## Archive for June 16th, 2009

### 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.

(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.

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.

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.

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 $cos(a-b)$.

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

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!