- You have a quiz today over the prime factorization (as well as the class discussions and other readings assigned last week).
- For the quiz, you may have only hand-written notes (e.g. example problems, summaries of readings, etc.). If you have a calculator, great; if you must use your phone, then you may only use it in calculator mode, and it will be on the table in front of you.
- I've assigned you some homework problems, just to check if you are getting it.
- Last class you met up with The Great Fraudini, who is able to
read minds!
This magic trick, carried out with the help of six simple cards, no doubt amazed and astonished you, and made you think "Hmmm... maybe there's money to be made here!"
- Did any of you make any money? Steal any candy from young
children? Astonish your friends, parents, or family members?
- Today we will dig deeper into the secret of the Great Fraudini's
trick ("Bad magician! Never give away your secrets!"); but I want you
to be able to confidently astonish and amaze all your friends and kids
and nieces and nephews, too, and you'll need a better understanding for
that.
- A reminder of the Great Fraudini's trick:
It turns out that The Great Fraudini can read anyone's mind -- so long as they're thinking of a number between 1 and 63, and they're willing to play a little card game with him. So let's recall how this works.
- Rules of the game: you show your "victim" every card. If they see their number on it, they say "Yes"; otherwise they say "No". Then you tell them their number!
- Here's a handout for the cards.
- Each card has a unique power of two in the upper lefthand
corner: add up the powers of two, and you've got their
number. But how do we know that there aren't two
numbers that appear on exactly the same two cards?
That would completely mess things up, right? We get to that in a moment.
- Some students noticed something last time that's
important: each card showed a different pattern;
- the "1" card is all the odds -- every odd number will have to include a 1. You can think of that as a skip of one -- because 2 picks up the slack.
- The "2" card starts 2,3, then skips two numbers (because 4 picks up the slack!).
- The "4" card starts 4,5,6,7, then skips four (because 8 picks up the slack).
- The "8" card starts 8,9,10,11,12,13,14,15, then skips eight (because 16 picks up the slack).
- The "16" card starts 16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31, then skips sixteen (because 32 picks up the slack).
- The "32" card starts (and ends), with 32 numbers from 32 to 63 -- at which point 64 would have picked up the slack, had we wanted to use more than six cards.
- We also discussed a related "trick": Primitive counting.
A method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht proposed at the 2011 National Math Meetings. Here's how it works:
- divide your sheep equally ("one for you, one for
me") into two pens: either there is one sheep left
over, or not.
You make a note of whether there is a leftover sheep or not -- maybe you make a mark, like a "1" or a "0"; or maybe you tie different knots in a string (more on knots in your reading). This is all you have to do to communicate the number to the King! You don't need to know how to count, to our way of thinking.
- Send all the sheep in pen two (and any "left over") out to pasture, and then
- You divide the remaining sheep into pens one and two: i.e., just do it again! And again, and again, and.... until you get down to a pen with just one sheep in it.
- Then we send our marks off to the King, perhaps on a tally stick.
- divide your sheep equally ("one for you, one for
me") into two pens: either there is one sheep left
over, or not.
- We used a ternary tree (three children) to carry out the process.
I say "Whoever's doing this..." -- I mean, "Whoever's doing this primitive counting...." This is presumably someone who doesn't know how to count -- at least not the way we do -- but they can tell if there's a sheep left over (and write "1") or not (and write "0").
Notice that we eventually have a single sheep in a pen, and that's when we're done. We have to write a "1" for the final sheep, to indicate that there's "one leftover".
Then the answer will be written as 1, 0, 1, 1, 0. That is, from the bottom up, left to right. This is important! We have to have a consistent scheme for writing.
So how do we write
- 27 sheep
- 390 sheep
- Can you go backwards? How many sheep is meant by
- 1,1,1,1
- 1,0,0,1,0,1,0
- 1,0,0,0,0,0,1,0,1
Can you rebuild the tree using these "tally sticks"? If so, you can get a job in the King's counting house. Let's rebuild some....
- Maybe you've seen this t-shirt somewhere: if so, you should get the joke now!
- If you're afraid of sheep, think about counting pennies. When I
was little I counted lots of pennies (I collected them, and
would get rolls from banks to look through -- when I was done I
had to put 50 pennies back into the rolls).
But rather than count out fifty pennies, I'd make a stack of 10, then four more stacks of exactly the same height (one-to-one correspondence). Five stacks of 10 makes 50. This is a similar idea....
If you have a lot of pennies, you could divide them in half, count half, and then multiply by two. But if you have one left over, you'd have to add that one.
Do it again, until you get to the point where you arrive at 1 coin. Suppose you start with 81 pennies:
- Divide in "half": \[ 2*40+1 \]
- Do it again, but with 40: \[ 2*(2*20+0)+1 \]
- Do it again, but with 20: \[ 2*(2*(2*10+0)+0)+1 \]
- Do it again, but with 10: \[ 2*(2*(2*(2*5+0)+0)+0)+1 \]
- Do it again, but with 5: \[ 2*(2*(2*(2*(2*2+1)+0)+0)+0)+1 \]
- Do it again, but with 2:
2*(2*(2*(2*(2*(1+1+0)+1)+0)+0)+0)+1
- Now why does the Fraudini trick work?
The secret of Fraudini's trick is another secret of the counting numbers, 1, 2, 3, 4, ....: the binary factorization of a number.
Theorem: Every counting number is either a power of 2, or can be written as a sum of distinct powers of 2 in a unique way.
This is the second great factorization of the counting numbers. So it turns out that the counting numbers and sums of distinct powers of two can be put into one-to-one correspondence (if we count single powers of two as "sums"...).
Here are the first few powers of 2:
\[ \begin{array}{l} 2^0 = 1 \cr 2^1 = 2 \cr 2^2 = 4 \cr 2^3 = 8 \cr 2^4 = 16 \cr 2^5 = 32 \cr 2^6 = 64 \cr 2^7 = 128 \cr 2^8 = 256 \end{array} \]
(Do you see why the Great Fraudini can only read numbers up to 63?)
- These powers of 2 illustrates what we mathematician types call
"exponential growth". Powers are what exponential growth is all
about. If you double something (as we're doing here, starting from 1),
you multiply by 2; and if you double 1 say every week for 20 weeks, you
get to
\[
2^{20}=1048576
\]
Over a million. So ask your friend if he or she can do a push-up; then
ask if they can double their number of push-ups every week for 20
weeks, and see how they do!
- The fact that there is a unique way -- exactly one way --
to write a counting number is the secret to Fraudini's trick. This
trick is more properly called the "Binary Card Trick", because what
we're really doing is using "binary" numbers (base 2) to represent the
numbers we usually think of in base 10.
So in base 10 we think of "17" as \[ 17 \textrm{ (one ten and seven ones) } = 1*10 + 7*1 = 1*10^1 + 7*10^0 \]
That last bit on the right shows 17 as a sum of "weighted" powers of 10 -- "weighted" because we have to say just how many ones we need (7 of them in this case). The weight could be any of the 10 digits we need in our system,
0,1,2,3,4,5,6,7,8,9 So if I want to pay you 17 dollars, I could give you one 10 and 7 ones.
For binary numbers the "weights" are easy: 1 or 0. Just two digits. And that's perfect for computers, because they only have two fingers:
-
On the other hand, $17=16+1=2^4+2^0$. Try to write 17 as a sum of
distinct powers of 2 in another way. (Good luck! You can't.)
You might try
\[
17=1+8+8=2^0+2^3+2^3
\]
and hope that we're not paying attention -- but those aren't
distinct powers of two. So the "distinct" part is really
important.
17 appears only on the 1 card and on the 16 card. We figure out how to write a number as a sum of powers of 2, and then we write that number on each of those cards. And then it will be the only number exclusively on those cards -- and the sum will tell me the number!
So we can write \[ 17=1*2^4+0*2^3+0*2^2+0*2^1+1*2^0 \] or, better yet, $17_{10} = 10001_2$ -- 17 base 10 is equal to 10001 base 2.
So if I want to pay you 17 dollars, I could give you one 16 and one 1 dollar note. I've always hoped that a student would design binary bills for me. In a way, the Fraudini cards could be considered bills, and if I need to pay you 17 dollars, I just give you each card that has a "17" on it. That would make them supremely easy to use, right?
-
I always tell people to write their salaries in base 2, because they
look a lot bigger! More impressive.... All of a sudden you find
yourself making six or seven figures -- per week!
If you're making 30,000/year, it's suddenly
111,010,100,110,000/year.
Wow! Does that feel better? You're a trillionaire!
- Much like the prime factorization, we can use a tree to break down
a number into its binary representation. Let's give it a go for a few
examples:
- Let's start easy with 97 (a big prime!).
- Now let's use a tree to find the binary factors of 437.
-
Fraudini used his amazing powers (his amazing powers of 2, that is) --
to "read peoples' minds!" I hope that you make a buck or two reading
minds this weekend....
- What else can you do with binary numbers? binary
hand dance, of course!
Vi Hart has the most amazing ways of showing us interesting mathematics. She shows us that we can count to 31 on one hand. And, if you'll use the 10 fingers of two hands, you can get all the way up to $2^{10}-1=1023$...
Some people can only count to 10 with their fingers....
