Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.
In this lesson, you will learn about some of the many patterns found within Pascal's triangle, a set of numbers that has been loved by mathematicians for centuries.
Pascal's triangle is a set of numbers, arranged in a triangle, that contains an amazing number of patterns within it. Pascal's triangle is used in the binomial theorem, a rule that allows you to raise expressions with two terms, like x+y, to high powers easily. But the triangle is also fun to study just for its many patterns.
The numbers are arranged in a bowling-pin pattern, starting with the number 1 at the top of the triangle and the two number 1s in the first row beneath. To find the numbers in the remaining rows, take the number at the left and add it to the number to the right. Here, the two numbers connected by an arrow (1 and 2) are added to get the number in the blue circle, which is 3.
This diagram only shows the first eight rows (we don't count the 1 at the top as a row), but Pascal's triangle goes on forever.
There are many, many patterns within the triangle. Let's look at a few examples:
Counting numbers (blue) and Triangular numbers (green)
In this image, the most obvious pattern is found in the blue rectangle that contains the numbers 1 through 5. In the green rectangle just below it to the left, you can see the triangular numbers, or those obtained by a certain number of objects that form a triangle. For example, you could make a triangle out of three dots and enlarge it by adding a row of three dots to get six, and a row of four dots, which would give you ten.
Also, if you add together the numbers in each row together, you get the following pattern: 2, 4, 8, 16, 32, and so on. That pattern shouldn't be too surprising if you've added the numbers correctly.
Now, if you create diagonals, as shown in red in this image, and add the numbers found on the diagonals, you get the numbers found in the Fibonacci sequence, where each number is the sum of its two predecessors.
The Fibonacci numbers
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To see how the sequence works, choose a number somewhere in Pascal's triangle. For example, suppose you chose the number 4 in the fourth row. There will be four paths you can take from the top of the triangle to that 4 (no backtracking allowed). The same is true of any number in the triangle. For instance, if you want to get to the 6 from the top, there are six paths from the top of the triangle that would work.
Triangle showing four paths to 4
You can also find some fun patterns within Pascal's triangle by highlighting only the numbers divisible another number. Here, in the triangles formed by the white rectangles, all the numbers in the rectangles are divisible by 5, such as 10, 15 and 20.
Multiples of 5 in white
The Pascal triangle is named after the mathematician, Blaise Pascal, who was born in 1623. However, the triangle was used around the world centuries before. The picture shows a representation of Pascal's triangle made in the 1300s in China.
Pascal's triangle has been studied for centuries. It is a triangular set of numbers that contains within it a lot of interesting patterns, like triangular numbers and the Fibonacci sequence. It is used in the binomial theorem, which is a rule that allows you to raise expressions with two terms, such as x and y.
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