# Unexpected Rewards

On a recent rabbit hole of learning, I have been consuming a ton of videos from the YouTube Channel Numberphile, which has ignited a curiosity and love of math that I didn’t know I had. Let me be straight, I have loved math for a long time, but this delightful journey into a variety of topics has proven to shed light on areas I once didn’t understand, connected previously disjoint topics, or shown whole new ideas and concepts that are fun and incredible. In this entry, I wanted to highlight some of that learning and showcase this application.

One on hand, I have learned the following four unique and interesting ideas I previously didn’t know.

1. Operations with shapes
2. Multiplication with halving and doubling
3. Heroes and Super Heroes
4. Ptolemy’s Theorem

On the other hand, I’ve learned a lot about different math ideas of which I will highlight some key features here. The first is understanding and accessing graph theory, which is a topic I first heard about during a talk in graduate school. I remember being completely lost in the talk and not understanding any of how the pieces connected or shared together. After watching a few episodes of Numberphile, where graph theory played a central role, I was able to access the understanding well enough to read a book I’d purchased years prior but was unable to access. Reading Richard Traduea’s “Introduction to Graph Theory” covered many of the concepts highlighted in the Numberphile Videos which was fantastic, but the Numberphile videos made the content seem easy and so accessible.

I will write a separate entry for both graph theory and abstract algebra as each warrants its own entry in upcoming discussions.

Returning to the ideas item 1 above, the two items I would like to highlight are the following: 1) Using circles to calculate square roots, and 2) Using parallel lines for multiplication.

Suppose you wish to calculate the square root of 9, you may know that the product of three and itself yields nine, so the square root is three. Here’s the steps:

1. Draw a number line starting at the origin
2. Count off, until you have reached the number you are looking to determine the square root of, in our case, we would count out 9 units.
3. Next, add 1 more to your location, and from 0 to this location (one more than the value you wish to calculate the square root of) will be the diameter of the circle we are constructing. In our case, our diameter is 9+1=10.
4. Now, we construct the center by marking the spot which is half the radius. For our example, we have the ordered pair (5,0), and a radius of 5.
5. Draw the circle.
6. From the location of the point, you wish to determine the square root of, draw a line perpendicular to the axis from the number you wish to know the root of, in our case, we draw a vertical line from 9.
7. When your perpendicular line intersects the circle you constructed earlier, the length of the line you just drew is the desired quantity. In our case, the length turns out to be 3 as expected.

The next example is for parallel lines and finding products. Construct two lines that intersect at any acute angle. Along each line, equally space the distance between integer values marking a progression of increasing values. Suppose you wanted to calculate that the product of 1 and 3, then draw a line connecting 1 from the “horizontal axis” to the 3 on the angled axis. To calculate, another multiple of 3, say 4 times 3, draw a line parallel to your first line through the point 4 on the “horizontal” axis. The point of intersection between the parallel line through 4 and the “acute angled” line will be the resulting product, 12 in our case.

Now comparing the products of two numbers using the halving and doubling strategy made famous by two different nationalities, which we will compare below.

The “Russian” format takes the idea of double and halving in the following format:

1. Write the numbers horizontally
2. Take the first of the two numbers from left to right, and divide it by 2
3. If you get a remainder from the step above, take only the whole number part
4. Continue this process on the left until you reach unity
5. On the right-hand side, double the process until you reach the same level as the one on the left
6. Look at the left column, throw out any even number rows
7. Add up the corresponding values on the right column of the remaining rows

For the Egyptian format, it is a similar process with a few key differences. The process is outlined below:

1. Write the numbers horizontally
2. Below the number of the left, write the number 1
3. Continue to double this number until the result is above the original number
4. Now on the right-hand number, rewrite the same number below it, the 5 in our case
5. Continue to double the number until you have reached the last row equal to the left-hand side
6. Now, use the numbers add up to the first number on the left, so we used 16 and 2, the numbers in the same rows as 16 and 2 are added together to create the resulting product, i,e. 16=80 and 2=10, so 80+10=90.

The Egyptian model is nice since it doesn’t require the halving to remove unwanted quantities and requires you to balance the addition of the two sides.

The unexpected rewards of watching Numberphile has opened up lines of inquiry, connections, and understanding I wouldn’t have expected and has me excited about learning a variety of types of math…though it hasn’t convinced me to say maths….yet.