14 - What is Euler's Number 'e', Ln(x) - Natural Log & e^x Functions?

The script introduces Euler's number e, a fundamental constant in mathematics, and its significance in the context of natural logarithms and exponential functions. It explains that e is an irrational number, much like pi, and is integral to various mathematical disciplines, including calculus. The video aims to explore the origin of e, emphasizing its natural emergence in exponential growth and decay processes, and its equivalence in status to pi, despite being introduced later in the mathematical curriculum.

This paragraph delves into the concept of exponential growth as the natural process from which the number e arises. It contrasts e with other special numbers like pi, which comes from geometry, and explains that e's significance is due to its prevalence in calculations involving exponential growth, such as compound interest. The script also begins to explore the compound interest formula as a means to understand the derivation of e, starting with an example of an annual 100% interest rate.

The script continues to explain how the number e is derived from the compound interest formula by increasing the frequency of compounding. It illustrates the process by comparing the outcomes of compounding annually, semi-annually, monthly, and daily, showing that as the number of compounding periods increases, the growth factor approaches a limit. This limit is identified as Euler's number e, highlighting the relationship between frequent compounding and the emergence of e.

The discussion advances to the concept of continuous compounding, where the number of compounding periods becomes infinitesimally small, approaching a limit. The script demonstrates that as the compounding periods increase without bound, the growth factor converges to the number e. It provides numerical examples, showing how the growth factor approaches e as the compounding intervals become more frequent, eventually discussing the mathematical concept of a limit as the number of compounding periods approaches infinity.

The script defines Euler's number e as the limit of (1 + 1/n)^n as n approaches infinity, emphasizing its irrational nature with non-repeating, infinite decimals. It explains that e is a fundamental number in advanced mathematics, particularly in calculus, and is approached by increasing the frequency of compounding in the context of exponential growth. The importance of e is further underscored by its role in exponential functions and natural logarithms, which are central to calculus.

This paragraph underscores the pervasiveness and importance of Euler's number e in advanced mathematical fields, suggesting that it appears even more frequently than pi in certain contexts. It outlines the properties of the exponential function with e as its base, highlighting its unique characteristic where the slope of the function at any point is equal to the function's value at that point. The script also touches on the significance of e in various mathematical equations and its foundational role in calculus.

The script concludes by introducing Euler's formula, which is often hailed as one of the most beautiful equations in mathematics due to its elegant integration of five fundamental mathematical constants: e, i, pi, 1, and 0. It explains that this formula encapsulates the harmony between these constants and provides a glimpse into the profound interconnectedness of mathematical concepts, leaving the audience with an appreciation for the elegance and depth of mathematical principles.