Quck answer
Pascal’s Triangle is a mathematical pattern that consists of numbers arranged in a triangular shape. Each number in the triangle is the sum of the two numbers directly above it. The triangle starts with a row containing only the number 1, and each subsequent row is created by adding the adjacent numbers from the previous row. Pascal’s Triangle has many interesting properties and applications in various fields of mathematics, such as binomial expansions, probability theory, and number theory. It is named after the French mathematician Blaise Pascal, who introduced the triangle in the 17th century.
While we were in the Wonderopolis cafeteria, we witnessed an amusing interaction between a square, a triangle, and a circle during their lunch break:
Square: What are you two eating?
Circle: A piece of pi.
Triangle: A slice of pizza.
Square: Cool…there’s nothing like a good square meal to keep you going throughout the day!
Triangle: See you later, Circle! It’s getting too hot in here.
Circle: You’re right, triangle. I’m 360 degrees!
These punny characters went on with their jokes for a while, but we couldn’t bear to listen to any more bad geometry jokes!
Putting jokes aside, today’s Wonder of the Day focuses on a special version of one of those shapes: the triangle. Specifically, we will be discussing Pascal’s triangle.
Pascal’s triangle is an infinite, equilateral triangle made up of numbers. The numbers in Pascal’s triangle follow a simple rule: each number is the sum of the two numbers above it.
When you look at Pascal’s triangle, you will notice that the top number is one. All the numbers on each side going down from the top are also ones. The numbers in the middle vary, depending on the numbers above them.
Since Pascal’s triangle is infinite, there is no bottom row. It goes on and on. Pascal’s triangle is named after Blaise Pascal, a French mathematician who used the triangle in his studies of probability theory in the 17th century.
However, Blaise Pascal did not actually “discover” the triangle named after him. It has been studied worldwide for thousands of years. Historians believe that ancient mathematicians in India, China, Persia, Germany, and Italy studied Pascal’s triangle long before Pascal was born. Pascal did, however, develop new applications for the patterns in the triangle, 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, Pascal’s triangle has continuously surprised mathematicians throughout history with its intriguing connections to various other areas of mathematics, such as probability, combinatorics, number theory, algebra, and fractals.
So why is Pascal’s triangle so captivating to mathematicians? The more you delve into Pascal’s triangle, the more fascinating patterns you discover. This is significant in mathematics because mathematics itself has been referred to as the “study of patterns” and even the “science of patterns.”
Many of the mathematical applications of Pascal’s triangle are difficult to understand unless you are an advanced mathematician. However, even young students can recognize a few of the simpler patterns found within Pascal’s triangle.
For instance, the left side of Pascal’s triangle consists of all ones. The next set of numbers in, known as the first diagonal, is the set of counting numbers: one, two, three, four, five, and so on. Additionally, if you add up the numbers in each horizontal row, starting from the top, you will notice an interesting pattern. The sums double each time you move down one row, making them the powers of the number two!
Try It Out
Are you ready to have some mathematical fun? Ask a friend or family member to help you explore the following activities:
- If you think you can create your own Pascal’s triangle, you can print this Pascal’s triangle Worksheet to practice filling in the missing numbers. The worksheet also includes other problems related to Pascal’s triangle. How many of them can you complete?
- For fun, you can check out The Twelve Days of Christmas and Pascal’s Triangle online. Do you think using Pascal’s triangle would be a simple way to keep track of all the gifts? Why or why not?
- In a sample Pascal’s triangle, can you find all the numbers that are multiples of 3, 5, or 7? Give it a try! You can go online and try the Coloring Multiples in Pascal’s Triangle activity. Just click “Roll Random Value” and then click on all the multiples of the given number. Challenge your friends and family to see who can do it faster!
Sources of Wonder
- http://www.livescience.com/51238-properties-of-pascals-triangle.html
- http://www.mathsisfun.com/pascals-triangle.html
- http://www.maa.org/press/periodicals/convergence/mathematics-as-the-science-of-patterns-mathematics-as-the-science-of-patterns
- http://mathworld.wolfram.com/PascalsTriangle.html
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