This essay explores how algebra and mathematics quietly underpin everyday activities, using the author's three-speed cruiser bicycle as a concrete, relatable example. The paper walks through the construction of a simple algebraic equation to calculate the pedaling effort required to travel one kilometer in different gears, demonstrating how variables such as tire circumference, gear ratio, and distance interact. By plugging in real numbers and comparing results across gears, the author illustrates the counterintuitive but mathematically sound principle that higher gears require less total pedaling effort over a given distance—a concept that also applies to automobile transmissions.
It is strange, though somehow comforting, to consider how deeply math in general—and algebra specifically—is woven into our daily lives. Strange, because we rarely have to perform mathematical operations explicitly in order to complete our daily tasks and routines. Comforting, because the concrete and unchanging nature of numbers adds a degree of certainty to a world that can so often seem chaotic and ungrounded. Even if algebra cannot predict what will happen to oil prices or mortgage markets, it can at least provide an explanation of what is happening, how it is happening, and perhaps even why.
The complex mathematics behind economic phenomena is best left to economists and members of the Federal Reserve. Most of us, however, encounter numbers in far smaller and more personal ways every day. One such way presents itself through the bicycle—a machine whose mechanics can be expressed clearly and elegantly through algebra.
I ride my bicycle almost everywhere, and several times I have experienced minor breakdowns on the road. These incidents have given me a basic understanding of how my bike and its various gears work together to move me forward. The functions of a bicycle and its gears can be expressed algebraically using gear ratios. The actual equations that describe a bike's travel in full detail would be quite complex and would require extensive measurement and experimentation. However, the basic equations needed to calculate effort, speed, and travel time across various distances and gears can be illustrated through a simple thought experiment using straightforward numbers.
A brief description of the bicycle is useful before proceeding. While I own a twenty-one-speed mountain bike, I typically ride a three-speed cruiser around town. For the sake of simplicity, the equations here pertain to that cruiser. The terms that need to be defined—and assigned numerical values—are: tire circumference (which also equals the linear distance traveled per revolution of the tire), the number of tire revolutions that result from each push of the pedals, and the effort required to turn the pedals. Many other variables would affect real-world riding speed, and the effort variable would itself be far more complicated than represented here, but this framework is sufficient for illustration.
Using these variables, the following equation can be written to calculate the effort needed to travel one kilometer (one thousand meters) in a given gear:
(M / T) / G = E
Where:
M = the distance of the journey in meters
T = the circumference of the tire (and therefore the linear distance traveled per revolution)
G = the number of tire revolutions per push of the pedal, which changes from gear to gear
E = the number of times the pedals must go around, representing the total effort required to travel the given distance
Plugging concrete numbers into the equation makes it easier to see how it works. Assume that the tire circumference is 1.5 meters and that in third gear the tire completes three revolutions for every push of the pedal. For a one-kilometer journey, the equation becomes:
(1000 / 1.5) / 3 = E
"Equation solved for first and third gear"
"Higher gears reduce total pedaling effort"
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