could find.

On to adding!  Some of this is easy:  If you have a number like 358 and you want to add 100, you just move one of the 100 beads [third column from the right].

458

But what if it’s not that simple?  If you have a number like 358 and you want to add 18, you could do it bead by bead (Slow.   Boring.  Lots of regrouping.)  or you could just add 2 tens and then remove 2 ones.  That’s the more authentic way.

376

Likewise, if you had 358 and you wanted to add 4, you could add a five bead and subtract 1 unit, then regroup, or you could add 1 ten bead and subtract the five and a one bead.    [Is that even clear?  You can find java versions online to follow along, like this one although it uses two decimal places on the far right instead of starting with the 1s.]

362

On to multiplication!  For single digits, well, you pretty much just need to know all your basic multiplication facts.  No way around it.  But this process will work for anything beyond that.

Let’s find 87×625.  Pick one of the numbers (87) and put it on the far left-hand side of the abacus.  Since 87 is a 2 digit number, we want to leave 3 blank spaces that’s (2+1) on the far right, and we’ll put 625 after that (where by “after” I guess I mean “before”, to the left).  Just look at the picture and it should be clear.

Mult Start

(Pause.)  Before we get started on the process, it’s helpful to take a look at the big picture.  We’re going to put our answer on the far right, which is why we needed those blank spaces.  But our answer will be more than 3 digits long, so we’re going to need more space.  We’ll create the extra space as we move along.  We’ll first multiply 87 by the 5 of 625 then we’ll get rid of that 5, freeing up an extra column.  then we’ll multiply 87 by the 2 of 625, freeing up yet another column, and finally we’ll multiply the 87 by the 6.

Let’s get started.

We want to multiply 87 by the 5.  We’ll multiply the digit 8 and then the digit 7 [left to right].  Yes, the is backwards from how we’re multiplying 625 [right to left], but that just keeps it fun.   Whenever we multiply by the 8 of 87 we’ll put the answer two columns over, and whenever we multiply by the 7 of 87 we’ll put the answer three columns over.   [No matter how many digits that number is, when you multiply by the largest place-value you put the answer two columns over; the next place value goes three columns over; the next would go four columns over, etc.]

So to start, we know that 8×5 is 40.  Since we’re using the 8, we’ll put the answer two columns over from the 5.  We always use two for the left-most column.

Mult step 1

Now we’ll move to 7×5.  This is 35, and we put the answer three columns over from the 5.

Mult Step 2 Since we’re done multiplying 87 by 5, we can remove the 5.

Mult Step 3

Notice the number on the far right, 435, is just 87×5.

We’ll do the same thing, multiplying 87 by 2.  First we find 8×2 and put that answer two columns over from the 2.  [I’m going to do that by adding 20 instead of 16, and then subtracting 4.]

Mult Step 4

Then we’ll multiply 7×2 and put that answer three columns over from the 2.  [Here I added 15 instead of 14, and then I subtracted 1.]

Mult Step 5

We’re done multiplying by 2, so we can remove it.  Except I forgot to take a picture with it removed.  If you ignore that 2 of 62 (formerly 625), you can see that the number on the far right is now 2175.  Sure enough, this is 87×25.

Nearly done, we’ll multiply the 8 of 87 by 6 and put the answer two columns over from 6.  Notice that this 6 originally stood for 600 (in 625), but we don’t need to keep track of which place value the 6 originally had.  By always putting the single-digit products two or three (or four for longer numbers) over, the place value takes care of itself.

Mult Step 6

And finally, we multiply the 7 of 87 by the 6 and put the answer three columns over from 6.

Mult Step 7

And remove the 6 because we’re done multiplying by it.  Here’s the final answer:

Mult Finished

Yes, it’s 54,375.  And a quick check shows that   I made a mistake so I had to redo all of the photos I took, and then I made another mistake and had to retake more photos, but on the third time It’s right!  And despite how long it takes to explain, it really isn’t too bad to do by hand even if you’re learning it for the first time.

The Fourth Bunch of Ways to Multiply

June 17, 2009

slide-0Just three more ways today, although with all the ones that have been suggested I think we’ll get to the 25!

(15) Multiplication with Log Tables.  People had already been using trig tables to multiply, but when logarithms were discovered they became THE way to multiply numbers.  The idea behind log tables is that logarithms turn multiplication (nasty) into addition (fun!) without having to derive a bunch off trig formulas.  In particular, we’re going to use the fact that log(x·y)=log(x)+log(y).

Here’s how we can find 875×978 with logarithm tables.  We’ll start by writing the two numbers in Scientific Notation:  875 would become 8.75×102 and 978 would become 9.78×102.   We’re going to multiply 8.75 and 9.78, and then adjust that product by the appropriate power of 10 (in this case, 104).

The next step for the multiplication is to look in your Table of Common [Base 10] Logarithms.  If you don’t have a copy handy, you can look here.  It turns out that log(8.75) is 0.9420081 [that’s our log(x)] and log(9.78) is 0.9903389 [that’s our log(y)].

Now we’ll add those together to get 1.9323470.  This must be the log of our product!  So we work backwards with the table, looking for the number whose log is 1.932347.  Unfortunately, the table only gives results between 0 and 1, so initially it seems like we’re stymied, but we can be sneaky and subtract 1.  We’ll come back to that in a moment.

So now we’re looking for a number whose log is 0.932347.

Looking back at our table, we see that log(8.55)=0.9319661 and log(8.56)=0.9324738, so the number we’re looking for must be between 8.55 and 8.56.  We can pick 8.56, which is the closer number, or do a little interpolation.  If we round, we’re looking for 0.9323 and instead we got 0.9320 and 0.9325.  The number we wanted was about 60% of the way from the smaller to the larger, and so instead of choosing 8.55 or 8.56 we could go about 60% of the distance between them, and guess that the product was 8.556.  Let’s do that.  [We could be even more accurate if we used a calculator to figure the exact percentage, but that seems to defeat the purpose of using the log table to multiply.]

So now we have a product of 8.556, but clearly that’s not exactly right.  We first have to account for the fact that we subtracted 1.  Notice that 1+log(blah) is the same as log(10)+log(blah) [because we’re using the common logarithm], and THAT is the same as log(10·blah).  Here log(blah) is 0.9323470, and we just saw that blah was approximately 8.556, so the product that we want — the product of 8.75 and 9.78 — is approximately 85.56.

We’re almost there!  Remember how originally we wanted the product of 875 and 978 but we first wrote those in Scientific Notation?  We need to adjust our answer of 85.56 by 104, leading us to the conclusion that 875×978 is approximately  (drum roll please) 855600.  The correct answer is 855750, so we’re certainly in the right ballpark and more accurate than we were with the trig tables, but, well, it’s a wonder to me that this method was so popular for so long when so many other ways are even more accurate.  Maybe I’m missing something.

(I’ve multiplied with log tables before, but refreshed my memory with this site on the Obsolete Skills Wiki, which also explains how to Get off the couch to Change Channels on the TV set and how to Make Change in [old] Shilling and Pence.)

(16) Slide Rule.  This takes the ideas of the log table, but bypasses the actual looking up.  Instead, the numbers are scaled on the slide rule in such a way that you don’t have to do much at all.  William Oughtred is credited in many places (online) with the slide rule, and the date 1622 shows up, so this happened just a few years after the invention of logarithms.  Pretty quick thinking!

Let’s do a simple example first:  2.5×3.   You can use an actual slide rule, or use the java version here.    The slide rule looks complicated because it can do a lot of things, but we’re mostly going to be looking at the bottom of the slidey part in the middle (C) and the fixed part at the bottom (D).

To multiply 2.5 by 3, you start with2.5  and align the 1 on C with the 2.5 on D.  Then look for 3 on C, and right below it will be the product (in this case, 7.5):

See how once you’ve aligned the 1 with the first factor 2.5, you can move the slider doohickey to the 3 on C so that you can see what’s right below it?

Now let’s look at a more complicated example:  back to 875×978.  As with log tables, we’ll start by writing the numbers in scientific notation (8.75×102 and 9.78×102) and then we’ll multiply 8.75 and 9.78 and adjust our final answer.

Normally we’d align the 1 of C with 8.75 beneath it on D, and then start looking to the right.  But 8.75 is so large, we quickly run out of room:

Slide 1

So we’re sneaky, and instead of aligning the 1 (on C) with the 8.75, we align 10 (on C) with 8.75.  We’ll have to adjust by multiplying by an extra power of 10 at the end, kind of like we did with the log tables before (and, really, it’s not a coincidence that this happened in both calculations).

So we’ve aligned the 10 on C with the 8.75 on the bottom.  Then we move the slider over to 9.78 on C, and look below to see the product!  The product looks like, ummm, it seems to be a bit over 8.55 but not yet at 8.56, so we’ll say 8.555.

As our final step, we need to multiply by 10 because we aligned with the 10 instead of with the 1 (as discussed above), and also by 104 because we’d had to write the numbers in scientific notation.  This means our final answer is 8.555×105, or 855500.  As we’ve seen before, it’s not exact but it is pretty close and it’s a lot faster than looking up in tables!

(17) The Gunter scale.  This is the precursor to the slide rule, invented by Edmund Gunter, and it didn’t slide at all but it was BIG:  two feet long was standard.  It was a big ole piece of wood with a logarithm scale on it, and if you wanted to measure a product like 2.5×3 you’d measure the physical distances to 2.5 and to 3, add them together, and see where you ended up on the scale. Essentially it was a Slide Rule that didn’t slide:  you had to do that part by hand.

(There’s a picture here and more information about Gunter here, including the fact that he coined the terms for Cosine and Cotangent, though Cosecant is a bit older.  I hadn’t known about this method until Pat Ballew brought it up in a recent comment:  thanks Pat!)

The Third Bunch of Ways to Multiply

June 16, 2009

Let’s multiply!  (The First Bunch had Egyptian, Babylonian, lattice, and current methods, and the Second Bunch had Greek, Napier and Genaille-Lucas, plus links to Vertically and Crosswise Multiplication and finger multiplication.)

We’ll just do two more today, starting with one I forgot we’d already talked about:

(13) The Method of the Cups, which was described by Friar Juan Diez in 1556 (see  this post from last year).  It’s done digit by digit, but multi-digit numbers aren’t always written on a single line: if you want a glimpse of what it’s like, here’s the image for 875×978.

1556 Multiplication by Friar Diez

(14) Prosthaphaeresis, or Multiplication with Trig Tables. This method dates back to the late 1500s, although logarithms essentially rendered this method obsolete.

Here’s the basic idea:  remember the sum and difference formulas for sine and cosine?  We’ll use those.  For example, since:

cos(a+b)=cos(a)cos(b)-sin(a)sin(b)

and

cos(a-b)=cos(a)cos(b)+sin(a)sin(b)

then by adding those together, we get

cos(a+b)+cos(a-b) = 2cos(a)cos(b)

which leads to the formula

cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)

And that’s the formula that we’ll use, although by also looking at sine we could just have easily come up with a different formula that had a mixture of sines and cosines.    When the last step of dividing by 2 is ignored, the resulting formulas are called the Werner formulas.

Here’s how we’ll do it.  Suppose we want to multiply 875 and 978, because those were the numbers that appeared in the previous example.  We’re going to be using a table of cosine values (like this one).  This table actually shows both sine and cosine, but cosine is read on the right.

Trig 0

Notice that 0.9997 is cos(1.4°), for example — we’re adding 0.2° to the angle for each column, going right to left.

Cosine values always fall between 0 and 1, so we need to scale our numbers to be between 0 and 1.  Thus 875 becomes 0.875 and 978 becomes 0.978, and we’ll have to multiply our final answer by 103·103=106.

Remember that our formula is cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b).  In this case, cos(a) is 0.875 and cos(b) is 0.978.  We need to find a and b.

Trig 1

Remember that we read the angles on the right.  Since 0.875 is between 0.8746 and 0.8763, our angle a must be between 28.8° and 29°.  We’ll use 29°, since that gives the best estimate.

Trig 2

Likewise, we need to find b knowing that cos(b)=0.978.  From the chart we can see that b is between 12.0° and 12.2°, but basically it’s 12°.  Yay!  We’re halfway there!

We have a and b,  so the sum a+b is 41° and the difference a-b is 17°.  Now we need to know cos(a+b) and cos(a-b).

Trig 3

This part is easy!  The cosine of 41° is just 0.7547

Trig 4

and the cosine of 17° is 0.9563.

Now we can plug all this into our formula!  Remember how 0.875 was cos(a) and 0.978 was cos(b)?  Using:

cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)

we get

0.875 \cdot 0.978 = \frac{1}{2}\cdot 0.7547 + \frac{1}{2} \cdot 0.9563

which simplifies down to 0.8555.  Multiplying that by the 106 from earlier gives 875×978 as approximately 855500.   We’re only off by 250, which feels like a lot but it’s less than ½% of the answer, so in the grand scheme of things it’s pretty good, and with more accurate tables we could have done even better.

Next up:  Log tables and slide rules!

The Second Bunch of Ways to Multiply

June 11, 2009

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!

The First Bunch of Ways to Multiply

June 10, 2009

Trevisio grid 5I blithely mentioned in yesterday’s post that I only knew about 13 ways to multiply [“only” because it would be great to write a book called Twenty-five ways to Multiply], and then Jason asked me to list them.  I was originally going to list them all, but then I started describing them which is taking a lot longer and I won’t have time to watch the next episode of Heroes on Netflix prepare for a committee meeting if I do that all today so I’ll do it in steps and you can read all about multiplication for a few days!   I’ll see if I can get Godzilla do to some demonstrations of the more complicated methods tomorrow.

These are carefully ordering according to The Order In Which I Thought of Them.

(1) Doubling and Halving, as described yesterday (used in Egypt, Ethiopia, and presumably Russia).  This is one of my favorite methods, because it’s surprising that it works.

(2) Duplation, which is a variant on the above method, in which you start with 1 and one of the the numbers and double both.  For example, to multiply 14 and 12 you’d start with 1 & 12, and double both until the left-hand column was going to be bigger than the other number, in this case 14.  That’s not at all clear, is it?  Here’s what I mean for 14×12:

1 & 12
2 & 24
4 & 48
8 & 96
16 & <— Oh, I can stop because 16 is bigger than 14.

Since 14 can be written as 8+4+2, put a little mark by those rows:
1 & 12
*2 & 24
*4 & 48
*8 & 96

and add up the corresponding numbers on the right:  96+48+24=168.  And there’s your product!  What you’re really doing is adding the appropriate doubles (96 is 8 12s, 48 is 4 12s, and 24 is 2 12s so when you add them you get 14 12s, as you wanted).  This was also used by the Ancient Egyptians, and it’s referred to on the video from yesterday.

One thing that neat about these methods is that all they use is adding, subtracting, doubling, and/or halving.   The folk who used them had to remember the process, but they didn’t have to memorize 45 separate single digit multiplication facts.

(3) and (4) Mesopotamian Multiplication.  Like the Egyptians, the Babylonians broke down multiplication into addition, subtraction, and halving.  They had one more trick, though:  they had tables that contained the squares of all the numbers from 1 to 60 [which was all they really needed, since they used a Base 60 system].   So to multiply a and b they used one of two formulas:

\frac{\left( a+b \right) ^2 - a^2-b^2}{2} or \frac{\left( a + b \right) ^2 - \left( a - b \right) ^2}{4}

In other words, to find 14×12, to use the first formula you’d look up 262, 142, and 122 in your table, subtract the last two numbers (196 and 144)  from the first one (676), and then cut that answer in half.  With the second formula you’d look up 262 and 22 in your table, take the difference, and cut that in half twice.

This method totally amuses me because although you can check the algebra to make sure it works (indeed, I usually introduce this by asking students to come up with a formula for ab that uses only addition, subtraction, doubling, halving, and a2, b2, (a+b)2, and/or (ab)2), it’s really an ancient plug-and-chug method.  You don’t have to think at all about why it works, you just do it.

Incidentally, if you walpha a number, Wolfram Alpha will give it to you in Babylonian symbols.  Here’s  34.  Bab 34But watch out: it’s sometimes wrong.

Edited 7/8 to add: Although I read in one book (I’m not sure which now; it’s been several years) that this was done by the Babylonians, Victor J. Katz’s A History of Mathematics doesn’t match that.   I’m not quite sure what to believe now.

(5) Grid or lattice multiplication.  A lot of schools are teaching this today because, although it takes a little while to draw the boxes, you do all your single digit multiplication first and then all your addition.  I’ve heard people complain about it being a new-fangled method, and I like to point out that it’s actually about a thousand years old (Doesn’t that sound snarky?  I try to say it in a friendly way.  I probably fail.).

It’s possible that it originated in India (I’ve seen sources claim this, but I think it’s not considered absolutely certain); it certainly appeared in Arabic books before it made its way to Western Europe.

Here’s how it works.  Suppose you want to multiply 345 and 12, which are conveniently the numbers that I found in this Wikipedia example.  You have a 3-digit number and a 2-digit number, so you make a 3×2 array of boxes.  Put the first number (345) on the top, and the second number (12) on the side.  One of those is the multiplier and one is the multiplicand, but I can never remember which is which.

Hindu_lattice
The picture above also shows the next step:  divide each little square in half, and then multiply each pair of digits.  For example, 5×2 is 10, so you write the 1 above the diagonal and the 0 below.

For the next step, you add along the diagonals, carrying as necessary.

Hindu_lattice_2

The reason this works is that the digaonals automatially take care of place value.  For example, look at the third diagonal, with the purple arrow (that ends up in the hundreds place).  It has the 4 of 4×1, which was really 40×10=400.  It has the 0 in the tens place of 4×2, which really would stand for the hundreds place of 40×2.  It has the 6 of 3×2, which really comes from 300×2=600.  Plus it has anything that was carried from the previous diagonal.

Writing the total flat along the bottom makes more sense if you’re teaching it to kids, but in some older books people wrote the answer on the bottom and left-hand side, like on this example from page 23 of The Treviso Arithmetic (Arte dell’Abbaco), an Italian textbook [published in the town of Treviso in Northeastern Italy] from 1478.  This example shows that 934×314=293,276.

Trevisio grid 1

When I do this in class I often decorate the sides when I’m done, because I’ve seen that done before.  Then it looks like this:

Trevisio grid 1 colored

(5½) There’s a variation of the above method in which the diagonals go in the other direction but you write the number on the right “upside down”.   Here’s an example from the same page of the Trevisio, again showing 934×314=293,276.

Trevisio grid 2

I didn’t want to count this as a separate method, because it has all of the same principles as the previous grid.  But I wanted to mention it because if you look at it, it’s a small step to getting the next version of 934×314, shown below, in which the boxes above have had the diagonals removed but you still add along the diagonal (although the total is written at the bottom):

Trevisio grid 3This is actually a little bit confusing because with the diagonals removed, the numbers don’t quite line up the way they’re supposed to.   It’d be clearer if the rows were shifted a bit, like this:

Trevisio grid 4

And THAT looks an awful lot like “traditional” multiplication, where in this case “traditional” means “the way I learned multiplication in in 1978”.  But this isn’t a new-fangled method  either:  this “traditional” multiplication ALSO appeared in the Trevisio!  And that leads us to….

(6) Multiplication like I first learned it, like this:

Trevisio grid 5

Phew!  That was more lattice multiplication than you ever wanted to see, wasn’t it?  But isn’t it neat how all those pictures (all on the same two pages) just lead into one another, even though they actually were in the reverse order in the original book?

Incidentally, while we’re on the subject of The Trevisio it might amuse you to learn that this 500 year old book also contains the problem “If 17 men build 2 houses in 9 days, how many days will it take 20 men to build 5 houses?”.  If you’re wanting to get yourself a full copy, here’s a link to all 24 Mb of it!  Or, if you don’t read Italian, you can learn a little more about it in this 1996 column by Ivars Peterson.

Next up, some different ways that they don’t teach in school.

Edited to add: Here is The Second Bunch of Ways!

Ethiopian Multiplication

June 9, 2009

plastic_numbersOne of our recent (oh my goodness has it really been seven years???) grads just sent me this Youtube video of Ethiopian Multiplication, with a note that this reminded her of History of Mathematics.  Which, of course, made me totally happy.

This method of multiplication is also called Egyptian Multiplication (because it was done in Egypt) and Russian Peasant Multiplication (although the Peasant part might be intended as a bit of a pejorative).

Here’s the basic idea:  Suppose you want to multiply two numbers like 14 and 12.  You could use your fingers, of course, but here’s another way:

Start with the two numbers on top.  Halve one, ignoring any remainders or fractions, and double the other, stopping when you get to 1.

14 & 12

7 & 24

3 & 48  [See how I ignored the fact that halving 7 leaves 1 left over?]

1 & 96  <— Stop here.

Now look at the numbers on the right.  Some are across from an even number: in this case, 12 is across from the original 14.  Ignore those, and add the rest.  So we’ll add 24, 48, and 96, which were across from odd numbers, and get 168.  And that’s the product!  Isn’t that cool?

(I think it would be fantastic to write a book called 25 ways to multiply.  I only have about 13 at the moment, though.)

Here’s the video!

Number photo from gokuro.

Cinco de Mayo Math

May 5, 2008

Today is the day many people (mostly in the US) celebrate Mexican Independence Day (nope that’s in September) the day in 1862 that the Mexican Army beat the French at Puebla, about 70-80 miles east of Mexico City. As part of this celebration, you might do some Cinco de Mayo Math by Paula J. Maida, which includes ideas like finding the proportion of colors on a Mexican Flag or writing the eleven letters CINCODEMAYO on wooden cacti and letting kids choose a letter randomly as a way to explore both probability and fractions.

Or, expanding beyond Mexico (as the celebration has in many places), you might read “Spanish Colonial Mathematics: A Window on the Past” by Ed Sandifer, which was published in the College Math Journal in September 2002 and is available as a .pdf file here from Ed’s homepage. As Ed explains in this interview:

[S]ome Latino students feel disconnected to math. But we can help make a connection by teaching about Spanish-colonial math and pointing to facts such as there were 11 math books published in Spanish in the New World before there were any in English, with the first being published in 1556, only 100 years after the printing press was invented.

The article above examines 7 of those math books. There’s the 1556 Sumario Compendioso de las quentas de plata y oro que in los reynos del Piru son necessarias a los mercadores: y todo genero de tratantes. Los algunas reglas tocantes al Arithmetica. Fecho por Juan Diez freyle. (Isn’t that a GREAT title!?! It translates as Compendious summary of the counting of silver and gold that are necessary in the kingdoms of Peru to merchants and all kinds of traders. The other rules touching on Arithmetic. Made by Juan Diez, friar.) which was full of all sorts of tables to help you out of you were a tax collector (Hey! More Tax Math!) and includes among other things the following example of how to multiply 875 by 978:

Isn’t this wild? Some of it is explained in the article: the initial 8×9 of 800×900 becomes the 72 of the upper left. Then 8×7 of 800×70 is 56, but this is written with the 5 below the 2 in 72 and the 6 next to the 72, making the 72 look like 726. Then 8×8 of 800×8 is 64, and the 6 of 64 is put next to the 5 while the 4 is next to the 726.

It continues in this fashion, with 7×9=63 of 70×900 put underneath the 56 and with all the numbers crammed together like Galley Division of the same time period. David Smith write in his book History of Mathematics (p. 119) that this method is essentially The Method of the Cup (per copa) because it looks like a goblet.

Several books followed the Sumario Compendioso…, including King Philip of Spain’s Pragmática sobre los diez días del año in 1584, which wasn’t so much about math as about how to deal with the fact that switching from the Julian to the Gregorian calendar involved skipping ten days, and several books that look at military mathematics and formations. In the latter category is the Breve aritmética por el mas sucinto modo, que hasta oy se ha visto. Trata en las quentas que se pueden ofrecer para formar campos y esquadrones by Benito Fernandez de Belo— another fabulous title which means Brief arithmetic for the most succinct method which has been seen up to today. Treating calculations that one can do for the formation of camps and squadrons — which shows how to align 278 men in a squadron into a regular pentagon and contains a woodcut doodle in the back. The final book in the article is the 1696 Cubus, et sphaera geometrice duplicata by Juan Ramón Coninkius about straightedge and compass constructons. Granted, the constructions (doubling the volume of a cube or sphere) were impossible, but that was still unknown at the time.

Happy Cinco de Mayo!

Multidigit multiplication: vertically and crosswise

February 15, 2008

I think I have a new favorite way to multiply numbers: the vertically and crosswise technique. I learned about it from George Gheverghese Joseph’s Crest of the Peacock [a book that is sadly out of print], and was recently reminded of just how cool it is.

Click here to find out how it is done and why it works

Three finger tricks for multiplying

December 31, 2007

hands-9x4.jpg

Many math sites teach the following method of using your fingers to remember the multiples of nine: to find the product of 9 times n, hold your hands out in front of you and fold down your nth finger from the left to separate the tens and the ones. For example, to find 9×4, you would hold down your 4th finger from the left as in the above photograph. The bent finger separates the tens and ones digits, so the configuration of 3 fingers (folded finger) 6 fingers gives the answer of 36.

While this method has enjoyed great popularity among students and teachers, there are two other lesser-known finger tricks for multiplying numbers. Click here to find out what they are!