As stimulating as this constant is to the retention of memory, it holds a special place in fields outside alphabetical membership. Particularly, there’s no arguing that Euler’s number has its fair set of applications. It’s the base of the natural logarithm, the limit of (1 + 1/n) n as n approaches infinity, and the component of an expression known as “the most beautiful equation in math”.
What is e?
The letter case of e can be definitive in the interpretation of its use. Uppercase E is often seen in calculator displays as scientific notation, that is, numbers that aren’t apt for longhand because of their extreme length. E can be placed in front of numbers raised to a power of 10. For example:
3E34 = 3×1034
On the other hand, lowercase e is used to depict Euler’s number. The everyday calculator will give the answer of “error” for the square root of -1. Here it is important to note that senseless does not mean worthless. Calculators cannot interpret certain irrational numbers because they only deal with rational math, that which can be represented as the ratio of two integers. Irrational numbers undergo a process that finds the resolve for the infinitely small and the infinitely large. This is the case with e.
How was it discovered?
The constant e was discovered by mathematician Jacob Bernoulli, a member of the infamous Bernoulli family, as he was implementing what is now known as compound interest, which allows for the sum of small amounts of money to accumulate over time. Bernoulli realized that starting a 100 percent interest rate account with a deposit of £1.00 at the beginning of the year would grant him a total value of £2.00 on year’s end. He noticed a pattern that increased as intervals fractioned. In that case, he wondered, what would happen if he received interest every month of the year? Numerically, this can be expressed as:
1 x (1 + 1/12) 12 = £2.61
This showed that upon increasing the frequency of your interest, the smaller intervals would create the accumulation of a greater total in the long run. If we considered an even smaller durations, such as one every week in the year, we would get:
£1 x (1 + 1/52) 52 = £2.69
Now, for every day of the year:
£1 x (1 + 1/365) 365 = £2.71
Then for minutes, seconds, nanoseconds and so on. This makes you wonder how frequently one would have to continue to reach monetary climax. Is infinity enough?
n→∞ (1 + 1/n) n
An infinite equation is bound to result in an infinite number. This is what, in their own respects, Leonhard Euler and Fredrich Gauss set out to find. The result was:
There’s a welcoming number. Math is known for those.
Why is it considered the most beautiful equation?
The usual calculator only known four mathematical operations: addition, subtraction, multiplication, and division. Our constant normalizes the inclusivity of one of these operations, addition. So, we start with:
1 + 1/1! + 1/2! + 1/3! + 1/4! + …
And as we add more terms into the equation, we get closer to the value of Euler’s number. If we apply this to sine and cosine, prophetic properties and infinite series persist.
Sin (x) = x/1! – x^3/3! + x^5/5! – x^7/7! + …
The sine of each term increases by 2, as is the power by which we are raising x, and the sign of each terms keeps alternating between two operations, addition and subtraction. If we apply this to cosine, something similar happens:
Cos (x) = 1 – x^2/2! + x^4/4! – x^6/6! + x^8/8! – …
Again, like the sine patterns, the denominator and power terms increase by 2 each time, only this time we begin counting from 2 rather than 1. The series for e, sine, and cosine look eerily alike. It makes you wonder if adding the results for sine and cosine results in e. Something like this:
Cos (x) + sin (x) = 1 + x/1! – x^2/2! – x^3/3! + x^4/4! + x^5/5! – x^6/6! – x^7/7! + …
At surface level, the combination might look right, but it is, unfortunately, not. The signs between the sine and cosine series keep alternating, while the signs in e don’t. But what if we found a way to make them equal by multiplying the x term? We would need to multiply x when it was squared by something that results in -1. The problem is, when we square a number, the result is always positive, not negative. What we need is a number that, when squared, is equal to -1. There is no such number, and it’s not like we can invent one, right? Right?
Now, we’re, by the art of rhetoric, allowed to do this:
i2 = -1
When we raise e to the power of i-x, and multiply the sine term by i, in order to find the relationship between sine, cosine and e, we make them equal. This is Euler’s formula:
eix = cos (x) + isin (x)
A specific case of Euler’s formula results in Euler’s identity. More specifically, when x is equal to pi.
Cos (π) = -1
Sine (π) = 0
So, ei-π = -1
Rearranging this, gives us Euler’s identity, the most beautiful equation in mathematics:
ei-π + 1 =0
This makes one wonder, what is beautiful under mathematical standards? In our case, mathematical beauty is the resonance of elementary patterns in the creation of a universal formula. Although it seems ocularly insignificant, it’s as if, upon being interpreted, the equation is suggesting parented consciousness. One that debunks individuality and suggests singularity. It’s an equation without starved meanings, one that has the potential of turning into an idea.
Why is it important?
This ultimately goes down to its applications, which determine its relevance in the long term. Aside from the regularities listed above, e is also familiar in probability theory, equations for waves in physics (ex. light, sound, and quantum waves), and in calculating the lifespan of radioactive chemicals. After all, there’s a reason why this constant has its own birthday.
Featured image retrieved from: https://www.thoughtco.com/the-number-e-2-7182818284590452-3126351