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1. PASCAL’S
TRIANGLE —
BLAISE-ING A
TRAIL OF
MATHEMATICS
EddieTchertchian
Los Angeles Pierce College
AMATYC 2018 – Orlando, FL

2. “All of men’s miseries
derive from not being
able to sit quietly in a
room alone.”
• BornJune19,1623inClermont-
Ferrand,Auvergne, France to
father Etienne Pascal &mother
Antoinette Begon
• Lost mother at the age of 3–
family relocated to Paris five
yearslater

3. “Do you wish people to
think well of you? Don’t
speak well of yourself.”
• Atage 16–Pascal’s theorem:
If six arbitrary points are chosen on a
conic (which may be an ellipse,
parabola or hyperbola in an
appropriate affine plane) and joined by
line segments in any order to form a
hexagon, then the three pairs of
opposite sides of the hexagon
(extended if necessary) meet at three
points which lie on astraight line,
called the Pascalline of the hexagon.

4. “I would prefer an intelligent hell to a
stupid paradise.”
• When Etienne became
the king’scommissioner
of taxes in the city of
Rouen, Pascal tried to
aid his father in doing
many computations by
constructing theworld’s
first mechanical
calculator, the
“Pascaline.”

5. “To make light of philosophy is to be
a true philosopher.”
• Pascal’scontributions to mathematics
were numerous, aswerehis contributions
outside ofmathematics:
– Philosophy (of mathematics –axiomatic
method; formalism basedon Descarte’swork)
– Literature & religion (ThePensées –
“Thoughts”)
– Physical sciences (pressure –Pascal’sprinciple)
– Probability theory/gambling (primitive form of
roulette wheel)

6. “You always admire what you
really don’t understand.”
• Pascal’striangle & binomial coefficients were
studied by Pascalin 1653,but had been
described and well-known centuries before
thataround the world:
– Indian studies of combinatorics & the numbers of
the triangle date back to Pingala(2nd century BC)
– Iran:Al-Karaji wrote anow lost book which
contained the first description of Pascal’striangle;
repeated later byOmar Khayyam(1048-1131) –
Khayyam’s triangle
– China:JiaXian (1010-1070);YangHui (1238-1298)
presented the triangle –Yang Hui’s triangle

7. “It is man's natural
sickness to believe
that he possesses
the truth.”
– Germany: Petrus
Apianus –fulltriangle
published (1527)
– Italy: Tartaglia’s
triangle (1556–first
six rows); Cardano
published the triangle
& additive and
multiplicative rulesfor
constructing it. (1570)

8. The entry in
the n-th row,
r-th column
is simply the
binomial
coefficient “n
choose r”

18. 𝟏 + 𝟔 + 𝟐𝟏 + 𝟓𝟔 = 𝟖𝟒
𝟏 + 𝟕 + 𝟐𝟖 + 𝟖𝟒 + 𝟐𝟏𝟎 + 𝟒𝟔𝟐 + 𝟗𝟐𝟒 = 𝟏𝟕𝟏𝟔
𝟏+ 𝟏𝟐 = 𝟏𝟑

19. 𝑛 +1
𝑚
=
𝑛
𝑚
+
𝑛 −1
𝑚 − 1
+ ⋯+
𝑛 −
𝑚
0
𝑚 + 𝑟+ 1
𝑚
=
𝑚 + 𝑟
𝑚
+
𝑚 + 𝑟− 1
𝑚 − 1
+ ⋯+
𝑟
0

22. If the 1st element in
a row is a prime
number (remember,
the 0th element of
every row is 1), all
the numbers in that
row (excluding the
1's) are divisible by
it. For example, in
row 7 (1 7 21 35 35
21 7
1) 7, 21, and 35 are
all divisible by 7.

23. The sum of the squares of the
elements of row n equals the
middle element of row 2n. For
example,
12 + 42 + 62 + 42 + 12 =70.
In general form:
𝑛
෍
𝑘 =0
𝑛
𝑘
2
=
2𝑛
𝑛
Row4
Row8 =2 x4

24. Sumof the first krows is the
Mersenne number
2𝑘 − 1

26. • There are infinitely many numbers that occur at least six times in Pascal’s(whole)
triangle, namely the solutions to:
𝑟 =
𝑛
𝑚 − 1
=
𝑛 −1
𝑚
given by
𝑚 = 𝐹2𝑘 −1 𝐹2𝑘
𝑛 = 𝐹2𝑘 𝐹2𝑘+1
where 𝐹𝑖is the 𝑖-th Fibonaccinumber.
• Thenumbers that occur fiveor more times in Pascal’striangle are 1; 120;
210; 1540; 3003; 7140; 11628; 24310; and the number
61,218,182,743,304,701,891,431,482,520
with no others up to 33 ⋅ 1016

29. Using Nilakantha’s infiniteseries
for 𝜋

36. “The last thing
one discovers
in composing a
work is what to
put first.”
• Thanksto:
– APieceof the Mountain:TheStory of Blaise Pascal
(Joyce McPherson;Greenleaf press; 1995)
– Pascal’sTriangle:AStudy in Combinations (Jason
VanBilliard; CreateSpace Publishing;2014)
– Blaise Pascal: Reasonsof the Heart (MarvinO’Connell;
Wm. B.Eerdmans PublishingCo;1997)
– http://mathworld.wolfram.com/PascalsTriangle.html and
associated linkswithin
– http://www.cut-the-
knot.org/arithmetic/algebra/PiInPascal.shtml
– https://en.wikipedia.org/wiki/Blaise_Pascal
– https://en.wikipedia.org/wiki/Pascal%27s_triangle
– https://www.goodreads.com/author/quotes/10994.Blaise
_Pascal
– https://www.mathsisfun.com/pascals-triangle.html

37. Thank you!
Eddie
Tchertchian
Los
Angele
s Pierce
College
• Thispresentationis availableonSLIDESHARE:
tchertea@piercecollege.ed
u
http://www.slideshare.net/EddieMath