How does the exponential function work
Hey martin f and welcome to the forums.
The exponential function works like any other normal function and has a specific value that is > 0 for the entire real line (all real numbers positive and negative).
The function though has special properties that make it useful for a variety of purposes including information theory, statistics, science (like physics, biology, and chemistry) as well as in mathematics (applied and also for more theoretical kinds).
Through complex numbers it allows one to relate the trigonometric functions to the exponential in the complex plane. It also has a natural analog to the hyperbolic functions that underly things like special-relativity as well as non-euclidean geometry. Together hyperbolic and trigonometric frameworks are used to generalize geometry.
As mentioned before, it comes up nearly everywhere in analysis and in various ways theoretically because of many reasons including the derivative properties that make it useful for considering linear differential equations as well as results in linear algebra.
It's naturally related to geometric algebra due to the nature of the geometric product that is now being strongly looked at in modern physics.
In statistics its used for the Gaussian, exponential, and other distributions. It's also the foundation for other Gaussian type representations like the heat equation in partial differential equations, and is related to the concept of frequency in integral transforms.
One really important thing though about the exponential function though is how it ends up giving us insights about exponentiation. This might seem kind of obvious, but this concept has helped solidify the understanding of numbers from the integers all the way to the complex numbers which underlies our current understanding of quantity and variability, because exponentiation symbolically can produce every kind of quantity and knowing how to classify these with respect to symbolic notation is what theoretical mathematicians work on when attempting to construct numbers algebraically (often as rational numbers or functions of rational numbers like an infinite series).
Because of this, it gives a lot of research and investigation into the construction of the real numbers and classification therein (an example is the classification of whether a number is transcendental or not, and many numbers that are calculable often with infinite-series definitions haven't been shown to be transcendental let alone intervals like [0,1] with all real numbers).
Please conduct your own research first before coming here and asking very general questions. There is a lot of information available on the internet which you can review. For example, go to http://www.khanacademy.org/
If you have researched the subject and have more specific questions, then you are welcome to post.
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