Pascal's triangle is really cool. It can go on forever. The way it forms is by adding the numbers next to each other and then writing it in the next row down, between the two numbers. This is just its basic function however. There are a lot of things Pascal's triangle reveals about numbers. First, as we have already seen, you can use Pascal's triangle to expand powers like (x + y)^5. You just take row 5 of Pascal's triangle and use it as the coefficients, and then have the power on x decrease as the power on y increases. Another cool thing to note about Pascal's triangle is how it works with the powers of 11. Take 11 to the zero power. That equals 1, row zero of Pascal's triangle. 11 to the first equals 11, row 1 of Pascal's triangle. 11 to the first equals 121, which is, yes, row 2 of Pascal's triangle. Think the pattern holds up all the way? 11 to the ninth power is 2357947691. Row nine of Pascal's triangle is 1 9 36 84 126 126 84 36 9 1. At first, that doesn't appear to follow the pattern at all, but take a closer look. You have to add the double digits to the number to the left of it. So 9+3 (from the 36)= 12. Take the 1 from 12 and add it to the 1. Now you have 2, 2, 84. Add the 8 and 4 and add the tens digit (1) to the 2. Now you have 2, 3 and 2 (remaining from the 12). Now add the numbers to the right of
it (1 and 2) and now you have 5. Continuing in this way, you will get 11 to the ninth power, or 2357947691. And yes, by the way, if you add the numbers in the ninth row of Pascal's triangle, you get 512, which is 2 to the 9th-- a much easier pattern to see.
There is a cool connection to how the triangle relates to geometry. If you set up a chart going from point to segment to triangle to quadrilateral, etc (you just add a side to the side of the polygon each time) and you laid out Pascal's triangle under those rows, it would provide you with information about the formations in circles. Taking row three of Pascal's triangle, in a circle with three POINTS, you would have three SEGMENTS, and one TRIANGLE. This ignores the 1 that is always the first digit in Pascal's triangle, but you could think of it being in 1 circle. Here is an image to help visualize (from
www.jamesrahn.com)
Blaise Pascal was a Frenchman who lived from 1623 to 1662. He did a lot with math, but he was also known for his work in chemistry. The unit pascal (Pa), a measure of pressure, is named after him.
