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A BBC Podcast on 'e'
Melvyn Bragg and his guests discuss Euler's number, also known as e. First discovered in the seventeenth century by the Swiss mathematician Jacob Bernoulli when he was studying compound interest, e is now recognised as one of the most important and interesting numbers in mathematics. Roughly equal to 2.718, e is useful in studying many everyday situations, from personal savings to epidemics. It also features in Euler's Identity, sometimes described as the most beautiful equation ever written.
With:
Colva Roney-Dougal
Reader in Pure Mathematics at the University of St Andrews
June Barrow-Green
Senior Lecturer in the History of Maths at the Open University
Vicky Neale
Whitehead Lecturer at the Mathematical Institute and Balliol College at the University of Oxford
Producer: Thomas Morris.
Ode to e
Euler's constant, mysterious 'e',
A number so simple, yet complex to see.
Its value transcends all numbers whole,
A mathematical gem, a limitless goal.
Defined by a series, it seems to go on,
Without end or pattern, forever drawn.
Yet it's woven into life's very fabric,
From the growth of populations to a rabbit's antics.
Compounding interest, exponential growth,
The power of 'e' is clearly shown.
A constant that defies all limits,
Infinite and unbounded, it never quits.
It's found in nature, in physics and space,
From the motion of particles to the speed of light's pace.
An underlying force, it drives us ahead,
A symbol of progress, of the future we tread.
So let us embrace this magical 'e',
A symbol of possibility, of all that can be.
A reminder that limits are just in our mind,
And that with perseverance, we can break through the bind.
- chatGPT
10000 digits of e
2.718281828459045235360287471352662497757247093699959574966967627724076630353 547594571382178525166427427466391932003059921817413596629043572900334295260 595630738132328627943490763233829880753195251019011573834187930702154089149 934884167509244761460668082264800168477411853742345442437107539077744992069 551702761838606261331384583000752044933826560297606737113200709328709127443 747047230696977209310141692836819025515108657463772111252389784425056953696 770785449969967946864454905987931636889230098793127736178215424999229576351 482208269895193668033182528869398496465105820939239829488793320362509443117 301238197068416140397019837679320683282376464804295311802328782509819455815 301756717361332069811250996181881593041690351598888519345807273866738589422 879228499892086805825749279610484198444363463244968487560233624827041978623 209002160990235304369941849146314093431738143640546253152096183690888707016 768396424378140592714563549061303107208510383750510115747704171898610687396 965521267154688957035035402123407849819334321068170121005627880235193033224 745015853904730419957777093503660416997329725088687696640355570716226844716 256079882651787134195124665201030592123667719432527867539855894489697096409 754591856956380236370162112047742722836489613422516445078182442352948636372 141740238893441247963574370263755294448337998016125492278509257782562092622 648326277933386566481627725164019105900491644998289315056604725802778631864 155195653244258698294695930801915298721172556347546396447910145904090586298 496791287406870504895858671747985466775757320568128845920541334053922000113 786300945560688166740016984205580403363795376452030402432256613527836951177 883863874439662532249850654995886234281899707733276171783928034946501434558 897071942586398772754710962953741521115136835062752602326484728703920764310 059584116612054529703023647254929666938115137322753645098889031360205724817 658511806303644281231496550704751025446501172721155519486685080036853228183 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905198704230017946553679...
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God as E
Sarah was a young mathematician studying at a liberal arts college. She majored in Mathematics and minored in English Literature, and had a special love for poetry. She often spoke of mathematics as the most poetic of disciplines. "I love its beauty," she said.
Sarah also took process theology under me. She liked the idea that in process theology mathematical entities reside in the very mind of God, which means that mathematical inquiry can be a way of exploring and experiencing the divine mind. Like Whitehead, she was Platonic in sensibilities. She didn't think we humans create mathematical entities, she thought we discover them, and that in some ways they were already "there" to be discovered. And she was among the few of my students who wanted to read Whitehead's mathematical works. (For some overviews, click here.)
Sarah was also attracted to the idea that mathematical entities, numbers for example, could be metaphors for God. She told me about friends of hers who have their doubts about God but not about numbers: "They believe in the beauty of equations. I think that's their way of believing in God."
One day she came into my office wanting to discuss God and the number 'e.' She found it amazingly beautiful in its own right, with its endless series of decimals and its transcendental status. I was unfamiliar with 'e,' and she tried to explain it to me. Scroll down for my own explanation, or, better, listen to the BBC podcast above in which some mathematicians discuss 'e' and its applications. Sarah suggested that just as 'e' is essential to many mathematical equations, so God is essential to the fabric of the universe. And just as 'e' is beautiful in its complexity and elegance, so too is God beautiful in the way God helps give the world its patterns. For her, mathematics was not at all about numbers alone; it was about pattern recognition. "Numbers," she said, "are the ways patterns are arranged."
She saw similar forms of arrangement in poetry, not only in the contents of words in relation to one another, but in their placement on pages and screens. "The white space is as important as the words." She gave me new eyes for Buddhist notions of Emptiness. I hoped that some day she might come back and we could talk about God and 0. And maybe God and 3.
I saw that, for her, mathematical metaphors for God could be as helpful, and sometimes more helpful, than personal metaphors. I began to think of God in e-like terms. God is E with an upper case E, and 'e' is e with a lower case e.
I see seven affinities between E and e. Each is transcendental, irrational, constant, functional, endless, non-repetitive, and amazing. The descriptors will differ in meaning relative to theology and mathematics, but the similarities are still present.
To say that E is irrational means that E cannot be represented as a simple fraction such as 1/2 or 11/5. God cannot be fractionalized.
To say that E is constant means that E's love is unchanging. E is present throughout the universe in a way that is homologous, the same throughout. In the words of Thomas Oord, God is essentially kenotic.
To say that E is non-repetitive means the applications of divine love in the world through 'lures' are changing from circumstance to circumstance, never quite repeating itself, because the circumstances are different. God is temporal as well as timeless.
To say that E is endless is to say that God is never finished, there is always more to E than is contained in its past, just as the number 'e' has no final decimal.
To say that E is transcendental means that E is not merely a "solution" to certain intellectual problems but rather, and more deeply, a beautiful presence, like 'e' itself.
To say that E is functional is to say that E is perpetually at work in the universe as an organ of order and novelty, connecting things.
To say that E is beautiful is to say that the power of this E is not force or compulsion, not violence or manipulation, but beauty. It is everywhere, in a non-coercive way, connecting things.
I well realize that none of this proves the existence of God. And for some the very word "God" has a sacredness to it that cannot be substituted by the letter E, even if capitalized. But for others, the word "God" carries overly heavy baggage, and E is better. For them there is something freeing and refreshing about E that cannot be conveyed by the word "God." Whereas some might say E is another word for God, they would reverse it: God is another word for E.
The universe is laced with the transcendental, irrational, constant, endless, non-repetitive, functional, and beautiful. I can well imagine a mathematical liturgy which begins with a shared conviction: Praise E from whom many blessings flow, some of which are in science, mathematics, and engineering, and some of which are in wonder. The scripture would be Euler's equation. The congregants would include mathematicians from different parts of the world. Sarah would be there, and maybe even Whitehead.
- Jay McDaniel
Sarah also took process theology under me. She liked the idea that in process theology mathematical entities reside in the very mind of God, which means that mathematical inquiry can be a way of exploring and experiencing the divine mind. Like Whitehead, she was Platonic in sensibilities. She didn't think we humans create mathematical entities, she thought we discover them, and that in some ways they were already "there" to be discovered. And she was among the few of my students who wanted to read Whitehead's mathematical works. (For some overviews, click here.)
Sarah was also attracted to the idea that mathematical entities, numbers for example, could be metaphors for God. She told me about friends of hers who have their doubts about God but not about numbers: "They believe in the beauty of equations. I think that's their way of believing in God."
One day she came into my office wanting to discuss God and the number 'e.' She found it amazingly beautiful in its own right, with its endless series of decimals and its transcendental status. I was unfamiliar with 'e,' and she tried to explain it to me. Scroll down for my own explanation, or, better, listen to the BBC podcast above in which some mathematicians discuss 'e' and its applications. Sarah suggested that just as 'e' is essential to many mathematical equations, so God is essential to the fabric of the universe. And just as 'e' is beautiful in its complexity and elegance, so too is God beautiful in the way God helps give the world its patterns. For her, mathematics was not at all about numbers alone; it was about pattern recognition. "Numbers," she said, "are the ways patterns are arranged."
She saw similar forms of arrangement in poetry, not only in the contents of words in relation to one another, but in their placement on pages and screens. "The white space is as important as the words." She gave me new eyes for Buddhist notions of Emptiness. I hoped that some day she might come back and we could talk about God and 0. And maybe God and 3.
I saw that, for her, mathematical metaphors for God could be as helpful, and sometimes more helpful, than personal metaphors. I began to think of God in e-like terms. God is E with an upper case E, and 'e' is e with a lower case e.
I see seven affinities between E and e. Each is transcendental, irrational, constant, functional, endless, non-repetitive, and amazing. The descriptors will differ in meaning relative to theology and mathematics, but the similarities are still present.
To say that E is irrational means that E cannot be represented as a simple fraction such as 1/2 or 11/5. God cannot be fractionalized.
To say that E is constant means that E's love is unchanging. E is present throughout the universe in a way that is homologous, the same throughout. In the words of Thomas Oord, God is essentially kenotic.
To say that E is non-repetitive means the applications of divine love in the world through 'lures' are changing from circumstance to circumstance, never quite repeating itself, because the circumstances are different. God is temporal as well as timeless.
To say that E is endless is to say that God is never finished, there is always more to E than is contained in its past, just as the number 'e' has no final decimal.
To say that E is transcendental means that E is not merely a "solution" to certain intellectual problems but rather, and more deeply, a beautiful presence, like 'e' itself.
To say that E is functional is to say that E is perpetually at work in the universe as an organ of order and novelty, connecting things.
To say that E is beautiful is to say that the power of this E is not force or compulsion, not violence or manipulation, but beauty. It is everywhere, in a non-coercive way, connecting things.
I well realize that none of this proves the existence of God. And for some the very word "God" has a sacredness to it that cannot be substituted by the letter E, even if capitalized. But for others, the word "God" carries overly heavy baggage, and E is better. For them there is something freeing and refreshing about E that cannot be conveyed by the word "God." Whereas some might say E is another word for God, they would reverse it: God is another word for E.
The universe is laced with the transcendental, irrational, constant, endless, non-repetitive, functional, and beautiful. I can well imagine a mathematical liturgy which begins with a shared conviction: Praise E from whom many blessings flow, some of which are in science, mathematics, and engineering, and some of which are in wonder. The scripture would be Euler's equation. The congregants would include mathematicians from different parts of the world. Sarah would be there, and maybe even Whitehead.
- Jay McDaniel
We can only reproduce the equation and not stop to inquire into its implications. It appeals equally to the mystic, the scientists, the mathematician." This formula of Leonhard Euler (1707-1783) unites the five most important symbols of mathematics: 1, 0, pi, e and i (the square root of minus one). This union was regarded as mystic union containing representatives from each branch of the mathematical tree: arithmetic is represented by 0 and 1, algebra by the symbol i, geometry by pi, and analysis by the transcendental e. Harvard mathematician Benjamin Pierce said about the formula, "That is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."
- Mudassir Ibrahim: http://mudassiribrahim.blogspot.com/2015/04/euler-rotation-and-eulers-formula.html
- Mudassir Ibrahim: http://mudassiribrahim.blogspot.com/2015/04/euler-rotation-and-eulers-formula.html
For Us Non-Mathematicians
A constant
"e" is a special number that appears in many areas of mathematics, science, and engineering. It is a mathematical constant in that, if used in an equation, it does not change. It is not a variable. It is one of the five most important mathematical constants, along with 0, 1, π (pi), and i (the imaginary unit).
Non-Repetitive and Endless
Another property is that its decimal expression is non-repetitive. It is approximately equal to 2.71828 if we need to simplify, but its decimal expression goes on infinitely without repeating the same pattern.
Irrational
Another important property of "e" is that it is an irrational number, which means it cannot be expressed as a fraction of two integers. In other words, "e" cannot be written as a simple whole number or fraction, like 2 or 3/4.
Transcendental
An additional important property of "e" is that it is transcendental. A transcendental number is a special kind of number that cannot be expressed as a solution to any equation that involves only addition, subtraction, multiplication, division, and exponentiation using whole numbers or fractions. Other famous transcendental numbers include π, √2, √3, and φ (the golden ratio).
This means that "e" cannot be expressed as a solution to any equation that only involves these basic mathematical operations using whole numbers or fractions. For example, the number 2 is not transcendental because it is a solution to the equation x - 2 = 0, which only involves addition and subtraction. The square root of 2 is also not transcendental because it is a solution to the equation x^2 - 2 = 0, which involves only addition, subtraction, multiplication, and exponentiation using whole numbers or fractions. However, "e" is transcendental because it cannot be expressed as a solution to any equation that only involves these basic mathematical operations using whole numbers or fractions.
Functional
Amid all this, "e" is immensely functional. One of the most significant applications of "e" is in the field of calculus. It arises naturally in the study of exponential functions and is used extensively in many areas of mathematics and science. For example, "e" appears in the formula for compound interest, which is used in finance and economics.
- chatGPT and Jay McDaniel combined
"e" is a special number that appears in many areas of mathematics, science, and engineering. It is a mathematical constant in that, if used in an equation, it does not change. It is not a variable. It is one of the five most important mathematical constants, along with 0, 1, π (pi), and i (the imaginary unit).
Non-Repetitive and Endless
Another property is that its decimal expression is non-repetitive. It is approximately equal to 2.71828 if we need to simplify, but its decimal expression goes on infinitely without repeating the same pattern.
Irrational
Another important property of "e" is that it is an irrational number, which means it cannot be expressed as a fraction of two integers. In other words, "e" cannot be written as a simple whole number or fraction, like 2 or 3/4.
Transcendental
An additional important property of "e" is that it is transcendental. A transcendental number is a special kind of number that cannot be expressed as a solution to any equation that involves only addition, subtraction, multiplication, division, and exponentiation using whole numbers or fractions. Other famous transcendental numbers include π, √2, √3, and φ (the golden ratio).
This means that "e" cannot be expressed as a solution to any equation that only involves these basic mathematical operations using whole numbers or fractions. For example, the number 2 is not transcendental because it is a solution to the equation x - 2 = 0, which only involves addition and subtraction. The square root of 2 is also not transcendental because it is a solution to the equation x^2 - 2 = 0, which involves only addition, subtraction, multiplication, and exponentiation using whole numbers or fractions. However, "e" is transcendental because it cannot be expressed as a solution to any equation that only involves these basic mathematical operations using whole numbers or fractions.
Functional
Amid all this, "e" is immensely functional. One of the most significant applications of "e" is in the field of calculus. It arises naturally in the study of exponential functions and is used extensively in many areas of mathematics and science. For example, "e" appears in the formula for compound interest, which is used in finance and economics.
- chatGPT and Jay McDaniel combined
Reading List (offered by BBC)
William Dunham, Euler: The Master of Us All (The Mathematical Association of America, 1999)
Tim Gowers, June Barrow-Green and Imre Leader (eds.), The Princeton Companion to Mathematics (Princeton University Press, 2008)
Jan Gullberg and Peter Hilton, Mathematics: From the Birth of Numbers (W W Norton & Co Ltd, 1997)
Julian Havil, John Napier: Life, Logarithms and Legacy (Princeton University Press, 2014)
Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (John Wiley & Sons, 2000)
Eli Maor, e: The Story of a Number (Princeton University Press, 2009)
Tim Gowers, June Barrow-Green and Imre Leader (eds.), The Princeton Companion to Mathematics (Princeton University Press, 2008)
Jan Gullberg and Peter Hilton, Mathematics: From the Birth of Numbers (W W Norton & Co Ltd, 1997)
Julian Havil, John Napier: Life, Logarithms and Legacy (Princeton University Press, 2014)
Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (John Wiley & Sons, 2000)
Eli Maor, e: The Story of a Number (Princeton University Press, 2009)