Ancient mathematicians provided the basis to bring Pascal’s Triangle into being before the birth of Blaise Pascal in 1623. One was the Chinese mathematician, Chu Shih Chieh, also known as Zhu Shijie, who was born about 1260 in Yan-shan, near Peking, China. He contributed to the formation of the triangle by computing its use for providing coefficients for the binomial expression of (a + b) n in his 1303 treatise, “The Precious Mirror of the Four Elements” (see figure 1 for Chinese version of Pascal’s Triangle in Chinese numerals). His studies helped create the basis of Pascal’s Triangle. He also wrote two books called the Suanxue qimeng and the Siyuan yujian, which were impressive works. They described polynomial algebra and polynomial equations, by the “coefficient array method,” that developed in northern China by the earlier thirteenth century. He died about 1320 and it is unknown where he was last seen. Pascal’s Triangle was his addition to the math discovery.

Another person who contributed to the Pascal’s Triangle was Omar Khayyam. He was a great eleventh-century Indian astronomer, poet, philosopher, and mathematician, who lived in what is now Iran. He was born May 18th in 1048 in Nishapur, Persia. Khayyam described the array of numbers in the future Pascal’s Triangle as a useful tool for representing the number of combinations of short and long sounds in poetic meters (see figure 2 for a glimpse of Pascal’s Triangle in Arabic numerals). He wrote several works, such as Problems of Arithmetic, a book on music and another on algebra. His studies of the future Pascal’s Triangle were the earliest records found of the triangle that would eventually bear Blaise Pascal’s name. He also contributed the Jalali calendar, new components to algebra, astronomical tables, and the Rubaiyat. He died December 4th in 1131 at Nishapur, Persia.

In the seventeenth century, Blaise Pascal helped develop the theory of counting in the Pascal’s Triangle (see figure 3 for a picture of Pascal). He was a child prodigy who became interested in Euclid’s Elements at the age of twelve. Four years later, he was conducting original research and wrote a paper of such quality that some of the leading mathematicians of the time refused to believe that a sixteen-year-old boy was the author. His knowledge and wisdom about mathematics gave him the strength to create new theories and equations for the future.

Pascal created a theory, known as “Pascal’s Theorem,” which stated that if a hexagon were inscribed in a cone, the points of intersection of the opposite sides will lie in a straight line. He employed his arithmetic triangle in 1653 (see figure 4 for the original triangle), but no account of his method was printed until 1665. The triangle was constructed with each horizontal line being formed from the one above it by making every number in it equal to the sum of those above and to the left of it the row immediately above it.

Pascal also made other contributions to mathematics. He created the first digital calculator, known as the Pascaline to help his father, who was a tax collector. Adding French currency was difficult because the currency consisted of different coins, worth different values. Pascal’s machine, however, was not a great success. The only function it could perform was addition!

Pascal abandoned mathematics in his later years and devoted his time and life completely to philosophy and religion. In 1658, however, while being unable to sleep because of a toothache, he decided to think about geometry to take his mind off the pain and surprisingly, the pain stopped! Pascal took this as a sign from God and heaven that he should return to mathematics. But for a short time he returned to his research until he was seriously ill with dyspepsia, a digestive disorder. He lived the remaining years of his life in excruciating pain, doing little work until his death at age thirty-nine in 1662.

Pascal’s Triangle is an array of numbers that has numerous applications in math. It is not a geometric figure, but an array of natural numbers shaped in the form of a triangle. The sum of two adjacent numbers is equal to the number directly below and between them. The triangle continues infinitely. The numbers in horizontal lines make up rows. Numbers in an oblique line on the diagonal are called diagonals, or columns. They are numbered from the zero diagonal to infinity. Numbers in the Pascal’s Triangle are referred to as elements.

In order to construct Pascal’s Triangle, one must first know that entries in the triangle are given the row number and place within that row. Starting with row zero and place zero, the number one will always be at the top of the triangle and at the first and last entries in each row. In rows zero and one, there will always be ones in all the entries. For row three, you must employ the rule, which states that any number within the triangle is the sum of the two numbers immediately above it. Knowing that rule will help one construct the triangle perfectly.

For example, the triangle will start with the number one in row zero. Then, two ones are located in row one because the blank spots next to the number one in row zero are considered to be zeroes. In row two, the sequential order will be one, two and one because the number one must always be on the sides of the triangle and the number two is the sum of the numbers directly above it, which are one and one.

As on would imagine, with all these numbers in the triangle, there are many patterns present. There are a few that are obvious. One could be found by summing the numbers in each row. The sum of the numbers in any row equals 2n, where n is the number of the row. For example, in use row four, the sum of the numbers is sixteen and 24 equals sixteen.

Another pattern involves the prime numbers. For any row whose second placed entry is prime, then all the numbers in that row, excluding the ones on the ends, are divisible by that prime number. For example, let’s use row seven. The numbers seven, twenty-one and thirty-five are all divisible by seven, the prime number. However, when the number is composite, the pattern will not work. In row eight, twenty-eight and seventy are not divisible by eight.

A third pattern within Pascal’s Triangle is known as the Hockey Stick Pattern (see figure 5 as an example). If a downward diagonal of numbers are selected, beginning with any of the ones bordering the sides of the triangle and ending as any number inside the triangle on that same diagonal, the sum of the numbers inside the selection is equal to the number below the last number in the selection that is not on the same diagonal. Using figure 5 as an example, one plus nine equals ten. One plus five plus fifteen equals twenty-one. One plus six plus twenty-one plus fifty-six equals eighty-four.

There are other types of sequences that can be found within Pascal’s Triangle. One such sequence is that the diagonals beginning from row one and extending downward (exclusive of the outer ones column) forms the counting numbers (ex. one, two, three, four, five…). Another sequence is demonstrated by the number of points required to make triangles of progressively greater sizes (ex. one, three, six, ten). That sequence can be found beginning with the “1” places within row two (one, three, six, ten, fifteen, twenty-one).

A special form of math that is patterned from Pascal’s Triangle is known as the Fibonacci sequence. This is applicable, in many natural mathematical situations ranging from sunflowers to pinecones. It is also the source of the Golden Ratio. The Golden Ratio is a special number approximated to 1.6180339887498948482. The Greek letter Phi represents it. Like Pi (approximates to 3.14), the digits of the Golden Ratio go on forever without repeating, also referred to as a nonrepeating decimal. The ratio of a term in the sequence to the term right before it approaches the Golden Ratio as the sequence reaches infinity.

Starting with one and adding the preceding numbers on the diagonal provide the next term for the Fibonacci sequence. For example, one plus zero equals one, one plus one equals two, one plus two equals three, two plus three equals five, and three plus five equals eight. In simpler form, the first six terms of the Fibonacci sequence, are one, one, two, three, five and eight. This sequence can be found in Pascal’s Triangle by taking the sums of the diagonals as shown in figure 6.

Pascal’s Triangle is used in many aspects of our life. One use is in probability, specifically within the field of medicine. Frequently doctors experiment with combinations of new drugs to combat hard-to-treat illnesses such as AIDS and hepatitis. Let’s assume that a drug company has developed five antibiotics and four immune-system stimulators to treat AIDS. Pascal’s Triangle can be applied to figure out how various treatment programs consisting of three antibiotics and two immune-system stimulators can treat the disease. Pascal’s Triangle can help speed our computations of various treatment protocols combinations.

To solve this problem, we must select the drugs in two ways: first the antibiotics and then the immune-system stimulators. With the antibiotics, we are choosing three drugs from five, which can be done in C (5,3) ways. Looking at the fifth row of Pascal’s Triangle, the third entry is ten, which makes C (5,3) equal ten ways. When choosing two immune-system stimulators from four, it can be done in C (4,2) ways. Looking at the fourth row of Pascal’s Triangle, the third entry is six, which makes C (4,2) equal six ways. Thus, to find the answer for the antibiotics and the immune-system stimulators together, the answer will be ten multiplied by six equals sixty ways.

Another application of Pascal’s Triangle involves formulation of combinations. Suppose that a person has five hats on a rack and they want to know how many different ways they can pick two of them and wear them. It does not matter to the person which hat is on top, it just matters which two hats are picked. We are trying to figure out how many ways that person can pick two objects from a set of five objects. The answer is the number in the second place in the fifth row of Pascal’s Triangle, the number ten. The binomial coefficient in this case is six to three or six choose three where the number ten is located. Pascal’s Triangle is like a grid, with ordered pairs, except the numbers are labeled with row, then entry.

Pascal’s Triangle is also used in algebra. Suppose that one has the polynomial of x plus one and they want to raise it a few powers, like one, two, three, four and five. If you make a chart of what you get when you do these power raisings (ex. (x + 1) 1 = 1 + x), it will look like Pascal’s Triangle only with coefficients. Because of this connection, the entries in Pascal’s Triangle are called the binomial coefficients.

A binomial is an algebraic expression with two terms, (ex. x + y, 3×2z – 2siny) having either one addition or one subtraction sign. A coefficient is a number that is next to a variable (ex. In x + y, the coefficients are one and one and in the algebraic expression 3×2z – 2siny, the coefficients are 3 and –2). When inside the Pascal’s Triangle, the terms are multiplied by themselves to create binomial coefficients.

In conclusion, Pascal’s Triangle is an array of natural numbers that are used in algebra and probability, mostly to find combinations. The dedicated works of ancient scholars and mathematicians have brought math to a whole new dimension. Without their studies and hypotheses, our world would not be as modern as it is today.