3. From Ancient Math to Modern Science: The Fantastic Journey of the Exponential Function

Abstract

For centuries, the exponential function has been a cornerstone of mathematics, and its influence has extended far beyond the confines of this discipline. This chapter will delve into the rich history of the exponential function, tracing its origins back to ancient civilizations and exploring its evolution over time. The chapter will also highlight some of the key mathematical breakthroughs that led to the exponential function. Finally, we will present a brief survey of how the exponential function is used in fields such as physics, engineering and biology.

Introduction¶

A story goes that once God Krishna took the form of a wise man and went to the king’s court. He challenged the king to a game of chess because he loved playing it. Before the game started, they had to decide what the prize would be if the sage won. The sage said that he only wanted a small amount of rice, but the amount had to be calculated using a chessboard. A grain of rice would be placed in the first square and the number of rice grains would double for each successive square on the board. The king lost the game and had to give the sage the rice. But as he started putting the rice grains on the board, the king realized that he wouldn’t have enough rice to pay his debt. Krishna then showed himself in his true form and told the king that he didn’t have to pay all the rice right away, but he could do it gradually. Every day, the king would serve free rice pudding to people who came to the temple until the debt was paid off.

This story has different versions. In some versions, God Krishna is replaced by a servant, the inventor of chess, or a craftsman who makes the best chessboards. In others, rice is substituted by wheat. The ending is also different. In some versions, the ruler has the person who should get the reward killed, while in others, the reward is given only if each grain is counted individually. But the message stays the same: the explosive increase in a pattern where each step is multiplied by the same number (geometric progression) instead of just being added by the same amount (arithmetic progression).

With a history dating back to the Greeks and possibly even the Sumerians, geometric progressions---a discrete form of the exponential function---boasts a rich legacy. Despite this, it wasn’t until the 18th century, with the contribution of some of the world’s most renowned mathematicians, that the exponential function was finally uncovered. Nevertheless, its influence has been tremendous and spans across numerous scientific domains. A key milestone in this history is the creation of logarithms, which was a seminal event.

In this chapter we briefly review their history. For those seeking a deeper understanding of the history of logarithmic and exponential functions, the History of the Exponential and Logarithmic Concepts series of articles, published in 1913 in the American Mathematical Monthly: Cajori (1913), Cajori (1913), Cajori (1913), Cajori (1913), Cajori (1913), Cajori (1913), Cajori (1913).

The Invention of Logarithms¶

Logarithms were independently developed by the mathematicians John Napier, from Scotland, and Jost Bürgi, from Switzerland, to simplify computations in spherical trigonometry, which is used in astronomy and celestial navigation. Despite Bürgi’s probable creation of his system around 1600, Napier’s discovery was published first in 1614 in the book Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms), making him widely known as the inventor of logarithms and greatly influencing its subsequent evolution.

Logarithms establish a connection between the operation of multiplication on the positive real numbers and addition on the real number line. Napier, in particular, viewed logarithms as the relationship between two particles moving along a line---one at a constant speed, the other at a speed proportional to its distance from a fixed endpoint. In current terms, Napier’s logarithm ($\mathrm{NapLog}$) can be related to the natural logarithm ($\ln$) in this way:

English mathematician Henry Briggs made two visits to Edinburgh to collaborate with John Napier in 1616 and 1617. During their conversations, they reached an agreement on Briggs’ proposed modification to Napier’s logarithms. After his second trip, Briggs released the first table of his improved logarithms, now known as common or base 10 logarithms ($\log_{10}$), in 1617. The widespread use of common logarithms grew rapidly due to their ease in performing complex calculations during a time when calculators were not available. This was mainly due to the fact that our numbering system is built on powers of 10. Nevertheless, the natural logarithm holds a more prominent place in the history of mathematics because of its impact on the discipline’s evolution.

In 1649, Alphonse Antonio de Sarasa, who was previously a student of Grégoire de Saint-Vincent, demonstrated that the area $A(t)$ of the region bounded by the hyperbola $xy = 1$ from $x = 1$ to $x = t$ obeys the following relation common to all logarithmic function:

It was soon realized that this characteristic could lead to the creation of a new type of logarithm. Mercator, in fact, published the first tables of what is now referred to as the natural logarithm in his book Logaritmotechnia in 1668.

From Logarithms to the Exponential Function¶

In 1748, Leonhard Euler’s classic book Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) marked the final step in the development of logarithms, exponential functions, and trigonometric functions. Prior to Euler, these mathematical concepts were usually defined using integral calculus. However, Euler changed this by introducing exponentiation $a^x$ for constant $a$ in the positive real numbers, leading to the creation of the logarithm to base $a$. He also named the natural logarithm, calling it the ``natural or hyperbolic logarithm" due to its connection to the quadrature of the hyperbola. Euler designated number $e$ as the base for the natural logarithm, i.e. the number whose natural logarithm is equal to 1.

Jacob Bernoulli’s groundbreaking work in compound interest paved the way for the discovery of the constant later named $e$ by Euler. Bernoulli’s contribution can be succinctly described as follows: he examined the growth of capital when invested at an annual interest rate of 100% and is compounded at $n$ intervals. At the end of the year, the capital would have multiplied by a factor of

Bernoulli showed that as the number of compounding intervals increases, this factor approaches a constant value between 2 and 3.

Euler expanded upon Bernoulli’s work by defining the exponential and natural logarithmic functions as follows:

Additionally, Euler established that the exponential function is a exponentiation function with base $e$ ($\exp(x) = e^x$) and that the exponential and logarithmic functions are inverse of each other.

Transcendence of the Exponential Function¶

Godfrey H. Hardy wrote in A Mathematician’s Apology that “a mathematical idea is significant if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.” Hardy used Pythagoras’s proof of the irrationality of $\sqrt{2}$ as an example, highlighting how a simple and elegant theorem can open up new avenues for the development of mathematics.

The exponential function is a prime example of a mathematical idea that has left a lasting impact, not just in mathematics but in other fields as well. It is considered a cornerstone for the advancement of all modern areas of mathematics. Its applications are diverse and can be seen in fields such as physics, where it explains processes ranging from radioactive decay to population growth. In finance, it is employed in modeling interest rates and stock prices. The exponential function is also a critical component in engineering, contributing to the development of electrical circuits, control systems, and communication systems, just to mention a few examples.

In Introductio in analysin infinitorum, Euler introduced the equation named after him:

Specifically, when $x=\pi$, this formula leads to the well-known Euler identity:

which is considered by many to be one of the most beautiful equations in mathematics, as it combines the three basic mathematical operations (addition, multiplication, and exponentiation) and relates five fundamental mathematical constants (0, the additive identity; 1, the multiplicative identity; the unit of imaginary numbers, $i$; $e$ and $\pi$). The Euler identity is a point of convergence of various mathematical disciplines, such as arithmetic, trigonometry, and complex number theory.

Euler formula has been a vital aspect in physics and engineering, particularly through Fourier analysis. This method allows the transformation of complex functions into sums of simple trigonometric functions. Some of the most significant uses include:

1. Signal processing: Fourier analysis is used to analyze and manipulate signals such as audio, images, and voice.

2. Image compression: Fourier analysis is used in image compression algorithms to reduce the amount of data needed to represent an image.

3. Spectral analysis: Fourier analysis is applied to study the frequency components of signals, such as those produced by vibrating objects or electromagnetic waves.

4. Filter design: Fourier analysis is used to create electronic filters that remove unwanted frequencies from signals.

5. Heat transfer: Fourier analysis is applied to examine heat flow in solids and fluids.

6. Quantum mechanics: Fourier analysis is used to analyze the behavior of particles in a quantum state.

7. Seismology: Fourier analysis is employed to examine the propagation of seismic waves.

8. Electromagnetic analysis: Fourier analysis is applied to study the behavior of electromagnetic fields.

9. Analysis of neuronal signals: Fourier analysis is used in the study of the electric and magnetic fields emanating from brain activity.

10. Analysis of protein structure: Fourier analysis is employed to determine the structure of proteins using X-ray crystallography.

Discussion¶

This chapter has chronicled the remarkable journey of the exponential function from its earliest origins to its central role in modern mathematics and sciences. While geometric progressions date back millennia in various ancient cultures, it was not until the foundational work of 17th century figures like Napier, Briggs, and Euler that the exponential and logarithmic functions emerged in their modern formalized forms.

Key developments included the independent invention of logarithms by Napier and Bürgi, establishing the logarithmic relationship between multiplication and addition. Common logarithms streamlined calculations and advanced sciences. Later, the work of figures like Bernoulli, Euler, and Mercator uncovered deeper properties, culminating in Euler’s definition of the exponential and natural logarithmic functions using limits.

In summarizing this long history, the chapter has underscored how a single mathematical notion can evolve enormously over centuries through the work of generations of thinkers. It has also conveyed the exponential function’s status as a prime example of an abstract mathematical idea that revolutionized scientific understanding through illuminating connections to the natural world. Its influence ensures the story of this fantastic function will undoubtedly continue expanding into new uncharted domains.