I How were "e" and "pi" found?

[mayflowers]

TL;DR Summary

Looking for a concise explanation from a member.

It's pretty incredible that e can be used in logs and continuous compounding without fail. And how pi can be so widely utilized. Could anybody explain how these came to be found? And why they work so consistently?

 

[Mark44]

Could anybody explain how these came to be found?

Here's one of many links on the history of the number $\pi$: https://en.wikipedia.org/wiki/Pi

If you do a search on the history of the natural number e, that should give you an idea of how this number came about.

And why they work so consistently?

Regarding $\pi$, it "works" because it is the ratio of two attributes of any circle: its diameter and its circumference. It "works" because no other number gives this ratio.

As far as e is concerned, anything involving exponential growth can be described in terms of some exponential function; e.g., $2^x, 10^x$, etc. One of the things about the function $e^x$ is that it is its own derivative, unlike any other exponential function.

 

[fresh_42]

$\pi$ is easy: it is the relation of circumference and diameter of a circle and known since ancient times. It occurs whenever a circle appears somewhere.

$e$ is a bit more complicated. Name and letter came after its usage. Wikipedia says:

The history of Euler's number e begins in the 16th century with three problem areas in which a number appears that mathematicians were approaching at the time and which was later called e:

As the basis of logarithms in the logarithm tables of John Napier and Jost Bürgi. Both developed their tables independently, incorporating an idea from Michael Stifel and using results from Stifel and other 16th-century mathematicians. Bürgi published his "Arithmetic and Geometric Progress Tabulae" in 1620. Bürgi apparently instinctively used a number close to e as the basis for his logarithm system. In 1614, Napier published his "Mirifici logarithmorum canonis descriptio" (Canonical Logarithms) using a base proportional to 1/e. Napier and Bürgi wanted to use logarithmic tables to reduce multiplication to addition, thus making complex calculations simpler and less time-consuming.

As a limit of a sequence in compound interest. In 1669, Jacob Bernoulli posed the problem: "Let a sum of money be invested at interest, so that at individual moments a proportional part of the annual interest is added to the capital." Today we call this proportional interest addition "continuous interest." Bernoulli asked whether arbitrarily large multiples of the original sum could be achieved through contracts in which the individual moments become increasingly shorter, and arrived at a number as a solution that we now know as Euler's number, e.

As an infinite series (area of the hyperbola of Apollonius of Perga). The question (in today's language) was how far an area extends to the right under the hyperbola $xy=1$ from $x=1$ which is the same size as the area of the unit square. The Flemish mathematician Grégoire de Saint-Vincent (Latinized Gregorius a Sancto Vincentino) developed a function to solve this problem, which we now call the natural logarithm and denote by $\ln$. He discovered interesting properties, including an equation that we now call the functional equation of the logarithm, which Napier and Bürgi also used to construct and apply their logarithm tables. It is not certain whether he was aware that the base of this logarithm is the number that was later called $e$. This was only noticed after the publication of his work. At the latest, his student and co-author Alphonse Antonio de Sarasa represented the relationship using a logarithmic function. In an essay discussing the dissemination of Saint-Vincent's ideas by de Sarasa, it is stated that "the relationship between logarithms and the hyperbola was discovered by Saint-Vincent in all its properties, except in name." Through the work of Newton and Euler, it then became clear that e is the base. Leibniz was apparently the first to use a letter for this number. In his correspondence with Christiaan Huygens from 1690 to 1691, he used the letter b as the base of a power.

https://de.wikipedia.org/wiki/Eulersche_Zahl#Die_Vorgeschichte_vor_Euler
Automatic translation with Google.Translate.

 

[DaveE]

Better explanations than you'll get from most of us:
https://en.wikipedia.org/wiki/Pi
https://en.wikipedia.org/wiki/E_(mathematical_constant)

You can follow up on the links in there or google. Honestly, we'll do better with more specific questions.

 

[fresh_42]

Here is a bit more about $e:$
https://www.physicsforums.com/insig...umber/#Eulers-Number-and-the-mathbfe-Function

The most interesting (for me) appearance of $\pi$ is I think Buffon's needle problem. You can find $\pi$ by a stochastic process:
https://en.wikipedia.org/wiki/Buffon's_needle_problem

 

[jedishrfu]

And more amazing still is the Euler's formula:

$e^{ix}=\cos(x) + i\sin(x)$

And when $\pi$ is inserted for x and with some rearrangement of terms we get:

$e^{i\pi}+1=0$

Where we have five of the most famous math constants in one equation.

https://en.wikipedia.org/wiki/Euler's_formula

 

[Klystron]

Logarithmic spirals include both numbers e and π and were studied and described by many of the mathematicians mentioned in above posts.

It's pretty incredible that e can be used in logs and continuous compounding without fail. And how pi can be so widely utilized. Could anybody explain how these came to be found? And why they work so consistently?

From the article:

A logarithmic spiral or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie").

More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

Self-similar spirals occur throughout nature in plants such as the pattern of leaf spacing, pinecone segments and sunflower seeds, in animals such as nautilus shell growth and certain spiral horns, and in the formation of certain galactic spiral arms. Less visible natural logarithmic patterns also occur studying bone growth, genetic inheritance and microcellular life forms.

Irrational number π appears in so many physical and mathematical applications that entire books are devoted to this enigmatic subject.

History of Pi by Petr Beckmann
Pi A Biography of the World's Most Mysterious Number

 

[haushofer]

Euler hid them in his closet.

 

[fresh_42]

I just found this article about precisely your question while searching for a particular definition of $e$:
https://pnp.mathematik.uni-stuttgart.de/iaz/iaz1/Rump/Euler.pdf

 

[dextercioby]

@fresh_42 Is there a purely geometric definition of $e$? If so, where can I find proof that this is equivalent to the infinite series one?

 

[mayflowers]

I just found this article about precisely your question while searching for a particular definition of $e$:
https://pnp.mathematik.uni-stuttgart.de/iaz/iaz1/Rump/Euler.pdf

Awesome! Will take a look at some point.

I appreciate everyone. Hopefully I find more questions after digging deeper into all the above.

 

[fresh_42]

@fresh_42 Is there a purely geometric definition of $e$? If so, where can I find proof that this is equivalent to the infinite series one?

Well, we have the area equation $F(x)=1\Longrightarrow x=e$ with $\displaystyle{F(x)=\int_1^x \dfrac{dx}{x}}.$ Whether this counts as a geometric definition is a matter of taste. I was looking for a proof for $$e=\lim_{n \to \infty}\sqrt[n]{n\#}$$
when I found the article linked above. $n\#$ is defined as $\displaystyle{n\# = \prod_{p\text{ prime}}^{p\leq n}p.}$ Hence, we can find a connection to $\pi$ via the primes. But that does Stirling's formula, too. I don't think one can get any closer. The behavior $y=y'$ is difficult to describe geometrically.

 

[mathwonk]

here's a try;
y=f(x) > 0 has the geometric property that, for all x, the (directed) area under its graph between the ordinates 0 and x, equals one less than the height at x, if and only if f(x) = e^x. Then e =f(1) = the area under the graph y = f(x) between x = -infinity and x = 1.

 

[dextercioby]

[...] I was looking for a proof for $$e=\lim_{n \to \infty}\sqrt[n]{n\#}$$
when I found the article linked above. $n\#$ is defined as $\displaystyle{n\# = \prod_{p\text{ prime}}^{p\leq n}p.}$ Hence, we can find a connection to $\pi$ via the primes. [...]

Can you explain that part I marked in bold? The previous text referred to $e$.

 

[fresh_42]

Can you explain that part I marked in bold? The previous text referred to $e$.

I haven't worked it out. I was just reminded of the following proof about the Riemannian zeta-function:

Let $\mathbb{P}$ be the set of all primes and $p\in \mathbb{P}.$ Then
\begin{align*}
\left(1-\dfrac{1}{p^s}\right)\zeta(s)&=\sum_{n=1}^\infty \dfrac{1}{n^s}-\sum_{n=1}^\infty \dfrac{1}{(pn)^s} =\sum_{\stackrel{n=1}{n\not\in p\mathbb{Z}}}^\infty \dfrac{1}{n^s}
=\sum_{\stackrel{n=1}{p\nmid n}}^\infty \dfrac{1}{n^s}\\
\prod_{p\in\mathbb{P}}\left(1-\dfrac{1}{p^s}\right)\zeta(s)&=\sum_{\stackrel{n=1}{n\not\equiv 0\mod p\,\forall \,p\in \mathbb{P}}}^\infty \dfrac{1}{n^s}=\sum_{n\in \{1\}}\dfrac{1}{n^s}=1\\
\sum_{n=1}^\infty \dfrac{1}{n^s}=\zeta(s)&=1/\prod_{p\in\mathbb{P}}\left(1-\dfrac{1}{p^s}\right)=
\prod_{p\in\mathbb{P}}\left(1/\left(1-\dfrac{1}{p^s}\right)\right)=\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}
\end{align*}
That connects a sum over all positive integers with a product over all primes and $\zeta(2)=\dfrac{\pi^2}{6}.$

 

[dextercioby]

I'm sorry. This is nice, but is there any way to link $e$ and $\pi$ through infinite series or/and products of natural/integer/rational numbers?

 

[fresh_42]

I'm sorry. This is nice, but is there any way to link $e$ and $\pi$ through infinite series or/and products of natural/integer/rational numbers?

You mean, besides Euler's formula?
$$
0=1+\sum_{n=0}^\infty \dfrac{(i\pi)^n}{n!}
$$
What are you looking for? $\pi$ is attached to circles, and $e$ to spirales. How do you want to bridge this gap?

 

[dextercioby]

Yes, that's just $e^{i\pi} + 1=0$ with $e$ written as an infinite sum, but I was thinking of something along the lines

$$\mbox{elementary function}(e,\pi) = f(n), {}{} n\in\mathbb{N}$$

where $f$ is a combination of infinite sums or products of natural (or possibly prime natural) numbers.

 

[Klystron]

Returning to the OP's basic question with a simple geometric explanation in English:

Pi was discovered studying circles, specifically comparing a circumference, distance around a circle, to its diameter, distance across a circle at its widest.

Notice the indefinite article "a". This ratio Pi applies to any circle.

Likewise, the base of natural logarithms e derived from studying spirals in pinecones, seashells and other natural phenomena. Hence, logarithmic spirals.

These geometrical discoveries likely preceded and motivated formal algebras, number theory, and calculus (beyond counting and basic arithmetic).

 

[fresh_42]

Yes, that's just $e^{i\pi} + 1=0$ with $e$ written as an infinite sum, but I was thinking of something along the lines

$$\mbox{elementary function}(e,\pi) = f(n), {}{} n\in\mathbb{N}$$

where $f$ is a combination of infinite sums or products of natural (or possibly prime natural) numbers.

It seems to be an open problem for $f\in\mathbb{Q}[x,y].$ We don't even know whether $e\pi$ and $e+\pi$ are transcendental (no reference, I only found a comment on MSE and didn't want to search for it further). See also the Schanuel conjecture or a Heegener number.

Hence, we are left with infinite series and infinite products. Euler's formula is such an infinite series. There are integral formulas that connect the two, e.g.
\begin{align*}
\sqrt{\pi}&=\int_{-\infty }^{\infty }e^{-x^2}\,dx\\
e\pi^2&=\int_0^{\pi} 2x e^{\cos(2x)} [1 − x \sin(2x)]\, dx
\end{align*}
in which you can substitute the corresponding series but I haven't found a formula without using $i$ or integrals.

 

[lavinia]

It seems to be an open problem for $f\in\mathbb{Q}[x,y].$ We don't even know whether $e\pi$ and $e+\pi$ are transcendental (no reference, I only found a comment on MSE and didn't want to search for it further).

epi and and pi + e cannot both be algebraic since e and pi are both roots of the polynomial x^2-(pi + e)x+epi

I think the search for pi goes back to ancient times. One problem was to use a ruler and compass to construct a square whose area is the area of a given circle. This was called squaring the circle. This would have involved constructing a line segment of length the square root of pi. I imagine this was not understood to be impossible until sometime after calculus was discovered.

Here is the introductory paragraph from the Wikipedia article "Squaring the Circle."

"Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square."

 

[lavinia]

When one defines e(x) as the inverse function of the natural logarithm, the Chain Rule gives e'(x)/e(x) = 1 and e is then defined as e(1).

 

[lavinia]

From the Wikipedia passage on the logarithm quoted in post #3 it seems that the ancient Greek problem of Appolonius was to square the hyperbola. That is: to find a region between 1 and some point on the hyperbola whose area was the same as a unit square. That point is the number, e. So it seems that e was defined in ancient Greek mathematics although finding it took hundreds of years. This idea of squaring a curve seems to have been prominent in Greek mathematics. I wonder why that was.

The Wikipedia article also says that the original log tables used a base that was close to e. According to a video that I watched, it seems that this may have come simply from the approximation ln(1+ε)^n=nln(1+ε) = nε up to first order so that if ε is very small nε would look like the natural log of the base number (1+ε). The illustrative example in the video was to choose ε to be 1/10^a for a in the range of 6 or 7 so that the table would look like a column of pairs (n,(1+ε)^n) where the power n is the natural logarithm scaled by 10^a. In this way, natural logs could have appeared as an unintended result of making log tables that were dense enough to insure that interpolations would not create unacceptably large errors.

The video also shows that if one approximates the area under the hyperbola so that each approximating rectangle has a the same area, then one gets a log table. One chooses a partition of [1,x] of the form xs^i where s is an n'th root of 1/x. Each rectangle has an area of (1-s) so the pairs ((1-s)i,xs^i) form a log table.

The log of the highest power counts the total number of rectangles so the sum of the areas of the rectangles is a logarithm. These partitions can be chosen to give an arbitrarily accurate approximation so in the limit one might call the area the log of x. I suppose this is like thinking of a circle as a regular polygon with infinitely many sides.

Whether the video was historically accurate, I don't know but working through the ideas gave me a picture of how one might solve the problem of squaring the hyperbola using approximation by log tables. The thought process seems historically plausible since log tables were already in use for practical purposes.


https://www.google.com/search?q=inv...ate=ive&vld=cid:e52f23b7,vid:habHK6wLkic,st:0

 

[fresh_42]

Here is another formula that connects the two:
$$
\dfrac{e^{\pi}-e^{-\pi}}{2}=\sinh \pi =\sum_{k=0}^\infty \dfrac{\pi^{2k+1}}{(2k+1)!}
$$
which again makes use of the hyperbola. I think that is close enough to

I was thinking of something along the lines

$$\mbox{elementary function}(e,\pi) = f(n), {}{} n\in\mathbb{N}$$

where $f$ is a combination of infinite sums or products of natural (or possibly prime natural) numbers.

 

[Klystron]

This idea of squaring a curve seems to have been prominent in Greek mathematics. I wonder why that was.

Accepting the hypothesis that hunting, warfare and weapons provided reason, focus and model for early mathematics even before agriculture and architecture, then fascination with curves and continuous curved paths becomes apparent.

Many famous mathematicians such as Archimedes of Samos and Galileo Galilei earned their crust of bread designing and improving weapon systems for their respective patrons. Early work on cycloids, to name one example, developed by observing the odd motion and improved performance from a catapult or trebuchet when wheels and axles attached to the horizontal frame for transport were left free to travel during the catapult shot.

Work with ballistics tends to focus on parabolas, particularly after the initial impetus to the missile from a lever or, later, gunpowder, but hand thrown weapons involve many interesting shapes.

A thrown round stone generally follows a parabolic path in the normal plane. Apply spin and/or throw a flat object and gyroscopic then aerodynamic forces alter travel in computable directions. Add a leather strap or other dried animal skin to amplify arm strength and little David's sling requires improved calculations.

A thrower spins a stone held within a sling in a circular pattern increasing angular momentum, releasing one end of the sling approximately tangent to line-of-sight to the target. Can early mathematics model this dynamic and improve performance?

Alongside stones, early Mediterranean cultures hunted and fought with sticks. Thrown sticks tend to flail about. Add spin with the thumb of the throwing hand to a spear or via curved fletching attached near the rear of an arrow and a fearsome accurate weapon is created.

Circles, cycloids, spheroids and hyperbolas provide useful models for improving slings and larger "lever" machines. Spirals provide models for improving spears, arrows and other spinning objects. Add mathematicians' search for root numbers and relationships and the discovery of transcendental functions and numbers such as π and ε seem inevitable.

 

[lavinia]

@Klystron

There is also this Medieval view of geometry and ruler and compass constructions.

https://old.maa.org/press/periodicals/convergence/mathematical-treasure-god-the-supreme-geometer

 

[Klystron]

@Klystron

There is also this Medieval view of geometry and ruler and compass constructions.

https://old.maa.org/press/periodicals/convergence/mathematical-treasure-god-the-supreme-geometer

This beautiful post resonates with several themes developed in this thread and reminds me of a related irrational number. The original post specified π and e, with π known from antiquity and e defined by Leonhard Euler who also standardized the symbol for pi, working during the 18th Century. Euler's prodigious contributions include functions with e a unique example as explained in earlier posts.

Discussion of these essential symbols would be incomplete without mentioning Euclid, "Father of Geometry", who studied and defined the golden ratio symbol phi (⁠φ or ⁠ϕ). As π is to circles and e is to logarithmic spirals, φ denotes a property of certain rectangles.

Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities ⁠a and ⁠b with ⁠a>b>0⁠, ⁠a⁠ is in a golden ratio to ⁠b if

Like many artists and architects, I find rectangles with these proportions pleasing to the eye, easily converting commercial boards such as 2x4' to a golden ratio using only a ruler to mark the cut. This is a photo of a golden ratio board divided into phi spiral rectangles illustrating intersecting spirals. From 2010.

Phi, like Pi and Epsilon, has its devotees:

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. ...
— The Golden Ratio: The Story of Phi, the World's Most Astonishing Number