Saturday, May 2, 2015

Everyday Algebra

Recently I have ran more and more often into the following statement.

“Well, another day has passed. I didn't use algebra once.”

I even saw it printed on a t-shirt! It seems an increasing number of individuals need justification to not care about, or bother learning algebra. I could similarly remark, “Another day has passed, and I didn't perform the Heimlich maneuver or a single chest compression!” Yet, universally most would agree they are useful skills to know. While most of us did not perform open heart surgery today, algebra is one of those fields that are so fundamental to the things we do everyday, that is is virtually impossible to not use algebra at least once in the course of a day. Like language, mathematics allows us to communicate with our peers and perform basic everyday tasks we likely take for granted.

What is Algebra? A quick 'Google' search, or any standard dictionary tells us, “In its most general form algebra is the study of symbols and the rules for manipulating symbols.” Wow! That doesn't sound like anything I did today, or want to do ever! However, Algebra originating from Arabic's Al-jebr, and meaning, “reunion of broken parts”, is about bringing ideas together. Yes, that is right, bringing ideas together, and making sense of them.

Let us briefly take a look at the simplest type of algebra problem, namely, solving the following equation for $x$,


Note: Don't let my use of symbols, or mathematical jargon scare you, you don't need to understand any of that, I am simply going to attempt to illustrate where we use these ideas in our everyday lives.

Now, to solve this equation for $x$
  we subtract b from both sides, and then divide both sides by a. Mathematically we say, “We use the additive inverse of 'b' followed by the multiplicative inverse of '$a$' together with the algebraic field properties over the reals to find an element $x$ such that $f(x)=c$, whenever $f(x)=ax+b$.” ... Whatever that means. What is important is that in essence we are taking inverses, i.e., doing and undoing things, and applying rules to functions.

Let us now look at a few examples of functions in everyday life. One common example is a soda machine. The buttons on the machine are its inputs(the domain), and the respective drinks are its output (the range). Mathematically,


 Another example involves driving. It turns out that the speed at which the vehicle travels depends on how far we press the gas pedal with our foot. Once again this is a function with its input(domain) being the amount the pedal is displaced from its normal position, and its output(range) being the speed at which it travels. Mathematically,


 Hence, anyone who drives uses algebra, regardless of whether they are aware of that fact or not! Even the act of stapling a stack of papers is an exercise in algebraic problem solving. Its domain being the size of the stack of papers, the rule being the act of stapling the papers, and its range the stapled stack of a particular size. Before you say I am pulling your leg, let us examine this situation. Can we evaluate an inverse operation, and apply the inverse operation to the original rule thereby undoing the operation and restoring the original domain? Allow me to direct your attention to the following Image retrieved from:



 i.e., the stapler applies a staple to the stack of papers, and staple remover restores the stack to its original unstapled condition--except for the holes of course. Then there are also finances, whether it is about buying our favorite snack to paying our bills every time. When we buy something for say, \$1.78, we know that \$1.78+[Ourchange]=[Billwepaidwith], assuming we didn't swipe our card. The budgeting of our expenses, and calculating how much money we can spare towards entertainment the following weekend is also an algebra problem. You may say, “Wait that is an arithmetic problem,” and you would be correct if you only did that problem once, or it was the exact same problem every time, but the values vary, introducing variables into your calculations, thus, making it an algebra problem which most people know how to deal with in spite of the changing values.

Fundamentally, this is all there is to algebra! The ideas of algebra are really this simple. They are not a scary monster we must avoid at all cost. They are the stuff of thought and the rules that govern our everyday lives--so much that it is impossible for anyone to say, “Well, another day has passed. I didn't use algebra once.” and be right--chances are they did even if they didn't realize it! The better we get at manipulating the essence of our ideas the better we become at bringing our ideas together and creating new ones. Our native language is our means of communicating our thoughts with others, while Mathematics is a language that allows us to manipulate and work with our ideas. Like any other language mastering its rules, syntax, and grammar allows us to put together an elegant flow of ideas and create beautiful prose in the language of mathematics.

As one example of communicating a more advanced idea with the language of mathematics consider the following. It turns out there are three primary factors determining the resistance to blood flow within a blood vessel, namely, vessel radius and length, and blood viscosity. We could write tirelessly page after page about how a blood vessel that is twice the length of another, but having equal radii will have twice the resistance to blood flow, or that if the viscosity of the blood doubles, then resistance to flow will also double, or that with increased radius comes reduced resistance, etc. Or we could mathematically say all that and more, like a master poet, with a single formula:

$$F\propto\frac{\triangle P\cdot r^{4}}{\eta\cdot L}$$

Mathematics allows us to become better at working with our thoughts, hence improving our rhetorical skills. It makes us better thinkers and better writers as a result of being able to take apart an argument, analyze it, and put it back together through both deductive and inductive principles which are at the foundation of mathematical thought. For example, we don't exercise because we may find the need to lift 400 pounds at work, but rather because a healthy body allows us to do every day things more efficiently. Similarly, exercising our brains by entertaining a mathematical thought or puzzle makes us better thinkers, and as a result better at making decisions. Those who communicate clearly, and can express their ideas better are likely to get further ahead in their careers, as well as closer to their dreams. This is why so many strive to be better writers, and better speakers--mathematics only facilitates this by making us better at logical thinking, and since it is already the stuff of thought, even unknowingly, our days can not pass without engaging in it at least once in our day--every day!

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