My Favorite Constant
I was really interested in Euler’s number and where it came from.
It was kind of an after thought in the other mathematics classes that I have taken.
The professor would just say that it is a certain exponential function that its derivative is itself. Along with 0, it is the only number that has that property.
As I was researching it, I found out that the number was figured out under many different scenarios. It was figured out through using limits of a certain function.
Stay tuned... this shows up later if you're clever enough to find it.
The constant was also figured out through using the compound interest equation.
Anywho...
I wanted to figure out the value of e using calculus. This is the mathematics that I used to solve for it. Knowing that the definition for e is a number that is a base of an exponential function whose derivative is itself.
If the variable (a) were equal to zero, then the value works.
Based on our definition of derivative, the equation becomes
Since a^x is a constant and not part of the limit, it can be factored out.
By dividing both sides by a^x, the following result occurs.
By multiplying both sides by
,
the following results occurs.
Then, 1 is added to both sides of the equation.
Then, I took the root of both sides of the equation.
Solving for the variable (a) gets the value of e. I attached an Excel table for the values to show that it reaches e.
The table shows greatly how as the value of h becomes closer and closer to zero, but never actually reaching it, a starts to become closer and closer to e. Problems start to occur as Excel starts to round the 1+h to just 1. I only kept the first 11 iterations. This number is quite intriguing. I got the idea to do the reflection on this topic from the fact that e was popping up in so many of the physics equations that I have been studying and I wanted to know its origin and why it is so important.
It was kind of an after thought in the other mathematics classes that I have taken.
The professor would just say that it is a certain exponential function that its derivative is itself. Along with 0, it is the only number that has that property.
As I was researching it, I found out that the number was figured out under many different scenarios. It was figured out through using limits of a certain function.
Stay tuned... this shows up later if you're clever enough to find it.
The constant was also figured out through using the compound interest equation.
Anywho...
I wanted to figure out the value of e using calculus. This is the mathematics that I used to solve for it. Knowing that the definition for e is a number that is a base of an exponential function whose derivative is itself.
If the variable (a) were equal to zero, then the value works.
Based on our definition of derivative, the equation becomes
Since a^x is a constant and not part of the limit, it can be factored out.
By dividing both sides by a^x, the following result occurs.
By multiplying both sides by
,the following results occurs.
Then, 1 is added to both sides of the equation.
Then, I took the root of both sides of the equation.
Solving for the variable (a) gets the value of e. I attached an Excel table for the values to show that it reaches e.
The table shows greatly how as the value of h becomes closer and closer to zero, but never actually reaching it, a starts to become closer and closer to e. Problems start to occur as Excel starts to round the 1+h to just 1. I only kept the first 11 iterations. This number is quite intriguing. I got the idea to do the reflection on this topic from the fact that e was popping up in so many of the physics equations that I have been studying and I wanted to know its origin and why it is so important.