We define this as the functional equation:It is a bit like cauchy's functional equation but with multiplication instead of addition.

So that he can work full time on OurBigBook.com and revolutionize advanced university-level science, technology, engineering, and mathematics eduction for all ages.

Donating to Ciro is the most effective donation per dollar that you can make to:

Ciro's goal in life is to help kids as young as possible to reach, and the push, the frontiers of natural sciences human knowledge, linking it to applications that might be the the next big thing as early as possible. Because nothing is more motivating to students than that feeling of:rather than repeating the same crap that everyone is already learning.

To do this, Ciro wants to work in parallel both on:

Ciro believes that this rare combination of both:produces a virtuous circle, because Ciro:

As explained at OurBigBook.com and high flying bird scientist, Ciro is most excited to make contributions at the "missing middle level of specialization" that lies around later undergrad and lower grad education:But on that middle sweet spot, Ciro believes that something can be done, in such as way that delivers:in a way that is:

Ciro believes that today's society just keep saying over and over: "STEM is good", "STEM is good", "STEM is good" as a religious mantra, but fails miserably at providing free learning material and interaction opportunities for people to actually learn it at a deep enough level to truly appreciate why "STEM is good". This is what he wants to fix.

The following quote is ripped from Gwern Branwen's Patreon page, and it perfectly synthesizes how Ciro feels as well:

In addition to all of this, financial support also helps Ciro continue his general community support activities:

Good list: en.wikipedia.org/wiki/Geologic_time_scale#Terminology

Jean-Baptiste L. Romé de l'Isle was a French mineralogist and geologist, best known for his work in crystallography during the 18th century. He made significant contributions to the understanding of crystal structures and the principles of crystallography. One of his notable works is the formulation of the "law of symmetry" in crystals, which contributed to the classification of crystals based on their geometric shapes and symmetry properties.

The differential equation that is solved by the exponential function:with initial condition:

TODO find better name for it, "linear homogenous differential equation of degree one" almost fully constrainst it except for the exponent constant and initial value.

The Taylor series expansion is the most direct definition of the expontial as it obviously satisfies the exponential function differential equation:

The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.

More precisely, we know that for any base b, exponentiation satisfies:And we also know that for $b=e$ in particular that we satisfy the exponential function differential equation and so:One interesting fact is that the only thing we use from the exponential function differential equation is the value around $x=0$, which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.

Now suppose that we want to calculate $e^1$. The idea is to start from $e^0$ and then then to use the first order of the Taylor series to extend the known value of $e^0$ to $e^1$.

E.g., if we split into 2 parts, we know that:or in three parts:so we can just use arbitrarily many parts $e^{1/n}$ that are arbitrarily close to $x=0$:and more generally for any $x$ we have:

Let's see what happens with the Taylor series. We have near $y=0$ in little-o notation:Therefore, for $y=x/n$, which is near $y=0$ for any fixed $x$:and therefore:which is basically the formula tha we wanted. We just have to convince ourselves that at ${\lim }_{n\to \infty }$, the $o(1/n)$ disappears, i.e.:

To do that, let's multiply $e^y$ by itself once:and multiplying a third time:TODO conclude.

Is the solution to a system of linear ordinary differential equations, the exponential function is just a 1-dimensional subcase.

Note that more generally, the matrix exponential can be defined on any ring.

The matrix exponential is of particular interest in the study of Lie groups, because in the case of the Lie algebra of a matrix Lie group, it provides the correct exponential map.