Pascal’s Higher Power
Imagine it’s Friday evening, and the whole family is getting hangry (that is a combination of hungry and angry), and they all want to order pizza. So, you call your local pizzeria and, hallelujah, they have a “Large Pizza Two Toppings for Two Dollars” deal. The restaurant offers four types of toppings. You want to get as many combinations of pizza with those four toppings because you have some hungry teenagers who are about ready to eat the moldy lasagna that’s been sitting in the refrigerator. How many different combinations of pizzas can you create with only two toppings? Well, the answer lies in mathematics, of course!
There is this very cool triangle known as Pascal’s triangle that helps us find the answer. Imagine that we have a one at the top of the triangle.
Below that one, we have two ones. The reason why is because each of those numbers are the sum of the two numbers above it.
So, in this case, for the second row, the two numbers above it are 0 and 1. As a result 0+1=1.
We’re going to start making the triangle. On the third row, the values we place are the sum of the two numbers above it. So, repeating what we did before, the third row will read 1, 2, 1
The fourth row will read 1, 3, 3, 1.
The fifth row will read 1, 4, 6, 4, 1
Using this method, we can make the largest triangle in the world, if we so choose!
So, for the pizza dillema, we find the answer by counting down four rows starting at the second row. Then, not counting that first value of 1, we count two values over. The value that we land on is the number 6. We can order six different combinations of two-topping pizzas with the four ingredients at the pizzeria.
But with this fantastic triangle, there is more than meets the eye. In mathematical terms, Pascal’s triangle is a triangular pattern of binomial coefficients. A binomial coefficient is a group of positive integers that occur as a constant value in the binomial theorem.
A binomial is a polynomial with two terms. For example:
a + b or 1 + 2 or x + y
Now let’s say we take a binomial and put it inside parentheses, like (x+y) and then we give that binomial an exponent of 2. We now have the following binomial expansion:
(x+y)^2=(x+y)(x+y)=x^2+2xy+y^2
The binomial theorem shows us how to expand expressions when they are in the form:
(x+y)^n
With n representing the value of the exponent. For example,
We notice that the constant in front of each of those variables coincide with the values on the triangle.
For example,
(x+y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5
Since the exponent is 5, the constants coincide with the values of the row that lead with 5. Additionally, the exponents for the variables in each term add up to five.
For x, the exponents count downward
x^5, x^4, x^3, x^2, x^1
For y, the exponents count upward
y^1, y^2, y^3,y^4,y^5
Mathematically, it is represented as
The binomial expansion is just one of many uses of Pascal’s triangle. It also allows us to solve combinatorial problems, statistical problems, find Fibonacci numbers, solve compound interest, and more. It is one of the most useful tools in mathematics.
The triangle got the name from the mathematician Blaise Pascal who wrote The Treatise on the Arithmetical Triangle in 1654, even though the triangle had been around long before he wrote about it. In the treatise that he wrote, he presented a tabular presentation of binomial coefficients created out of columns and rows.
Pascal was born in Clermont-Ferrand, France, on June 19, 1623. Pascal’s mother died when he was only three years old. His father raised him and his two sisters. They eventually moved to Paris in 1631. Throughout his entire life, Pascal suffered from poor health. Regardless, he was determined to live a full, intellectual life. Pascal was so brilliant that he was considered a prodigy.
His father, Étienne, was a tax collector, a counselor to the French King, and a civil servant. Étienne chose to homeschool him and his sisters because he did not trust the rigor of the educational system in France. Even though his father was a mathematician, he discouraged his son from learning math at a young age. His father knew how fulfilling math was, and he felt that Pascal could immerse himself in the joy of math when he was older. But first, he wanted to teach his son humanities, philosophy, literature, and other classical studies.
Nevertheless, despite his father’s resistance to introducing him to math, Pascal decided to teach himself math. This story is quite similar to the story of Agatha Christie, whose mother did not want to teach her how to read until she was at least eight. Despite her mother’s discouragement, Agatha learned how to read by the time she was five. Sometimes you can’t stop brilliance!
By the time Pascal was twelve years old, his father would bring him along when he made his weekly visits to the Society of Mathematicians at Academy Libre. Here they would discuss current topics in science and math. At these meetings, Pascal had the opportunity to meet well-known mathematicians, including Marin Mersenne, Girard Desargues, Pierre de Fermat, and René Descartes.
Pascal was so inspired and motivated that by the time he was 16, he had published his first essay on conics.
He was truly ahead of his time. When he was sixteen years old, he designed and created a calculating machine. For three years, he designed 50 prototypes and 20 finished machines that he called Pascal calculators.
“All of humanity’s problems stem from man’s inability to sit quietly in a room alone.”
Pascal
When he was 23, he became fascinated with physics. A family friend introduced Pascal to Torricelli’s experiment, which involved a tube full of Mercury immersed in a bowl full of Mercury. What the experiment showed was that when the tube was placed in a bowl full of Mercury, the Mercury would fall in the tube to 760 mm. No more and no less. Even when the tube was moved, shaken, or tilted, it always remained at 760 mm. This consistency was due to the influence of atmospheric pressure. Pascal was fascinated with this experiment and, as a result, decided to immerse himself in the studies of physics.
By 1651, Pascal wrote A Treatise on the Vacuum, whichreferred to the Torricelli experiment. Sadly, this was also the same year that Pascal’s father died.
By 1654, Pascal published his Treatise on the Arithmetical Triangle. Meanwhile, he continued to struggle with poor health, as well as contend with religious beliefs. In 1654, he fully committed himself to God. As a result, from that point forward, most of his writings were philosophical. He wrote a work called Provincial Letters, which was a series of pastoral letters. He also started to compile a collection of writings. His writings were posthumously titled Pensées, which, in French, means Thoughts. Pensées was considered to be a preparation of Christian apologetics. It was a collection of about 1,000 fragments of his writings based on his beliefs about miracles and God’s proof of existence.
“Man’s greatness lies in his power of thought.”
Pascal
Pascal understood the value of thinking and mastered the art of deep contemplation. He lived an intellectually fulfilling life, despite his lifetime of physical ailments. On August 19, 1662, Pascal died from a malignant stomach tumor. He was only 39 years old. However, because of his ability to immerse himself in his academic studies, in his brief life, Pascal left a profound intellectual imprint in the fields of math, physics, and philosophy.