1. Good evening everyone!
It's you're favorite french speaking Filipino
giving you the low down on today's class.
(I really can't speak french)
To begin, please select your
preferred language:
French English
X
1
2. Thank you for selecting English...
Philosopher, physicist, mathematician.
Discoverer of Probability.
The impossible search for Truth.
His name was Blaise Pascal.
Blaise Pascal has led to many
expositions including the famous
Pascal Triangle! This triangle is an
inventory of Binomial Coefficients,
and the patterns, and relationships
it contains.
2
3. Pascal 1
Pascal's Triangle simplifies grunt work
for solving binomial equations that
Triangle 1 1
can take a huge amount of time
121
that everyone doesn't have.
1331
(Thank you Pascal)
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Let's do some binomial
equations to show you
how this triangle relates
to these equations...
3
4. (a + b)0 = 1
(a + b)1 = 1a + 1b
(a + b)2 = 1a2 + 2ab + 1b2
(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3
(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 +1b4
(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5
WOH! Check hizzout the
1
coefficients of the
11
equations above, and the
121
1331 numbers in the triangle:
14641
Strikingly similar!
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
4
5. All right! Let's look for
some creepy patterns!
Creepy
Pattern #1
4 +6
=
1
11
121
10
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Take any number, any number at all!
(From the table) Then take the 2 numbers
above it, and add it! oOoOo I'm not even
at the good part yet.
5
6. Creepy
Pattern #2
110 = 1 1
11
111 = 1 1 121
1331
112 = 1 2 1 14641
113 = 1 3 3 1 1 5 10 10 5 1
1 6 15 20 15 6 1
114 = 1 4 6 4 1 1 7 21 35 35 21 7 1
This is when I started to think
witchcraft had something to
do with it... hmm
6
7. Creepy
Pattern #3 Okay NOW this was
starting to look like a David
Blane trick. Make a
diagonal line going
downwards starting
1 anywhere for as long as you
11 want it to be. Then bend
121 the diagonal line the other
1331 direction for two rows as
14641 shown in the diagram to the
1 5 10 10 5 1 left. The LAST number is the
1 6 15 20 15 6 1 sum of all the number's
1 7 21 35 35 21 7 1 selected before it!! AHH!
It's a hockey stick!!
7
8. Creepy (a + b)0 = 1
Pattern #4 (a + b)1 = 1a + 1b
(a + b)2 = 1a2 + 2ab + 1b2
(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3
(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 +1b4
(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5
In each increasing
From the last term,
term, the variable
Look at the variable 'a,'
the variable 'b' is
'a' is decreasing by
and its exponents. The
1. increasing as you
exponent of 'a' of the
move up the
first term is the same
terms, and is also
number as the number
the same number
you're quot;poweringquot; in
you're quot;poweringquot;
(a+b).
in (a+b)
8
11. Phew!
That took longer than it needed it to be. Too
many magic tricks in this unit. Everything is just
beginning to unfold though, can't wait to see what
Monsieur Kurapatwa has to reveal tomorrow.
Wohoo! scribe picking time!
I chchchoooose:
Haiyan! !nayiaH
11
