University of Chicago long essay
Here's my long essay for UChicago.
Prompt: The Cartesian coordinate system is a popular method of representing real numbers and is the bane of eighth graders everywhere. Since its introduction by Descartes in 1637, this means of visually characterizing mathematical values has swept the globe, earning a significant role in branches of mathematics such as algebra, geometry, and calculus. Describe yourself as a point or series of points on this axial arrangement. If you are a function, what are you? In which quadrants do you lie? Are x and y enough for you, or do you warrant some love from the z-axis? Be sure to include your domain, range, derivative, and asymptotes, should any apply. Your possibilities are positively and negatively unbounded.
Inspired by Joshua Nalven, a graduate of West Orange High School, West Orange, NJ
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Since my first exposure to the Cartesian coordinate system and more generally, to algebra, I have learned that the “correct” way to address mathematics is through a specific routine-to follow specific steps for each problem. And since my first exposure to that methodical approach, I have despised it. Instead, I have always elected to think logically and develop my own way to solve each problem (though sometimes my methods coincide with the “correct” one). Even given my mathematical individuality, I must succumb to the general rules of mathematics-that is, while I can invent a method to find a derivative, I can’t re-define a derivative. Similarly, while I can graph a function in a different way than my teacher does, I still must use the universally accepted Cartesian coordinate system and come to the same visual image of the function in question. This forum provides a unique opportunity to avoid that ever-present constraint.
The first limitation that I will eliminate is the idea that axes on a Cartesian plane must be defined logically. Rather than defining x as an independent variable and y as some variable dependent on x, the two variables will be almost meaningless, or rather, I will change their meanings rather arbitrarily. I don’t see how I, or any person, can be defined by a typical two, three, or hypothetical four dimensional graph. By avoiding the specific labeling of axes, I hope to create a depiction which would not be possible given a well-defined system.
To begin, let’s define the placement of my graph. Because I’m a generally positive person, I hesitate to use anything but Quadrant I. However, other issues seem to outweigh this tendency. As you’ve probably gathered, I am not the type that falls into any standard categorization. The first quadrant, to me, symbolizes normality-too many graphs use solely the first quadrant. Therefore, I need to temporarily define an axis so as to allow myself to be placed in another quadrant. I have decided to initially define the x-axis as a non-quantitative representation of political ideology (that is, left of 0 will denote liberal, and right of 0 will denote conservative) and the y-axis as a portrayal of general attitude (like the x-axis, the y-axis will not be quantitative-above 0 will signify a positive mindset, and below 0 will signify a negative mindset).
Without a doubt, that definition puts me firmly in the second quadrant. By any definition, I am very liberal. Economically, I am somewhere between Karl Marx and reality, and socially, I’m left of any politician I’ve ever encountered. In terms of attitude, I’m always positive. I’m the guy who lives in the past only to remember the good times. If a plan goes awry, while other people are pointing fingers, I’m the guy who tries to get back on track. Therefore, my graph will be limited to solely Quadrant II.
This provides an interesting dilemma for defining other variables. As I explained before, the labeling of the axes will rarely be consistent. From now on, the axes will have little to do with their aforementioned meanings. The dilemma comes primarily from quantitative definitions of the x-axis. In a normal, first quadrant graph, moving to the right corresponds to an increase in the x-axis. In my graph, however, the opposite will be true-in times when the x-axis denotes some sort of time-related variable, the values of the x-axis will be treated as absolute values. That is to say, as time passes, we will move farther to the left on the x-axis. Fortunately, the y-axis will be treated just as it would be in a standard Cartesian plane.
Now that the plane has been defined and explained (for the most part), we can proceed to actually creating an image to place on this graph. There will be four key features on this graph. First, there will be a so-called “smiley face,” to signify my general attitude. Although this has already been included in the quadrant limitations, I feel it necessary to remind everyone that I’m a happy, easy-going guy. Plus, who doesn’t want a smiley face in Quadrant II?
The second facet is arguably the most meaningful, in the traditional sense of the word-a curve to represent my level of knowledge. To avoid confusion, I’ll first explain this curve in non-mathematical terms, and then convert into a function. From birth until death, my knowledge will undoubtedly be constantly increasing. Initially, it seemed that my education was increasing faster and faster as time went on-that is, as I learned more, it became easier to learn even more. However, as I near the end of high school, this phenomenon seems to be slowing. It seems now that my education is being limited by classmates, class selection (or lack thereof), and other random factors. In short, for the last few years, my knowledge has been increasing, but not as fast as it was before. I imagine that during college, particularly at an intellectual school such as University of Chicago, my knowledge will increase at an unbelievable rate. After college, I will continue to learn, though at a slower rate, due to issues such as a family and a career. Remember, when I am explaining the visual aspects of this curve, that time progression signifies a movement to the left on the x-axis. The curve begins at (0,0) with a negative slope and positive concavity. The concavity changes to negative at around x = -14 (-years), to positive around x = -18, and back to negative near x = -22. Eventually, the graph will approach (but obviously never reach) a horizontal asymptote of y = enough.
Having defined one substantive curve on my graph, it seems reasonable to return to the less intensive ones. Specifically, the third component of my graph will be a sine wave. To constrict it to Quadrant II, it will be defined as y=sin(x)+1 for x<0. This wave will have no significance other than to show my fondness for math.
The fourth and final aspect of this graph is dots. The pattern will be entirely nonexistent. In fact, I think the most fair way to dot this graph is by using my calculator’s random generator: (randint(0,-1*10^99,100), randint(0,1*10^99,100)). This will give 100 randomly generated points, anywhere in Quadrant II. “Why?” you ask? No reason.
Prompt: The Cartesian coordinate system is a popular method of representing real numbers and is the bane of eighth graders everywhere. Since its introduction by Descartes in 1637, this means of visually characterizing mathematical values has swept the globe, earning a significant role in branches of mathematics such as algebra, geometry, and calculus. Describe yourself as a point or series of points on this axial arrangement. If you are a function, what are you? In which quadrants do you lie? Are x and y enough for you, or do you warrant some love from the z-axis? Be sure to include your domain, range, derivative, and asymptotes, should any apply. Your possibilities are positively and negatively unbounded.
Inspired by Joshua Nalven, a graduate of West Orange High School, West Orange, NJ
----------------------------------------------------------------------------------------
Since my first exposure to the Cartesian coordinate system and more generally, to algebra, I have learned that the “correct” way to address mathematics is through a specific routine-to follow specific steps for each problem. And since my first exposure to that methodical approach, I have despised it. Instead, I have always elected to think logically and develop my own way to solve each problem (though sometimes my methods coincide with the “correct” one). Even given my mathematical individuality, I must succumb to the general rules of mathematics-that is, while I can invent a method to find a derivative, I can’t re-define a derivative. Similarly, while I can graph a function in a different way than my teacher does, I still must use the universally accepted Cartesian coordinate system and come to the same visual image of the function in question. This forum provides a unique opportunity to avoid that ever-present constraint.
The first limitation that I will eliminate is the idea that axes on a Cartesian plane must be defined logically. Rather than defining x as an independent variable and y as some variable dependent on x, the two variables will be almost meaningless, or rather, I will change their meanings rather arbitrarily. I don’t see how I, or any person, can be defined by a typical two, three, or hypothetical four dimensional graph. By avoiding the specific labeling of axes, I hope to create a depiction which would not be possible given a well-defined system.
To begin, let’s define the placement of my graph. Because I’m a generally positive person, I hesitate to use anything but Quadrant I. However, other issues seem to outweigh this tendency. As you’ve probably gathered, I am not the type that falls into any standard categorization. The first quadrant, to me, symbolizes normality-too many graphs use solely the first quadrant. Therefore, I need to temporarily define an axis so as to allow myself to be placed in another quadrant. I have decided to initially define the x-axis as a non-quantitative representation of political ideology (that is, left of 0 will denote liberal, and right of 0 will denote conservative) and the y-axis as a portrayal of general attitude (like the x-axis, the y-axis will not be quantitative-above 0 will signify a positive mindset, and below 0 will signify a negative mindset).
Without a doubt, that definition puts me firmly in the second quadrant. By any definition, I am very liberal. Economically, I am somewhere between Karl Marx and reality, and socially, I’m left of any politician I’ve ever encountered. In terms of attitude, I’m always positive. I’m the guy who lives in the past only to remember the good times. If a plan goes awry, while other people are pointing fingers, I’m the guy who tries to get back on track. Therefore, my graph will be limited to solely Quadrant II.
This provides an interesting dilemma for defining other variables. As I explained before, the labeling of the axes will rarely be consistent. From now on, the axes will have little to do with their aforementioned meanings. The dilemma comes primarily from quantitative definitions of the x-axis. In a normal, first quadrant graph, moving to the right corresponds to an increase in the x-axis. In my graph, however, the opposite will be true-in times when the x-axis denotes some sort of time-related variable, the values of the x-axis will be treated as absolute values. That is to say, as time passes, we will move farther to the left on the x-axis. Fortunately, the y-axis will be treated just as it would be in a standard Cartesian plane.
Now that the plane has been defined and explained (for the most part), we can proceed to actually creating an image to place on this graph. There will be four key features on this graph. First, there will be a so-called “smiley face,” to signify my general attitude. Although this has already been included in the quadrant limitations, I feel it necessary to remind everyone that I’m a happy, easy-going guy. Plus, who doesn’t want a smiley face in Quadrant II?
The second facet is arguably the most meaningful, in the traditional sense of the word-a curve to represent my level of knowledge. To avoid confusion, I’ll first explain this curve in non-mathematical terms, and then convert into a function. From birth until death, my knowledge will undoubtedly be constantly increasing. Initially, it seemed that my education was increasing faster and faster as time went on-that is, as I learned more, it became easier to learn even more. However, as I near the end of high school, this phenomenon seems to be slowing. It seems now that my education is being limited by classmates, class selection (or lack thereof), and other random factors. In short, for the last few years, my knowledge has been increasing, but not as fast as it was before. I imagine that during college, particularly at an intellectual school such as University of Chicago, my knowledge will increase at an unbelievable rate. After college, I will continue to learn, though at a slower rate, due to issues such as a family and a career. Remember, when I am explaining the visual aspects of this curve, that time progression signifies a movement to the left on the x-axis. The curve begins at (0,0) with a negative slope and positive concavity. The concavity changes to negative at around x = -14 (-years), to positive around x = -18, and back to negative near x = -22. Eventually, the graph will approach (but obviously never reach) a horizontal asymptote of y = enough.
Having defined one substantive curve on my graph, it seems reasonable to return to the less intensive ones. Specifically, the third component of my graph will be a sine wave. To constrict it to Quadrant II, it will be defined as y=sin(x)+1 for x<0. This wave will have no significance other than to show my fondness for math.
The fourth and final aspect of this graph is dots. The pattern will be entirely nonexistent. In fact, I think the most fair way to dot this graph is by using my calculator’s random generator: (randint(0,-1*10^99,100), randint(0,1*10^99,100)). This will give 100 randomly generated points, anywhere in Quadrant II. “Why?” you ask? No reason.