Happy π Day math lovers!
We here at Say It With Science would like to celebrate it with you by sharing some interesting trivia about one of our favorite physicists, Richard Feynman, and one of our favorite constants, π (pi). The Feynman Point is a sequence of six 9’s beginning at the 762nd decimal place of π, named after Nobel Prize winning physicist Richard Feynman. Feynman had memorized π to this point so that he could end his recitation of the mathematical constant by saying “nine nine nine nine nine nine and so on…”. At this point someone less knowledgeable about mathematics might assume the number continues this way forever, however we know better. It is believed that π is a normal number, meaning that its digits are as uniformly distributed among the digits 1 through 9 (or the digits of any other base you choose to use). If π is a normal number then the chances of coming across six 9’s in a row is 0.08%. Strange occurrences like this are what makes math beautiful. π Day is a perfect reason to start memorizing as many digits of π as you can! Happy π Day!
Gabriel’s Horn is a three dimensional surface that contains a finite volume but has an infinite surface area. It is made by taking the two dimensional graph of y=1/x and revolving it around the x-axis (with the domain of x ≥ 1). If we look at x coordinates from 1 to a, the volume can be found by the equation: Which is really just sum of the area of each circular cross section, hence it is the integral of πr2 (with r being the distance from the x-axis to the function). The surface area of the horn can be found by the equation: (Which is slightly more complicated to derive but a full explanation can be found here.) As you can see if we let a approach infinity, the surface area diverges, whereas the volume converges to π.
(Technically called Nikhilam Navatashcaramam Dashatah) This is a quick and simple way to multiply any two numbers. It’s easiest when the numbers are both close to a power of ten, but it will always work. The first step is to chose a power of ten that the numbers are closest to. In my example I will find the product of 14 and 12. Since 12 and 14 are close to 10 I will chose 10. 14 is 4 more than 10, and 12 is 2 more than 10, so I will write these numbers off to the side, as shown.
+4 times +2 is 8 so I write this number on the right. Then I cross add the 14 and the 2 or I add 12 and 4 to get 16. I write this number to the left, and put these two numbers together to get the right answer 168. (Although I say “put these numbers together” what is actually going on is that 16 is being multiplied by 10 then 8 is added. Knowing this will be helpful when the number on the right is larger than the chosen power of ten.)
Here’s an example with larger numbers. Since they are closer to 100, 100 is used instead of 10. This time the numbers are less than the chosen power of ten, but the same method can be used. Multiply -8 by -11 to get 88 (write that on the right), and add 89 to -8, or 92 to -11 to get 81 (write that on the left). 81 is then multiplied by 100 (since that is the power of ten we chose) and 88 is added. Hence the correct answer to 92x89 is 8188. This is a neat trick, but why does it work? consider the following algebra:
Say x is the power of ten we chose. Then a and b are the the two numbers that represent how far our factors are from the chosen power of ten.