Day 12, MAT115

Here's a key.

Let's do one more base example: Write 4021 in base 7. (If you can do this one, then you should be able to do any base!)

I want to note, however, that when you get beyond base 10, you need more digits than we have. Base 16 required 6 more digits, and we used the letters A-F. The Mayans are using base 20, so we can use their digits:

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19

The Babylonians needed 60 digits (but only had 59, because they didn't know about 0, sadly enough). Their digits are made up of boomerangs and martini glasses....

• At the heart of Egyptian multiplication is this fact (we might call it the "Fraudini fact", or the "binary factorization"):
  Every natural number is either a power of two, or can be expressed as a sum of distinct powers of two in a unique way.

  This is reminiscent of the prime factorization:

  Every natural number is either prime, or can be expressed as a product of primes in a unique way.

  Why binary? Because the Egyptians could double numbers, and they could halve numbers. We are good about doubling and halving, too. Where do you see doubling and halving our our lives?

  • Quantity
  • Size
  • Time

• Let's do another Egyptian multiplication: 39*72 (=2808):

  1	72
  2
  4
  8
  16
  32

• Recall that the order using the technique doesn't matter: that is, 72*39 would give the same answer.

  We use the larger number to double to reduce the size of the table, and that leaves the smaller number to recreate using binary representation.

  Now add up those rows marked with an asterix (*), and you'll get the answer.

• First of all, division is just multiplication backwards, right?

  Let's look at one of the simplest examples imaginable:

  divide 32 by 8.

  We can actually do it by Egyptian multiplication, since 8 divides into 32 evenly:

  1	8
  2	16
  4	32	*

  So the answer is 4 (how do we get 4?)

  In this problem you see both 8 and 32, plus the answer (4).

• Similarly we could divide 40 by 8, using the same table (again easy, since 8 divides into 40 evenly):

  1	8	*
  2	16
  4	32	*

  So the answer is 5 (how do we get 5?)

  In this problem you see neither 40 nor the answer (5) -- except that they are sums of other things that you see in the problem (from those lines indicated by the asterix). And that's how we're going to do these problems in general.

• When the denominator doesn't divide the numerator evenly, fractions make it more interesting (here they come!):

  Let's look at an example: divide 35 by 8.

  In a way we turn it into a multiplication problem: what times 8 equals 35? So we know the 8, and use it to "double" -- but then to "halve", when 8 won't go evenly into 35:

  1	8
  2	16
  4	32	*
  1/2	4
  1/4	2	*
  1/8	1	*

  So the answer is 4+1/4+1/8

  (Where have we seen those fractions before? Look into the Eye of Horus!)

  Again, in this problem you see neither 35 nor the answer (4+1/4+1/8) -- except that they are sums of other things that you see in the problem. And notice that the answer is literally a sum of other things!

• Now here's the really curious thing: the Egyptians restricted themselves to the so-called "unit fractions", which are fractions of the form 1/m: unit fraction table, which is found on the Rhind Papyrus (which dates to around 1650 BCE).

  But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:

  1	7
  1/2	3+1/2	*
  1/4	1+1/2+1/4	*
  1/7	1
  1/14	1/2	*
  1/28	1/4	*

  So the answer is 1/2+1/4+1/14+1/28

• Why did Egyptians do things this way? (an example division problem, using binary)

     Dominic Olivastro, "Ancient Puzzles", suggests a third reason why 
  this use of unary fractions is good. Consider the problem Ahmes poses 
  of dividing 3 loaves of bread between 5 people. We would answer "each
  person gets 3/5-ths of a loaf". If we implemented our solution, we might
  then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then
  cut one of the smaller pieces in half, giving the other two people 
  2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and
  they will insist that it is not an even division. Some have larger pieces,
  some have smaller. Ahmes would calculate 3/5 as :
     3/5 = ()3 + ()5 + ()15    [ = 1/3 + 1/5 + 1/15 ]
  Now cut one loaf into fifths, cut two more into thirds, then take one of
  the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you
  can now distribute everyone's 3/5-ths share in a way that _looks_ equal,
  since they will have exactly the same size pieces. (And no, I don't want
  to argue about the crust.)
  

• Here's a relatively easy one: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?

  (Think about what the answer means, in terms of bread.)

• A little trickier:
  1. How would you divide 5 by 7?
  2. How can we use the unit fraction table to get the same answer?

• How would you like to do story problems like this one?!