Hello once again!!! Have you ever wondered why patterns occur? Well... we don't have the answer for all patterns in the world, but we can explain one pattern that occurs in the math world. The following problem is based on the multiplication of "repunits" whose digits are compromised of the number 1.
Compute the following:
1 × 1 =
11 × 11 =
111 × 111 =
1111 × 1111 =
How far does this pattern go?
If we multiply out the four expressions above we get the following results:
As we can see the number in the center is the number of 1's that exist in the original number of the original equation. For example, 121 has two as its middle number because 11 has two 1's. Also, as the number of 1's increases in the equation, so does the pyramid that occurs in each line. For instance, 11 × 11 is 121 and when we multiply 111 × 111, we get 12321, an up and down pyramid that goes forward until it gets to three and then back down after three. It is a symmetric pyramid of numbers.
If you would like to see this pattern as it develops, watch it here in a video posted by James Sousa: http://www.youtube.com/watch?v=k1X1JCNHjRw
James' YouTube channel and webpage mathispower4u offer more mathematical resources, including short video explanations (similar to Khan Academy) and free downloadable open source textbooks.
Look out for patterns, they are everywhere!!