Understanding Numbers
An Expository Analysis of Pascal’s Triangle
Emily Pihl
MA 165
Dr. Rosentrater
25 March 2014
Pihl 1
Emily Pihl
Dr. Rosentrater
MA 165-1
24 March 2014
Understanding Numbers: An Expository Analysis on Pascal’s Triangle
According to Dr. Benjamin, a professor of Mathematics at Harvey Mudd, “one of
the most beautiful objects in mathematics [is ] Pascal’s triangle,” a tool so wonderful that
“You could spend your life looking and studying patterns that live inside of this beautiful
triangle.”1 Indeed, over the four centuries since Blaise Pascal published his work, his
infamous triangle has become a mathematical tool that has shaped, redefined, and
facilitated growth in the areas of Algebra, Calculus and Probability. The “patterns” of
which Dr. Benjamin describes have since allowed mathematicians to develop newer and
even easier computations, calculations and explanations for otherwise very challenging
problems. Specifically, Pascal’s triangle “works with symmetry and nth series… [that]
allow a mathematician to successfully create an object with ‘numbers in it [that] also
satisfy a number of interesting properties.’” 2 Such functional computations commonly
derived from this triangle include the development of counting techniques and probability
with combinations. Thus, because of the unique properties and the numerical patterns
found within Pascal’s Triangle, this triangle becomes a helpful and important tool for
understanding counting and quantitative summations and formulas, especially pertinent to
Probability and Geometry units addressed in sections of MA 165.
1
Benjamin, Arthur. "The Joy of Pascal's Triangle." The Joy of Mathematics: Course Guidebook. Chantilly,
VA: Teaching, 2007. 132.
2
Uspenskiĭ, V. A. Pascal's Triangle. Chicago: University of Chicago, 1974. Print.11.
Pihl 2
The modern known and studied Pascal’s triangle originally stems from
rudimentary mathematics developed over 800 years ago, among many Persian and Asian
cultures. The binomial identity, (𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2 , originally proven by Euclid
around 4th century BC, became the foundational base for many mathematicians in the
early AD, who developed their own methods and triangular formations before Pascal.
According to The History of Mathematics, the first connection of the binomial theorem to
the triangle appeared in The Precious Mirror of the Four Elements, a 1303 treatise where
Chu Shih-Chien creates “a triangular arrangement of binomial coefficients through the
eighth power, written in rod numerals and a round zero sign.”3 [Figure 1] Throughout the
16th century, the development of arithmetic triangles became popularized in Europe, and
Italian scholars like Girolamo Cardano adapted the use of binomial coefficient in a
practical sense, to find “progressions of higher order and music theory and harmony.”4
However, it was not until Blaise Pascal, a French mathematician and philosopher,
published his Traité du Triangle Arithmetiqu that he revealed the true complexity of the
binomial theorem. Pacal’s genius and theoretical mind broke down the theorem into
patterns, numerical sequences, and relative principles which became the guidelines to
forming his triangle. In his original diagram, Pascal creates a model of parallel and
perpendicular ranks which according to his Traité, can better be explained as : “the
number of each cell is equal to that of the cell which precedes it in its perpendicular rank,
added to that of the cell which precedes it in parallel rank.” 5 [Figure 2] This basic
3
Burton, David M. "The Development of Probability Theory: Pascal, Bernoulli, and Laplace."The History
of Mathematics: An Introduction. Boston: Allyn and Bacon, 1985. Print. 429.
4
Alberti, Furio. "Pascal's Triangle." Historical Topics for the Mathematics Classroom. By John K.
Baumgart. Washington: National Council of Teachers of Mathematics, 1969. Print. 157.
5
Burton, The History of Mathematics: An Introduction. 431.
Pihl 3
understanding of ranks and cells has since become the precedent to the triangle used
today with the only difference being the 45° rotation modern mathematicians have made.6
With the creation of such a well-developed triangle comes the beauty of its many
properties—patterns that simplify and explicate otherwise extremely complicated
concepts. The triangle is developed of an infinite number n of rows, beginning with the
0th row; each succeeding new row has one more number adds to it. For instance the 0th
row has 1 digit, the 2nd row has 2 digits, the 3rd has 3 digits and the 4th row has 4 digits.
From these we can say that each row has n has n digits in its sequence. The first proven
property, and the most visible, is that “all of Pascal’s sequences are symmetrical… about
the bisector of the angle whose interior it fills.”7 Thus, cutting the triangle or any of the
single rows in half vertically will result in lines with symmetric numerations. Another
important and easily recognizable property is that “the sum of the numbers in Pascal’s nth
sequence is 2n,” because as we progress from one row to the next row, “the sum of the
numbers is doubled.”7 This simple property, also Pascal’s 8th corollary, allows a
mathematician to then understand that “the sum of the numbers in a [triangular] base is
equal to the sum of the sums of the numbers in the lower [triangular] bases, plus 1” which
according to Pascal is a “‘doubling’ progression’” mathematically portrayed as: 2𝑛 =
2𝑛−1 + 2𝑛−2 + ⋯ + 2 + 1 + 1. 8 Here Pascal references the many arithmetic triangles
which are built up inside of his main triangle, set upon one another to create an infinite
stack of triangles [Figure 3]. These interior shapes become significant to the many
numbered sequences that develop out of Pascal’s triangle.
Uspenskiĭ, Pascal's Triangle.11.
Uspenskiĭ, Pascal's Triangle.10.
8
Edwards, A. W. F. Pascal's Arithmetical Triangle. London: C. Griffin, 1987. Google Books. Web. 23
Mar. 2014.
6
7
Pihl 4
Number sequencing, especially within this model, derives from many clearly
determined and often rather uniquely surprising patterns. By looking at the diagonals,
(either side can be used, as a result of the symmetric property) one recognizes many
special arrangements. The first diagonal consists of all ones, which is significant to the
binomial expansion which always begins with a simple variable preceded by a 1. The
second diagonal introduces whole, counting numbers (1,2,3,4…).9 These counting
numbers help a person to determine the row number, because the two correlate exactly.
The third diagonal consists of triangular numbers, which graphically are “figurate
number[s] that can be represented in the form of a triangular grid of points” [Image 4]
and algebraically are “numbers obtained by adding all positive integers less than or equal
to a given positive integer n.”10 These triangular numbers are important to the fourth
diagonal, which consists of tetrahedral numbers—numbers used in the layers of the
tetrahedron—because the tetrahedron shape is inclusive of these triangular numbers. By
looking at Table 1 [Image 5] one can recognize how the sequences of diagonal 4 and 5
are in essence the same numerical pattern used to describe the conditions of base and
volume of a tetrahedron. Overall, these numerical patterns make Pascal’s triangle useful,
by creating opportunities for further mathematical development.
Each of these patterns created by Pascal’s triangle are not solely purposeless; they
assist in the development of methodical formulas found in advanced mathematics of
probability, some of which are pertinent to Math 165. One of the primary utilizations is in
combinations and subsets. As proven, sets are “any collection of objects…[where] the
object is called an element of the set” and “a set which consists of n elements is called an
9
"Patterns Within the Triangle." Diagonals. Math Is Fun, 2011. Web. 23 Mar. 2014.
Weisstein, Eric W. "Triangular Number." From Wolfram Math. MathWorld, n.d. Web. 23 Mar. 2014.
10
Pihl 5
n-element set.”11 Pascal works with these subsets and the 0th row property of his triangle
𝑛
to discuss combinations. Usually, “Mathematicians will define ( ) as the number of size
𝑘
k subsets of the numbers 1…n; [or] the number of ways to choose k objects from a group
𝑛
of n objects when order is not important.” 12 Simply stated, this formula, ( ) gives the
𝑘
answer to how many subsets or choices are available out of a certain group or population;
of course the population (n) must also be greater than 0. This formula relates to Pascal’s
𝑛
formula through the numerical sequencing of each row, which will justly prove that C( )
𝑘
𝑛
= T ( ) with C being a traditional combination, and T being the combinations derived
𝑘
from the Triangle. 13 For instance, row four (remembering the 0th row property), written
4 4 4 4 4
in combinations would be ( ) ( ) ( ) ( ) ( ) . When calculating one combination, as
0 1 2 3 4
4
seen in taking the combination ( ) we get
2
4!
=
2!(2!)
24
2(2)
= 6.12 This number, 6, is the same
sum as the middle digit of row four. If one continued to compute the combinations the
results 4(1,4,6,4,1) would be found, which exactly correlates with the 4th row of Pascal’s
triangle. This proof is essential to probability, especially when questions are asked
pertaining to how many combinations of elements and subsets can be made out of a set.
When working with the probability of heads and tails for instance, this technique allows
someone to pre-determine how many combinations of {H,T} can be made based on the
number of tosses that are played upon.14 For example, to find the probability of getting
exactly two heads with 3 coin tosses, one looks at the third row of Pascal’s triangle and
Uspenskiĭ, Pascal's Triangle. 26.
Benjamin, The Joy of Mathematics: Course Guidebook. 133.
13
Uspenskiĭ, Pascal's Triangle. 31.
14
"Using Pascal’s Triangle." Heads and Tails. Math Is Fun, 2011. Web. 23 Mar. 2014.
11
12
Pihl 6
sees the sequence (1,3,3,1) that equals 8 signifying 8 subsets. Written out, it’s confirmed
that indeed there are 8 subsets {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. There
are 3 of these 8 that have exactly two heads, so the probability is
3
8
or .375. In Math 165,
Probability is a core part of the curriculum, and to have Pascal’s triangle to help predetermine how many results exist can make finding the probability of the event much
easier.
Another very interesting use of the Pascal triangle is the connection between his
sequenced rows and geometry. Already discussed in terms of tetrahedrons and triangles,
surprisingly, Pascal’s work also examines foundations of the circle. In this geometric
model “by connecting points from 1 to n on a circle, and counting the number of points,
segments, triangles up to the number o n-gons,”15 the rows of the Pascal Triangle begin to
appear, as better modeled in Figure 6. Pascal’s sequence alleviates the burden of
counting every component of the individual circle and its parts; instead of counting, one
simply finds the row which corresponds to how many points the circle has, and then thus
automatically knows how many segments, and triangles, etcetera also exist. According to
Katie Asplund’s Pascal’s Triangle Patterns and Extensions, this same geometric idea can
be used to construct multi-dimensional shapes “by continually adding an additional
vertex then connecting each of the original vertices to the new vertex.”15 In Math 165,
counting points and working with 3D structures, such as discussed by Asplund, is a key
section of the class. Pascal’s methodology clearly explains how simple the relation
between points and numbers can be; following patterns helps simplify tedious work.
15
Asplund, Katie. Pascal's Triangle Patterns and Extensions. Publication. Ed. Irvin Hentzel. Iowa State
University: MSM Creative Component, 2009. Web. 23 Mar. 2014.
Pihl 7
Overall, Pascal’s triangle is indeed one of the most beautiful objects of the
mathematical realm, as Dr. Benjamin proposed, simply because its complex nature
encompasses more than a system of numbers, but a meticulous and extraordinary
foundation for complex mathematical concepts. Pascal creates a tool with both
remarkable patterns, such as seen in the first five symmetrical diagonals, and excelled
theoretical application, as exemplified through the triangle’s utilization of combinations
and probability. The triangle expands beyond linear mathematics and analysis by
exploring geometric and 3D figures and their construction relative to number sequences.
Uniquely, even with the complex formulas and proposals, Pascal’s work is clear and a
simple tool to use. In relation to Math 165, Pascal’s triangle offers alternative methods
for solving probability problems including determining subsets and combinations of
elements. Furthermore, the triangle proposes counting techniques to help understand
geometric figures—tetrahedrons, triangles, circles— and their properties. There are
seemingly no limitations to Pascal’s work, or his infamous triangle. Indeed, as a tool and
a mathematical wonder, Pascal’s triangle will forever be the key to help students unlock
and discover the beauty of mathematics and all its complexity.
Pihl 8
Index of Figures
Figure 1:
Shi-Chieh, Chu. Old Method Chart of the Seven Multiplying
Squares. 1303. Ssu Yuan Yü Chien. MathisFun.com
Figure 2:
Pascal, Blaise. Pascal's "Arithmetic" Triangle. 1665. Public Domain. Figurate Numbers
and Sums of Numerical Powers: Fermat, Pascal, Bernoulli. Mathematical
Association of America. Web. 23 Mar. 2014.
Pihl 9
Figure 3:
“Pascal’s Triangle.” Onlinematcircle.com. 27 June 2011. 23 March 2014. [Blue marks
edited and added by Emily Pihl to show emphasis and clarification]
Figure 4:
Weisstein, Eric W. "Triangular Number." From Wolfram Math. MathWorld, n.d. Web.
23 Mar. 2014. <http://mathworld.wolfram.com/TriangularNumber.html>.
Figure 5.
"Comparison of Triangular Number and Tetrahedral Number." Tetrahedral Number
Sequence. Math Is Fun, 2011. Web. 23 Mar. 2014.
<http://www.mathsisfun.com/tetrahedral-number.html>.
Pihl 10
Figure 6:
Asplund, Katie. Pascal's Triangle Patterns and Extensions. Publication. Ed. Irvin
Hentzel. Iowa State University: MSM Creative Component, 2009. Web. 23 Mar.
2014. <http://www.math.iastate.edu/thesisarchive/MSM/AsplundCCSS09.pdf>.
Pihl 11
Works Cited
Alberti, Furio. "Pascal's Triangle." Historical Topics for the Mathematics Classroom. By
John K. Baumgart. Vol. 31. Washington: National Council of Teachers of
Mathematics, 1969. 156-57. Print. Yearbooks.
Asplund, Katie. Pascal's Triangle Patterns and Extensions. Publication. Ed. Irvin
Hentzel. Iowa State University: MSM Creative Component, 2009. Web. 23 Mar.
2014. <http://www.math.iastate.edu/thesisarchive/MSM/AsplundCCSS09.pdf>.
Benjamin, Arthur. "The Joy of Pascal's Triangle." The Joy of Mathematics: Course
Guidebook. Chantilly, VA: Teaching, 2007. 132-40. Google Books. Web.
Burton, David M. "The Development of Probability Theory: Pascal, Bernoulli, and
Laplace." The History of Mathematics: An Introduction. Boston: Allyn and
Bacon, 1985. 411-42. Print.
"Comparison of Triangular Number and Tetrahedral Number." Tetrahedral Number
Sequence. Math Is Fun, 2011. Web. 23 Mar. 2014.
<http://www.mathsisfun.com/tetrahedral-number.html>.
Edwards, A. W. F. Pascal's Arithmetical Triangle. London: C. Griffin, 1987. Google
Books. Web. 23 Mar. 2014.
Pascal, Blaise. Pascal's "Arithmetic" Triangle. 1665. Public Domain. Figurate Numbers
and Sums of Numerical Powers: Fermat, Pascal, Bernoulli. Mathematical
Association of America. Web. 23 Mar. 2014.
Pihl 12
<http://www.maa.org/publications/periodicals/convergence/figurate-numbersand-sums-of-numerical-powers-fermat-pascal-bernoulli>.
"Patterns Within the Triangle." Diagonals. Math Is Fun, 2011. Web. 23 Mar. 2014.
<http://www.mathsisfun.com/tetrahedral-number.html>.
Shi-Chieh, Chu. Old Method Chart of the Seven Multiplying Squares. 1303. Ssu Yuan Yü
Chien.
"Using Pascal’s Triangle." Heads and Tails. Math Is Fun, 2011. Web. 23 Mar. 2014.
Uspenskiĭ, V. A. Pascal's Triangle. Chicago: University of Chicago, 1974. Print.
Weisstein, Eric W. "Triangular Number." From Wolfram Math. MathWorld. Web. 23
Mar. 2014. <http://mathworld.wolfram.com/TriangularNumber.html>.
Wheeler, Ed, and Jim Brawner. "6.3 Pascal's Triangle and Binomial Coefficients."
Discrete Mathematics for Teachers. Boston: Houghton Mifflin, 2005. Google
Books. Web. 22 Mar. 2014.