In Mathematics, What Is Pascal's Triangle?

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  • Written By: Tara Barnett
  • Edited By: Melissa Wiley
  • Last Modified Date: 16 May 2017
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Pascal's triangle is a triangular array of numbers that relate to one another in interesting and useful ways from the perspective of mathematics. Forming the triangle is a relatively straightforward affair in which two adjacent numbers are added together to form a number directly below in the triangle. This expands the triangle at a regular rate and creates rows and diagonals of numbers that can be analyzed in many ways. The relationship between the numbers can be expressed as a mathematical formula, but constructing the triangle does not require this formula even though the reasons why patterns evolve are much more complex. Not only do the individual numbers relate to one another in the triangle, but the sums of the numbers and the patterns they form also have interesting applications in mathematics.

Most people in the West associate Pascal's triangle with Blaise Pascal due to his detailed investigation into the mathematical properties of these numbers, but this triangle and the acknowledged relationship of these numbers predates Pascal by centuries. In China, Yanghui studied and described this series, so it is more strongly associated with that mathematician. Pascal arranged his investigations into this topic into a treatise, creating a unified evaluation of the many complexities of this triangle.

The connection between Pascal's triangle and other mathematical concepts is complex. It has relations to the Fibonacci numbers, the Sierpinski triangle, and many other established mathematical patterns. It also has a number of practical uses, such as calculating combinations. Mathematicians can extend Pascal's triangle into negative numbers, revealing even more interesting patterns.

Some of the most interesting aspects of Pascal's triangle involve calculating sums of numbers across rows or shallow diagonals. The patterns of these sums relate to various other sequences. Along the diagonals, numbers progress in significant patterns as well. Many interpretations of Pascal's triangle label the point of the triangle as zero and the line with two numbers as one. Given the way the triangle is formed, the outer edge of the triangle is always one.

There are many different ways in which Pascal's triangle can be used. In very basic math classes, it is often used to think about interesting patterns relating to mathematics and to encourage people to think about the relationships between numbers. For more advanced mathematicians, the triangle presents a number of different patterns that can be useful when thinking about statistics and probability. One of the most common exercises performed in basic math classes using this triangle involves shading even and odd numbers different colors in order to point out the patterns that form.


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