Block B

Multiply

IIIIII * III = IIIIIIIIIIIIIIIIII

It's easy!!! For each mark in the second make a copy of the first. Glue the copies together. No wormwoodean times tables!

Integer Division and Mod

If you don't know mod (%), it is computed by dividing the right-hand operand into the left and computing the remainder.

8 % 3 = 2

Integer division is division that discards any remainder. Its symbol is //.

 8 // 3 = 2

What is the siginificance of 365% 7 = 1? Think about your birthday!

So let's do these operations with tally marks.

16 // 5

We now compute

IIIIIIIIIIIIIIII// IIIII

1. Make groups of 5.

IIIII IIIII IIIII I

Make one tally mark for each full group.

III

Et Voila! There's your integer quotient.

The remainder is what's left after you make the groups.

IIIIIIIIIIIIIII % IIIII = I

Arithmetic is butt-simple. But visual representaion is awful.

Roman Numerals

These borrowing conventions are all of the legal ones.

IX 9
IV 4
XL 40
XC 90
CM 900
CD 400

We convert 298 here.

CC       200
CCXC     290
CCXCV    295
CCXCVIII 298

Here is a little arithmetic.

Decimal      Roman
  1
  298        CCXCVIII
  521        DXXI
  ---        -------------
  819        DCCCXVIIII  → DCCCXVIX

PITA

Arithmetic is awful, but numbers are compact. People known as calculators did calculations as a profession.

Roman numerals are a denominational numbering system.

Another denominational system is US paper currency.

You were asked to write down a procedure to count out an integer number of dollars using as few bills as possible.

We discovered the "greedy algorithm." Use as many bills as possible with the largest denomination. Repeat this with each smaller denomination until the amount has been counted out.

Now imagine a country whose "bills" come in denominations that are powers of 2. Let's count out 221 units.

Begin by listing denominations until they are > 221.

1
2
4
8
16
32
64
128
256

In the second colum, you put how many bills you used, and in the
third puts what's left.  We can't use any 256s.  

1
2
4
8
16
32
64
128
256  0   221

We can use a 128

1
2
4
8
16
32
64
128  1    59
256  0   221

We can't use a 64.

1
2
4
8
16
32
64   0    52
128  1    59
256  0   221

Continue up the chain.

1    1     0
2    1     1
4    0     3
8    1     3
16   1    11
32   1    27
64   0    52
128  1    59
256  0   221

Observe that you never use more than one bill of any denomination. Read up the middle column and you will see the binary representation of 221: 0b10111011. The 0b prefix says, "this is a binary number."

This works for any base. Let's try 3.

We turned it upside down. Now read up the down the middle column.

243  0  243
81   2   59
27   2    5
9    0    5
3    1    2
1    2    0

221 = 22012₃.