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.
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.
