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Posted by on Feb 19, 2017 in Tell Me Why Numerous Questions and Answers |

What Are Some of the Unique Properties of Pascal’s Triangle?

What Are Some of the Unique Properties of Pascal’s Triangle?

In mathematics, Pascal’s triangle is a triangular array of the binomial coefficients. Since Pascal’s triangle is infinite, there’s no bottom row. It just keeps going and going. In the Western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy.

Blaise Pascal didn’t really “discover” the triangle named after him, though, he did develop new uses of the triangle’s patterns, which he described in detail in his mathematical treatise on the triangle.

The basic pattern of Pascal’s triangle is quite simple. Despite its simplicity, though, Pascal’s triangle has continued to surprise mathematicians throughout history with its interesting connections to so many other areas of mathematics, such as probability, combinatorics, number theory, algebra, and fractals.

So why is Pascal’s triangle so fascinating to mathematicians? The more you study Pascal’s triangle, the more interesting patterns you find. This is important in mathematics, because mathematics itself has been called the “study of patterns” and even the “science of patterns.”

Many of the mathematical uses of Pascal’s triangle are hard to understand unless you’re an advanced mathematician. Even young students, however, can recognize a couple of the simpler patterns found within Pascal’s triangle.

For example, the left side of Pascal’s triangle is all ones. The next set of numbers in, known as the first diagonal, is the set of counting numbers: one, two, three, four, five, etc. You’ll also notice an interesting pattern if you add up the numbers in each horizontal row, starting at the top. The sums double each time you descend one row, making them the powers of the number two!

Content for this question contributed by Jeff Hirst, resident of Bethel Park, Allegheny County, Pennsylvania, USA