Pascal’s Triangle

Pascal’s Triangle

by Eugene Toth

Mathematicians named Pascal’s triangle after the French mathematician Blaise Pascal.    Pascal’s triangle is a triangular graph.

In Pascal’s triangle each number is the sum of the two directly above it.  The numbers in each row are numbered beginning with 0 for the first row. Each number is positioned either to the left or to the right of the numbers in the rows above. The sum of the elements of a single row is twice the sum of the row preceding it.  A Pascal’s triangle can expand infinitely.

Mathematicians centuries ago in India, Greece, Iran, China, Germany, and Italy studied this triangle.  In India a 10th century book showed a triangle of numbers.   The triangle has surprising properties.  Pascal and others associated these triangles with religion or magic.

You can construct a Pascal’s triangle as follows.   On row 0, write only the number 1.   Then, in the following rows, add the number above to the left with the number above to the right to find the next number below.   If there is no number to the right or left, put a zero in its place. For example, the first number in the first row is 0 + 1 = 1.  In the third row add the numbers 1 and 3 to produce the number 4 in the fourth row.

Surprisingly, the value of a row is a power of 11.  For example row 2 shows 121 which is 112.   The first diagonal row on the left of the triance shows only one.  The second diagonal row shows counting numbers. The third row shows triangular numbers

Triangular Numbers

A triangular number counts the objects that can form an equilateral triangle.  The triangular numbers are 1, 1+2, 1+2+3, 1+2+3+4.  Some triangular numbers are even and some are odd.  Two even triangular numbers follow two odd triangular numbers.  One can calculate the triangular numbers by adding up consecutive numbers.  For example, the seventh triangular number is equal 1+2+3+4+5+6+7 which comes to 28.

Pascal’s Tetrahedron

Mathematicians call the three dimensional version Pascal’s Pyramid or Pascal’s tetrahedron.  The pyramids of ancient Egypt had a square base and four triangular sides.  “Pascal’s tetrahedron” has four triangular surfaces.

3 thoughts on “Pascal’s Triangle”

1. Thank you. Look at Fibonacci Sequence to learn more about it. The two are actually connected.

1. Rebeka

Yeah, when we were learning about it, the first thing I asked my teacher was, after asking if it was a tessellation, if it was related to the Fibonacci Sequence. She said, in a way it is similar, as each new number builds off the previous two. I also was really intrigued by your article on the Fibonacci Sequence, but of course being the clumsy person I am, I accidentally deleted my comment before posting it. Keep up the great work! 🙂

2. Rebeka

I love Pascal’s triangle and find it very interesting. I learned a little bit about it in school a couple weeks ago