Pascal's triangle
Pascal's triangle is a triangular arrangement of numbers that illustrates various mathematical patterns and relationships. Named after the French mathematician Blaise Pascal, who described it in the 17th century, the triangle features numbers organized in such a way that each number is the sum of the two directly above it. While Pascal is credited with its formal description in his 1654 work, the triangle was known and utilized by mathematicians in China and Persia centuries earlier.
In its modern representation, each row starts with the number 1, and interesting patterns emerge, such as the first diagonal consisting of all 1s, the second diagonal representing counting numbers, and the third diagonal showcasing triangular numbers. Furthermore, the sums of numbers in each row double as one progresses down the triangle, revealing a notable symmetry. The triangle's properties extend to various fields, including algebra and probability, making it a significant topic in mathematics. The exploration of Pascal's triangle continues to reveal new insights, reflecting its enduring relevance in the study of numbers and their relationships.
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Pascal's triangle
Pascal's triangle is a triangular arrangement of numbers named for French mathematician and philosopher Blaise Pascal, who described it in the seventeenth century. The way the numbers in the triangle are organized reveals several mathematical patterns and relationships among numbers. These patterns and relationships may occur across rows, columns, or diagonals.
Overview
Blaise Pascal was born June 19, 1623. He was the son of Étienne and Antoinette Pascal. Pascal grew up in Paris, France, where he was surrounded by some of the most brilliant scientific and mathematical minds of the period. He developed an interest in mathematics at an early age and wrote his first essay on mathematics when he was an adolescent. Pascal first described an "arithmetical triangle" in his 1654 work Traité du Triangle Arithmétique, which was published in 1665, after his death. For this reason, Pascal's triangle now bears his name.
Although the triangle is named for Pascal, it was discovered and in use by mathematicians in both China and Persia about five hundred years earlier. Writing in Traité du Triangle Arithmétique, Pascal described the triangle this way: "The number in each cell is equal to that in the preceding cell in the same column plus that in the preceding cell in the same row." Modern representations of Pascal's triangle look somewhat different than the original arrangement, but the patterns are still evident.
In the modern arrangement, beginning with the number 2 in the third row, each number is the sum of the two numbers directly above it. For example, the two numbers directly above the number 2 in the third row are 1 and 1, and 1 + 1 = 2. The pattern works the same way for other numbers in the triangle. In the last row, the third number is 10. The numbers directly above 10 are 4 and 6, and 4 + 6 = 10.
Other patterns are also evident in the triangle. For example, the first diagonal is all 1s. The second diagonal is the counting numbers: 1, 2, 3, 4, 5, and so on. The third diagonal represents the triangular numbers: 1, 3, 6, 10, and so on. The triangular numbers are derived from patterns of dots that form triangles. Another pattern emerges from the horizontal sums of rows. The sum of the numbers in each row is double the sum of the numbers in the row before it. For example, the sum of the numbers in the third row is 4 (1 + 2 + 1 = 4). The sum of the numbers in the fourth row is 8 (1 + 3 + 3 + 1 = 8). The fifth row's numbers sum to 16. Another interesting aspect of the triangle is its symmetry. When an object has symmetry, a line can be drawn directly through its center, and the left and right halves will mirror each other.
Mathematicians have identified several other patterns within Pascal's triangle. In addition, the triangle has numerous applications in specific mathematical fields, such as algebra and probability.
Bibliography
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