- Announcements:
- Roll
- Your next assignment is due
- Back to the bases!
- Last time we learned a trick, of the Great Fraudini!. Hopefully you'll be able to win when you play it on your first exam (which is coming up at the end of the month).
- Counting in binary: let's do some counting, as something of a warm-up. Let's do only the first four place values (so how many numbers are we going to count up to? How many fingers do we need?).
- So why do we care about binary? Where do we encounter binary in real life?
- How about at the grocery. Let's start with gallons....
- How about on the construction site?
- Turns out that cutting things in half (as well as doubling
things) is generally easy, which is why binary is useful.
- Now who would ever need this in real life? Ancient Egyptians!
- The Rosetta Stone:
- Egyptian multiplication
- Multiplication is accomplished through binary decomposition (successive doublings). Consider, for example, 57*63:
| 1 | 63 | * |
| 2 | 126 | |
| 4 | 252 | |
| 8 | 504 | * |
| 16 | 1008 | * |
| 32 | 2016 | * |
| 64 | too big! | |
Now add up those rows marked with an asterix (*). You'll get the answer (3591).
- How would the Egyptians do 8*17?
- Egyptian division:
- fractions -- binary decimals, oh Ra!
- Division is also carried out in binary, but
fractions make it more interesting: they restricted
themselves to the so-called "unit fractions", which are
fractions of the form 1/m
- the unit
fraction table, which is found on the Rhind
Papyrus (which dates to around 1650 BCE).
- Here's a relatively easy one: Suppose Fatima had 3
loaves to share between 4 people. How would she do it?
- A little trickier: How would you divide 5 by 7?
- How would you like to do story problems like this
one?!
- Why did
Egyptians do things this way? (an example division problem, using binary)
- More practice translating between bases
- Write 53 (base 10) in base 8
- Write 347 (base 8) in base 10
- One of my favorite early introductions to bases was Tom Lehrer's New
Math.
Website maintained by Andy Long.
Comments appreciated.
1 gallon = 2 pottles = 4 quarts = 8 pints = 16 cups = 32 gills = 256 Tablespoons. (what about the missing units of a quarter cup and eighth of a cup? Where did they go?)
