Mathematics in Real Life: Numerical Analysis
I studied math in school. No surprise given my content (see here), but I guess it makes sense to come out and say it. Thus it undoubtedly has shaped the way that I see the world and make decisions. Everyone sees the world through a different lens and getting another’s perspective outside of one’s experience and modality is important for growth. When it comes to my perception it usually is useful for me to describe things in terms of mathematical concepts and models. As I’m sure anyone can guess, this can be very frustrating for others who don’t share my passions.
One of the concepts that I’ve seen over and over is an application of numerical analysis. This branch of mathematics deals with finding approximate solutions to problems. It is useful when it isn’t easy to find an analytical solution to a problem. For example, most people that have completed high school in the US will remember factoring algebraic equations or finding the value of x. This is a basic form of an analytical solution:
x^2 - 1 = 0x^2 = 1x = 1 or x = -1
Simple, there was (hopefully) no need for a calculator and you could find the answer just with pen, paper, and symbolic manipulation. On the other hand, imagine the following:
sin(x) + x^5 - log(tan(y) - x^2) = e^(x^2)
Even at a first glance you can decide that this is more suited to a computer. And this is where numerical analysis comes in handy. Using several different algorithms a mathematician can quickly find a very accurate approximation of the true solution of a given function or model. An algorithm is really just a set of instructions that we follow to reach a given solution from an input.
One very basic but very effective algorithm is the bisection method. Imagine you have a dictionary and you need to find the page that has the definition of “parsimonious”. How do you proceed? Without getting pedantic you can say that for every pair of words, there is a word that is first in alphabetical order. That being said, open the dictionary to a random page. Usually you’d try to open it as close to halfway as possible, but that’s more a stylistic choice than anything. Next, look at the page and see what letter you are in. If the page you are on has letters that come before choose a new random page in the right half of the dictionary, else choose a random page in the left half. Repeat the process until you find your word.
Congratulations! You have just used bisection to find your word, and have thus used mathematics in real life.
The previous example was very concrete, and even a little simplistic. I like to see the more philosophical sides of mathematics in my life. Any time that you try a new activity you are correcting yourself to perfect your technique. You might be practicing a golf swing or shooting hoops. Maybe you are drawing or taking pictures. Whatever the task you are using some sort of optimization algorithm to refine your technique without even knowing it. How do you adapt your basketball shots? Do you add one inch more of force to the shot every time until it goes in the basket? No, you shoot a shot, then probably over correct, and then dial it back.
When I was in Army we learned about calling for fire. This is the method that we used to request cannons to drop bombs on the enemy in the battlefield. Most of the time the gun-bunnies are pretty good at what they do. Sometimes it takes a bit of adjustment to get it right. A common technique to adjust the attack is bracketing, which is just the Army version of bisection: add a certain amount of oomph to the shot, then split the difference until you kill them all.
Without getting too high in the clouds, I see that refinement comes everywhere in our lives. We act, then refine, but it is usually in the form of a swinging pendulum. Every action and adjustment take you swinging around your intended target and it seems like erratic motion that will never settle where we want it to. We feel that we are out of control and that our efforts are useless to control our outcomes, but this is the crux of science and mathematics. We Keep trying and refining until we get it stable.
I see it all the time in people that I train in weightlifting. Dialing in a diet and exercise regimen is very difficult to start. At first you start working out and feel great. Then you get sore and frustrated that you are not getting the results you want when you want them. But if you stick with it through the rocky times eventually you settle into a rhythm. That rhythm and stable point is where you can actually start growing. Once you have the foundation and can stick to a repeatable process you can really make progress.
Just like bisection, the results in any new endeavor can be wildly inaccurate at first. Over time, however, you refine the technique to get closer and more stable results. Once you have that stable, fixed point you can actually start working and improving on your situation.
In my mind’s eye I tend to see this as a plucked string. You pull the string and let it go and watch it flutter. At first it swings widely top to bottom and looks like it might not stop, yet eventually it starts to slow and ends up humming back to the stable start point.
Our lives are similar. When we start something new it feels crazy, fresh, uncomfortable. It feels like we will never get back to a stable point and that we have been foolish to get out of our comfort zone. And every time we feel like it’s all gone terrible it ends up smoothing out and calming down. We can rest and actually enjoy what we are doing. The worst thing we can do is invest the time to try something new and then abandon it all just as it is starting to smooth out. The string always calms down and we always end up adapting to the stress, we just have to let it.
Seth Godin wrote a short book called The Dip in the late 00s that expressed a similar concept. At the start of any activity or venture it is fun and easy and exciting. Eventually, that all fades and you begin the long trek up the mountain towards perfection. It is easy to become distracted from the long term goal because of the diminishing returns, but the grit to stick through the low points is what is important to long term success.
I’m not saying that it is easy or that it is faster by understanding, but it certainly helps to know going into any new venture that you will experience the wild swing of the pendulum or the violent fluctuations of the string. Throughout the process remember to use your bisection to dial into the best technique and to remember that eventually you will hit a stable point from which you can truly grow.