My first year of high school math was marked by advanced applications of mathematical patterns, and a breakaway from my traditional beliefs on how to approach the daunting school subject. One of the hardest days in math for me was trying to understand why there were other letters in my math problems. I recall yelling at my teacher one time, “What the heck are the letters e, l and n doing in my math homework?!” This question was the start of a new perspective towards mathematics, and showed me that the universe has more to offer in mathematics than the usual y=something*x equation.
I particularly felt that the most interesting and mind-blowing theme I encountered this year was the use of the letter e and its unique properties. The idea of seeing how this seemingly random number unites all branches of mathematics together is simply extraordinary, and has truly changed the way I think about nature and mathematics in tandem. Doing further research on this fascinating number showed that e appears everywhere, and that it is one of the most essential numbers for the field of mathematics today. The fact that e has a connection to nearly every type of math I learned showed me that these are not simply coincidences, but part of a mathematical code that preserves harmony in the laws of the universe.
To clarify, I am just the typical math student you would find at school, trying to finish the homework before the period bell rings. However, my discovery of this number has dramatically improved my approach towards mathematics and my view on it beyond the school classroom. If this number could turn me from a regular bored math student to a genuine enthusiast about understanding the why behind these patterns, I believe anybody can be fascinated by the wonders behind this simple letter in mathematics.
To understand what e is, it is important to understand its origins. Swiss mathematician Leonhard Euler, who lived from 1707-1783, is regarded as the “King of Mathematics” and one of the most influential figures for modern math. Euler is most recognized by his discovery of the number “e,” an irrational number that is called “Euler’s Number” today.
Euler’s number is equal to approximately 2.71828… Like pi (π), e is an infinitely long number with some special attributes. Mathematicians and students likely begin to encounter the letter “e” in their academic year of taking Algebra 2, where they learn how it is an exponential used for calculating continuous compounding interest, and is the only number to have its own unique logarithm function, “ln.”
Students explore the concept of e further in calculus, where they would find that e^x is the only function in which its derivative is equivalent to itself. Another contribution e has to calculus is that it can be derived using the sum of an infinite series, following the pattern of
1/1! + 2/2! + 3/3! + 4/4! + 5/5! …..
At this point, you may think that is all, but we have not even discussed the most (in my opinion) mindblowing aspect of this interesting number. Students who have studied even more advanced levels of math may know of Euler’s identity, which states that (e^iπ)-1=0. I believe that this is genuinely the most shocking math concept I have ever encountered, and it breaks my brain every time I try to understand this. I recall the first time I learned this I was so surprised by this relation of e to mathematics I stared at the wall in shock for nearly half an hour.
Imagine there being a formula that combines the 2 most basic integers (0 and 1) used in binary, the base imaginary number that is the square root of -1 (i), the universal geometric ratio of the circumference of a circle to its diameter (π), all being influenced by the number e, arguably the most influential number to all fields of mathematics. At this point, one would see that there are too many coincidences to occur in which this is true by luck: It is absolutely remarkable in which there is a connection unifying the 5 most important numbers in all mathematics. When Sal Khan, the creator of Khan Academy, proved the identity was true, he stated that if one’s isn’t simply blown away after this, they “have no emotion.”
I truly believe that after all of these facts, e is not any number that wondrously has interesting properties attached to it, but a crucial discovery in mathematics that is shown to keep the universe’s language in balance. I invite you to do the same and to continue exploring the mysteries of this mathematical foundation, and to see how it impacts all parts of math, and even patterns seen in nature and the daily world. If e has not blown you away, I also agree that you may have no emotion.
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