Euler Quotes

Nothing takes place in the world whose meaning is not that of some maximum or minimum.

Read Euler, read Euler, he is the master of us all.

You look like Euler's equation," he murmured as he looked me up and down. Nerd translation: Euler's equation is said to be the most perfect formula ever written. Simple but elegant. Beautiful.

Logic is the foundation of the certainty of all the knowledge we acquire.

When Euler died, he simply said I am finished and collapsed, to which someone in the audience muttered darkly" Another conjecture of Euler is proved

I could solve all the Diophantine equations, extend Newton’s work on infinite series, complete Euler's analysis of prime numbers, and it wouldn’t matter.” She looked at Isabelle. “Ella is the beauty. You and I are the ugly stepsisters. And so the world reduces us, all three of us, to our lowest common denominator.

Madam, I have just come from a country where people are hanged if they talk.

e^(iπ)+1 = 0

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.

Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were, sitting in the dining car, having a nice conversation, and that train is being pulled along at a terrific clip by certain locomotives named The Bertrand Russell and Riemann and Euler and others. And our friend Lawrence is running alongside the train, trying to keep up with us—it’s not that we’re smarter than he is, necessarily, but that he’s a farmer who didn’t get a ticket. And I, Rudy, am simply reaching out through the open window here, trying to pull him onto the fucking train with us so that the three of us can have a nice little chat about mathematics without having to listen to him panting and gasping for breath the whole way.

But that would mean it was originally a sideways number eight. That makes no sense at all. Unless..." She paused as understanding dawned. "You think it was the symbol for infinity?" "Yes, but not the usual one. A special variant. Do you see how one line doesn't fully connect in the middle? That's Euler's infinity symbol. Absolutus infinitus." "How is it different from the usual one?" "Back in the eighteenth century, there were certain mathematical calculations no one could perform because they involved series of infinite numbers. The problem with infinity, of course, is that you can't come up with a final answer when the numbers keep increasing forever. But a mathematician named Leonhard Euler found a way to treat infinity as if it were a finite number- and that allowed him to do things in mathematical analysis that had never been done before." Tom inclined his head toward the date stone. "My guess is whoever chiseled that symbol was a mathematician or scientist." "If it were my date stone," Cassandra said dryly, "I'd prefer the entwined hearts. At least I would understand what it means." "No, this is much better than hearts," Tom exclaimed, his expression more earnest than any she'd seen from him before. "Linking their names with Euler's infinity symbol means..." He paused, considering how best to explain it. "The two of them formed a complete unit... a togetherness... that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It's a beautiful concept." He paused before adding awkwardly, "Mathematically speaking.

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Certainly not! I didn't build a machine to solve ridiculous crossword puzzles! That's hack work, not Great Art! Just give it a topic, any topic, as difficult as you like..." Klapaucius thought, and thought some more. Finally he nodded and said: "Very well. Let's have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit." "Love and tensor algebra?" Have you taken leave of your senses?" Trurl began, but stopped, for his electronic bard was already declaiming: Come, let us hasten to a higher plane, Where dyads tread the fairy fields of Venn, Their indices bedecked from one to n, Commingled in an endless Markov chain! Come, every frustum longs to be a cone, And every vector dreams of matrices. Hark to the gentle gradient of the breeze: It whispers of a more ergodic zone. In Reimann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face. I'll grant thee random access to my heart, Thou'lt tell me all the constants of thy love; And so we two shall all love's lemmas prove, And in bound partition never part. For what did Cauchy know, or Christoffel, Or Fourier, or any Boole or Euler, Wielding their compasses, their pens and rulers, Of thy supernal sinusoidal spell? Cancel me not--for what then shall remain? Abscissas, some mantissas, modules, modes, A root or two, a torus and a node: The inverse of my verse, a null domain. Ellipse of bliss, converge, O lips divine! The product of our scalars is defined! Cyberiad draws nigh, and the skew mind Cuts capers like a happy haversine. I see the eigenvalue in thine eye, I hear the tender tensor in thy sigh. Bernoulli would have been content to die, Had he but known such a^2 cos 2 phi!

I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ... But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty.

Just before they boarded the yacht, Cassandra glanced at Tom and reached up to a delicate necklace she'd worn constantly since the day he'd given it to her. She touched the little charm, made in the shape of Euler's infinity symbol, that hung at the hollow of her throat. And as always, the private signal made him smile.

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsot has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error.

According to my almond-eyed little spy, the great surgeon, may his own liver rot, lied to me when he declared yesterday with a deathhead's grin that the operazione had been perfetta. Well, it had been so in the sense Euler called zero the perfect number. Actually, they ripped me open, cast one horrified look at my decayed fegato, and without touching it sewed me up again.

The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.… The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It

It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.

The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal.

A thought had come to Thozan more than once: the ultimate reason we are set in this world is to break each other's hearts...

As Boy had once heard his father say: Your desire will tell you who you are, you can’t reason with a boner – it’s as simple as that!

That this blind and aging man forged ahead with such gusto is a remarkable lesson, a tale for the ages. Euler's courage, determination, and utter unwillingness to be beaten serves, in the truest sense of the word, as an inspiration for mathematician and non-mathematician alike. The long history of mathematics provides no finer example of the triumph of the human spirit.

Only one number can stake any claim to any special status, and that is zero – the origin – upon which all other numbers depend. It is the perfect balance point of all the other numbers, which is why the monad is the “container” of all other numbers, their source. There it is, slap bang in the middle of the Euler unit circle, controlling all. It’s the SOUL of the circle.

You look like Euler's equation,' he murmured as he looked me up and down. Nerd translation: Euler's equation is said to be the most perfect formula ever written. Simple but elegant. Beautiful.

In retrospect, Euler's unintended message is very simple: Graphs or networks have properties, hidden in their construction, that limit or enhance our ability to do things with them. For more than two centuries the layout of Konigsberg's graph limited its citizens' ability to solve their coffeehouse problem. But a change in the layout, the addition of only one extra link, suddenly removed this constraint.

In his eulogy, the Marquis de Condorcet observed that whosoever pursues mathematics in the future will be "guided and sustained by the genius of Euler" and asserted , with much justification, that "all mathematicians...are his disciples.

Euler’s Formula encapsulates the whole of existence. It contains 0, the number of the monad (ontological zero); the number e that determines exponentiation; the number i that determines the imaginary domain (time); the number 1 that determines the domain of counting numbers (and with 0 creates the binary system of computing), and real numbers (space); -1, the number of the negative domain (antimatter); and the number π that determines the world of the circle and geometry. Euler’s Formula is the unquestionable God Equation.

In his life of seventy-six years, Euler created enough mathematics to fill seventy-four substantial volumes, the most total pages of any mathematician. By the time all of his work had been published (and new material continued to appear for seventy-nine years after his death) it amounted to a staggering 866 items, including articles and books on the most cutting-edge topics, elementary textbooks, books for the nonscientist, and technical manuals. These figures do not account for the projected fifteen volumes of correspondence and notebooks that are still being compiled.

Illuminism is based on Euler’s Formula. Euler’s Formula is the basis of Fourier mathematics. Fourier mathematics is the basis of quantum mechanics. Quantum mechanics is the basis of the scientific world. Therefore Euler’s Formula is the basis of the scientific world, and the basis of everything.

Linking their names with Euler’s infinity symbol means . . .” He paused, considering how best to explain it. “The two of them formed a complete unit . . . a togetherness . . . that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It’s a beautiful concept.” He paused before adding

The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered.

You can save nobody from himself, not even yourself

Never was his remarkable memory more useful than when he could see mathematics only in his mind's eye.

Die Mathematik ist es, die uns vor dem Trug der Sinne schützt und uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.

If you were given to thinking of numbers as having human-like qualities, you might picture e^i*pi as a guru into transcendental meditation who'd achieved infinite enlightenment. But there's a problem with that-Euler's formula shows that e^i*pi can never free itslef from worldly concerns. Recall, e^i*pi is really -1 in disguise, and -1 is just a mathism for owing a dollar to your friend, Steve. One hand clapping.

Feynman said, “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied” Our sentence would be: “The Monadology asserts that the fundamental units of existence are INFINITE, dimensionless, living, thinking points – monads, ZEROS, souls – each of which has INFINITE energy content, all controlled by a single equation – Euler’s Formula – and the collective energy of this universe of mathematical points creates a physical universe of which every objective value is ZERO, but, through a self-solving, self-optimizing, dialectical, evolving process, the universe generates a final, subjective value of INFINITY – divinity, perfection, the ABSOLUTE.” For ours is the religion of zero and infinity, the two numbers that define the soul and the whole of existence. As above, so below.

Quand mon cerveau excite dans mon ame la sensation d'un arbre ou d'une maison, je prononce hardiment, qu'il existe réellement hors de moi un arbre ou une maison, dont je connois même le lieu, la grandeur ou d'autres propriétés. Ainsi ne trouve-t-on ni homme ni bête qui doutent de cette vérité. Si un paysan en vouloit douter ; s'il disoit, par exemple, qu'il ne croyait pas que son baillif existe, quoiqu'il fut devant lui, on le pretendroit pour un fou et cela avec raison : mais dès qu'un philosophe avance de tels sentimens, il veut qu'on admire son esprit et ses lumières, qui surpassent infiniment celles du peuple.

In 1994, Karl Sims was doing experiments on simulated organisms, allowing them to evolve their own body designs and swimming strategies to see if they would converge on some of the same underwater locomotion strategies that real-life organisms use.5, 6, 7 His physics simulator—the world these simulated swimmers inhabited—used Euler integration, a common way to approximate the physics of motion. The problem with this method is that if motion happens too quickly, integration errors will start to accumulate. Some of the evolved creatures learned to exploit these errors to obtain free energy, quickly twitching small body parts and letting the math errors send them zooming through the water.

Reality is based on unobservable mathematical points (singularities). That’s the secret of existence. What’s at the centre of a black hole? – a singularity. What was the Big Bang? – a singularity event. What is the Big Crunch? – when spacetime returns to a singularity. What is light made of? – photonic singularities (immaterial and dimensionless; according to Einstein’s special theory of relativity, photons have no mass, are maximally length contracted to zero, and time has stopped for them). The whole universe is made of light. It comes from light and returns to light. Light is all about points – singularities. Light is the basis of thought, the basis of mind, and the basis of matter. Everything is derived from light, and light is nothing but mathematical points defined by the generalised Euler Formula, and it creates the visible world via Fourier mathematics.

Why is there something rather than nothing? Only because something can exist as nothing – via the mathematical capacity to express “ nothing ” in non-zero terms, e.g. e iπ + 1 = 0. In other words, wherever you see “ nothing ” (zero), you might in fact be confronting e iπ + 1 (“something” ), without knowing it. Only mathematics has this unique capacity to the ground state of the universe. The compulsory ground state of the universe is zero, the minimum value possible. There is no sufficient reason for any arbitrary non-zero number to serve as the ground state.

Euler calculaba sin aparente esfuerzo como los hombres respiran o las águilas se sostienen en el aire

Podczas pobytu na dworze carycy Katarzyny II wielki szwajcarski matematyk Leonhard Euler wdał się w dyskusję na temat istnienia Boga. Aby pokonać przeciwników, poprosił o tablicę, na której napisał: (x + y)^2 = x^2 + 2xy + y^2, a więc Bóg istnieje! Nie mogąc zakwestionować adekwatności wywodu, którego nie rozumieli, i nie chcąc okazać swojej niewiedzy, oponenci przyjęli ten argument za rozstrzygający.

Euler fue uno de los grandes matemáticos que podía trabajar en cualquier condición. Amaba los niños (tuvo trece, aunque cinco de ellos murieron siendo pequeños), y podía dedicarse a sus trabajos teniendo a alguno de sus hijos sentado sobre sus rodillas y a los restantes jugando en torno de él.

De Morgan cuenta lo sucedido en su clásico Budgel of Paradoxes, 1872: "Diderot fue informado de que un docto matemático estaba en posesión de una demostración algebraica de la existencia de Dios, y que la expondría ante toda la corte si él deseaba oírla. Diderot consintió amablemente... Euler avanzó hacia Diderot y dijo gravemente en un tono de perfecta convicción:   “Señor, , por tanto Dios existe. Replique”.

Euler añadió gravemente las pruebas solemnes de que Dios existe, y de que el alma no es una sustancia material. Se dice que ambas pruebas fueron incorporadas a los tratados de teología de la época.

Durante la invasión rusa en 1760, la casa de campo de Euler fue saqueada. El general ruso declaró que "no hacía la guerra a la ciencia", e indemnizó a Euler con una cantidad superior a la que representaba el verdadero daño.

En el gran fuego de 1771 su casa y todos sus muebles quedaron destruidos, y gracias al heroísmo de su sirviente suizo (Peter Grimm, o Grimmon), Euler logró salvarse con el riesgo de su propia vida, Grimm pudo arrastrar a su amo, ciego y enfermo, fuera de las llamas. La biblioteca se quemó pero gracias a la energía del Conde Orloff fueron salvados todos los manuscritos de Euler.

Perhaps Cardano's curious combination of a mystical and a scientifically rational personality allowed him to catch these first glimmerings of what developed to be one of the most powerful of mathematical conceptions. In later years, through the work of Bombelli, Coates, Euler, Wessel, Argand, Gauss, Cauchy, Weierstrass, Riemann, Levi, Lewy, and many others, the theory of complex numbers has flowered into one of the most elegant and universally applicable of mathematical structures. But not until the advent of the quantum theory, in the first quarter of this century, was a strange and all-pervasive role for complex numbers revealed at the very foundational structure of the actual physical world in which we live-nor had their profound link with probabilities been perceived before this. Even Cardano could have had no inkling of a mysterious underlying connection between his two greatest contributions to mathematics-a link that forms the very basis of the material universe at its smallest scales.

4

The PSR is not an abstract principle. It is embodied ontologically by way of Euler's formula, which is what the PSR reduces to mathematically. Anyone who denies that the universe is made of reason is automatically an irrationalist, and their irrational opinions can be dismissed. There is nothing more ironic, and irrational, than irrationalists trying to give reasons why the universe is not made of reason.

Euler’s formula – although deceptively simple – is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed.

The PSR is reflected in points traveling in complex-numbered Euler circles where no point is privileged over any other. From this motion, we get sine and cosine waves, even and odd functions, symmetry and antisymmetry, orthogonality and non-orthogonality, phase, straight-line radii, right-angled triangles, Pythagoras’ theorem, the speed of mathematics (c), π, e, i, Fourier mathematics … and from all of that we get the whole of mathematics (eternal, necessary and mental; Being), and thus the whole of science (temporal, contingent and material; Becoming). And that is the whole universe explained. Nothing else is required. The PSR gives us mathematics, mathematics gives us science, and that’s all we need for the universe: science with a mathematical and rational core rather than with a material and observable core. What could be more rational and logical?

EULER CALCULATED WITHOUT APPARENT EFFORT, as men breathe, or as eagles sustain themselves in the wind” (as Arago said), is not an exaggeration of the unequalled mathematical facility of Leonard Euler (1707-1783)

The PSR is equivalent to a generalized version of Euler’s Formula, the most important analytic formula of mathematics, which is in turn ontologically conveyed through mental, metaphysical, mathematical points (monads: eternal sinusoidal energy systems, each of which constitutes an autonomous mind). Despite what science says using the fallacies and incongruities of correspondence, the whole scientific world is in fact rooted in total coherence, in the generalized Euler Formula, the God Equation. The God Equation is ontologically conveyed not by a single eternal God, but by a myriad of eternal minds. All of these minds considered collectively constitute “God”, and they have a net result of zero.

Ontological mathematics is based on light. Light is eternal (it does not experience time and it does not experience space and is therefore indestructible); light is mental (it is massless and immaterial), light is absolute (it provides the absolute reference frame – the ether – for all spacetime reference frames). Light corresponds exactly to the immaterial, unextended mind posited by Descartes. Have you seen the light? Once you realize that light is nothing but sinusoidal waves, as per Euler’s Formula, you have the means to understand the whole of reality. Light is God. Light is the substance of an intelligent, living, thinking mathematical organism, calculating its own perfection. All of the great ancients understood this type of picture of reality. No modern scientist does.

There’s nothing mysterious about souls. They are simply mathematical singularities outside space and time. They are autonomous Fourier frequency domains. They are Leibnizian monads and Cartesian minds. They are the quintessence of ontological mathematics and the basis of everything. We can even write a precise equation for souls = singularities = minds = monads, namely, the God Equation, which is just the generalised Euler Formula, the centrepiece of mathematical analysis and fundamental to physics.

There’s nothing mysterious about souls. They are simply mathematical singularities outside space and time. They are autonomous Fourier frequency domains. They are Leibnizian monads and Cartesian minds. They are the quintessence of ontological mathematics and the basis of everything. We can even write a precise equation for souls = singularities = minds = monads, namely, the God Equation, which is just the generalised Euler Formula, the centrepiece of mathematical analysis and fundamental to physics.

The commonplace idea that math is a manmade language is comically dumb. We know what manmade languages are like: English, German, French, Spanish, Japanese, and so on. None of these has any resemblance whatsoever to math. How can a human being invent π or e? How can a human pluck Euler’s Formula out of nothing? The reason why people are so keen to say that math is manmade is because if they’re wrong then the converse is true: man is mathmade! That is, of course, exactly the position of ontological mathematics, and it represents the highest possible wisdom

Finally, in 1748, Euler published the explicit formula in his book Introductio in Analysis Infinitorum.

Euler’s constant, which is γ = 0.577215664901532 …. After π and e, γ is perhaps the most important mathematical constant not appearing in elementary arithmetic.

Consider Euler’s identity: eiπ + 1 = 0. Here we have the exact situation where a something – the expression on the left – is exactly equal to zero (nothing). The expression on the left is not nonexistence. It has properties, capacities, potentialities, the ability to interact with others of its kind, indeed, to contribute to the entire infinite system of mathematics, which, in the end, reduces to nothing but a set of infinite tautologies expressing 0 = 0. Since we know that eiπ + 1 = 0, we can substitute 0 for eiπ + 1, leaving 0 = 0. So, here we have something whose essence is to exist and yet be nothing. This is true of the whole of ontological mathematics, and it’s the only system of which this is true, hence it’s the only true system.

Consider Euler’s identity: eiπ + 1 = 0. Here we have the exact situation where a something – the expression on the left – is exactly equal to zero (nothing). The expression on the left is not nonexistence. It has properties, capacities, potentialities, the ability to interact with others of its kind, indeed, to contribute to the entire infinite system of mathematics, which, in the end, reduces to nothing but a set of infinite tautologies expressing 0 = 0. Since we know that eiπ + 1 = 0, we can substitute 0 for eiπ + 1, leaving 0 = 0. So, here we have something whose essence is to exist and yet be nothing. This is true of the whole of ontological mathematics, and it’s the only system of which this is true, hence it’s the only true system.

To this day the existence of odd perfect numbers remains unsolved.

The number of God is Pi and Euler number is evil.

This quest to take a problem and see what happens in different situations is called generalizing, and it is this force that drives mathematics forward. Mathematicians constantly want to find solutions and patterns which apply to as many situations as possible, i.e. are as general as possible. A maths puzzle is not complete when you merely find an answer, a maths puzzle is complete when you've then tried to generalize it to other situations as well-and minds including Leonard Euler and Lord Kelvin have excelled in mathematics by displaying just this kind of curiosity. Because mathematicians like the puzzles which work on the pure number rather than the symbolic digit and the system we happen to be writing our numbers down in, there is a sense that, when a puzzle works only in one given base, there is something rather, well, 'secind class' about it. Mathematicians do not like things which work only in base-10; it is only because we have ten fingers that we find that system interesting at all. Mathematics is the search for universal, not base-specific, truth.

The first scientist to contemplate the significance of places where things apparently cease to exist or become infinite ('singularities' that we would Now call them) in Newtonian Theory where the 18th century scientist leonhard Euler and Roger boscovich.

Objects in mathematics are named after the first person after Euler to discover them.

Multiplying an infinitesimal times an infinitely large number yields a finite number. There's no analogue to this rule in regular arithmetic. However, it accords with the intuitive idea that when the infinitely large is pitted against the infinitely small, the two basically cancel each other out in a titanic clash, and after the dust clears a finite number remains.

When Euler discovered that i^i is real, he exclaimed in a letter to a friend that this "seems extraordinary to me"-part of his genius, as well as of his charm, was an inexhaustible capacity to be surprised and delighted by his discoveries.

By the early 1600s mathematicians had managed to crank out an approximation of pi that was accurate to 35 digits, which is far more than needed for any earthly purpose. With only 39 digits, you could calculate the circumference of the observable universe to within the diameter of a hydrogen atom.

But just what are imaginary numbers, you may now be asking yourself, and what on earth could it mean to raise e to an imaginary-number power? This chapter concerns mathematicians' long struggle to answer the first of these two questions. Later we'll take up the second one , which inspired Euler to devise the most radical expansion of the concept of exponents in math history. At this point, suffice it to say that affixing an imaginary exponent to a number has a dramatic effect on it-something lime what happens to a frog when it's tapped by a standard-issue magic wand.

The imaginaries finally lost their air of impossibility when nineteenth-century mathematicians realized that they're actually perfectly ordinary, law-abiding numerical beings-it's just that they hail from a different dimension. We'll go there later on.

In my view, Euler's tranquil temperament, fairness, and generosity were integral to his greatness as a mathematician and scientist- he was never inclined to waste time and energy engaging in petty one-upmanship (like his mentor, Johann Bernoulli, who was known for getting into the eighteenth-century version of flame wars with his older brother, mathematician Jakob Bernoulli, and even with his own son, Daniel, over technical disputes), brooding about challenges to his authority (like Newton), or refusing to publish important findings because of the fear that they might be disputed (like Gauss).

Now be honest-wouldn't you have expected e^i*pi to be (a) gibberish aling the lines of "elephant inkpie," or, if it were mathematically meaningful, to be (b) an infinitely complicated irrational number? Indeed, e^i*pi is a transcendental number raised to an imaginary transcendental power. And if (b) were the case, surely e^i*pi would not compute no matter how much computer power were available to try to pin down its value. As you know, neither (a) nor (b) is true, because e^i*pi = -1. (I suspect the fact that both (a) and (b) are provably false is the reason that Benjamin Peirce, the nineteenth-century mathematician, found Euler's formula (or a closely rekated formula) "absolutely paradoxical.") In other words, when the three enigmatic numbers are combined in this form, e^i*pi , they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers. It's as if greenish-pink androids rocketing toward Alpha Centauri in 2370 had hit a space time anomaly and suddenly found themselves sitting in a burger joint in Topeka, Kansas, in 1956. Elvis, of course , was playing on the jukebox.

Chinese philosopher Tsang Lap-Chuen is a leading modern exponent of the idea that the sublime involves this kind of experience. In The Sublime: Groundwork towards a Theory, published in 1998, he wrote that the sublime "evokes our awareness of our being on the threshold from the human to that which transcends the human; which borders on the possible and the impossible; the knowable and the unknowable; the meaningful and the fortuitous; the finite and the infinite." In his view, there is no single essential common property possessed by sublime works or sublime natural objects, nor is there a single emotional state evoked by all of them. But he argues that there's a common thread in experiences of the sublime, which is that they take us "to the limit of some human possibility.

Here's just one of Ramanujan's many provocative formulas: (1+1/2^4) * (1+1/3^4)*(1+1/5^4)*(1+1/7^4)*(1+1/next prime number^4)x...= 105/pi^4. The infinite product on the left side of this equation is based on successive prime numbers raised to the 4th power. Primes are integers greater than 1 that are evenly divisible only by themselves and 1. Thus, 3 is a prime, but 4 isn't because it's evenly divisible by 2. The first nine primes are 2,3,5,7,11,13,17,19, and 23. The primes go on forever, which accounts for the ellipsis at the end of the product in Ramanujan's formula. This formula shows a deep connection between pi and the prime numbers.

Euler's general equation stands out because it forged a fundamental link between different areas of math, and because of its versatility in applied mathematics. After Euler's time it came to be regarded as a cornerstone in "complex analysis," a fertile branch of mathematics concerned with functions whose variables stand for complex numbers.

First, let me frame what I'm calling beautiful. It's not simply the equation's neat little string of symbols. Rather, it's the entire nimbus of meaning surrounding the formula, including its funneling of many concepts into a statement of stunning brevity, its arresting combination of apparent simplicity and hidden complexity, the way its derivation bridges disparate topics in mathematics, and the fact that it's rich with implications, some of which weren't apparent until many years after it was proved to be true. I think most mathematicians would agree that the equation's beauty concerns something like this nimbus.

Leonhard Euler (pronounced “oiler”, 1707–1783) is judged by all to have been the most productive, and by many to have been the best, mathematician of modern times. He was Swiss, but spent much of his life in Russia because he had a big family and Catherine the Great offered him a lot of money. His paper “The Seven Bridges of Königsberg” (1736), which we will discuss in Chapter 8, is the earliest known work on the theory of graphs. The theorem now known as Euler’s Formula was proved by Euler in 1752. It is one of the classic theorems of elementary mathematics and plays a central role in the next three chapters of this book.

It’s possible to prove mathematically that the image on the signs could never be a ball. There is something called the Euler characteristic of a surface, which describes the pattern behind how different 2D shapes can join together to make a 3D shape. In short, a ball has a Euler characteristic of two, and hexagons on their own cannot make a shape with a Euler characteristic of more than zero.

We always talk about the incredible relationship between zero and infinity. It’s Euler’s Formula that controls this relationship ontologically. You can have infinite things that are completely different and unique and yet are all exactly the same – ZERO. Every Euler circle, with its different Euler numbers and different energy characteristics, is, at one and the same time just zero energy because all the different elements ALWAYS cancel to zero. This is simply breathtaking. Only mathematics can deliver this and, in particular, only Euler’s Formula. That’s why it’s the all-controlling God Equation.

The world has now arrived at a place where we can do something with our own data—we don’t have to wait for a Milgram, let alone an Euler, to teach us about ourselves.

Kolmogorov – Poincaré – Gauss – Euler – Newton, are only five lives separating us from the source of our science.

A mathematical proposition expresses a certain expectation. For example, the proposition, “Euler constant C is rational” expresses the expectation that we could find two integers a and b such that C = a/b. Perhaps, the word “intention”, coined by the phenomenologists, expresses even better what is meant here.

Reality is based on unobservable mathematical points (singularities). That’s the secret of existence. What’s at the centre of a black hole? – a singularity. What was the Big Bang? – a singularity event. What is the Big Crunch? – when spacetime returns to a singularity. What is light made of? – photonic singularities (immaterial and dimensionless; according to Einstein’s special theory of relativity, photons have no mass, are maximally length contracted to zero, and time has stopped for them). The whole universe is made of light. It comes from light and returns to light. Light is all about points – singularities. Light is the basis of thought, the basis of mind, and the basis of matter. Everything is derived from light, and light is nothing but mathematical points defined by the generalised Euler Formula, and it creates the visible world via Fourier mathematics.

When we press into service Euler’s Formula – a formula used throughout physics – to explain everything, we get mocked by the cultists of scientism. You always mock genius, until you are forced to submit to it! Our vindication will be total and absolute. No one can defeat reason and logic.

today they are usually called the Fresnel integrals. One does still see them also called the Euler integrals, however, and it was Euler who first evaluated them.

77 LEONHARD EULER 1707-1783

Euler's proof that in Konigsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all the bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Konigsberg graph had four such nodes, one could not find the desired path.

The house is a normal-sized house, but once you step foot in the door, you are confronted with “The Dome.” Perfectly round, this room is one continuous curved wall of books. A copper dome sits on top with four stained glass windows fitted tight to allow for natural light to stream in. The four stained glass windows offer portraits of the four greatest mathematicians in history: Newton, Euler, Gauss, and Archimedes, though they are ordered alphabetically from left to right on the dome.

Along with working on the Basel problem, Euler realized that adding an infinite sequence of reciprocal powers for all whole numbers will give you the same answer as multiplying together an infinite sequence of fractions which use only the prime numbers. So the zeta function can be written as two different equations, one of which relies only on the prime numbers. The one which uses all the whole numbers gives the same result as the prime fractions, but it's easier to work with. We know what all the whole numbers are, but we don't know what all the primes are. So we can substitute one for the other.

Luckily, there's a neat approximation formula for just this sum. The higher the number n, the better the estimate will be. It is mathematically proven that as n grows to infinity, the approximation formula converges to the true value. Here it is: H(n) ≈ ln(n) + 0.58 The value 0.58 comes from rounding off the Euler-Mascheroni constant, which should be where the 0.58 is now. But since we just want to approximate, there's no need to be overly precise. For our purposes the rounded off value will do just fine.

How should you go about claiming your award money? There are two possible directions to take. You can try to prove that P = NP or you can aim to show that P is not equal to NP. To show that P = NP, all you have to do is take one of your favorite NP-Complete problems and find a polynomial algorithm that solves it. As we have seen, if you do find such an algorithm, then all NP problems will be solvable in a polynomial amount of operations. It might seem strange to think that a problem that demands an exponential or factorial amount of operations can be done in a polynomial amount of operations. It might seem strange to think that a problem that demands an exponential or factorial amount of operations can be done in a polynomial amount of operations. However, we saw something similar with the Euler Cycle Problem. Rather than look through all n! possible cycles to see if any are Euler cycles, we used the trick of checking if the number of edges touching each vertex is even or not. Does a similar trick for the Hamiltonian Cycle Problem exist? For many years, the smartest people around have been looking for such a trick or algorithm and have not been successful. However, you might possess some deeper insight that they lack. Get to it! On the other hand, you can try to show that P is not equal to NP. One way to do this is to take an NP problem and show that no polynomial algorithm exists for it. It so happens that it is very hard to prove such a claim: there are a lot of algorithms out there. This has turned out to be one of the hardest problems in mathematics. As a final hint, it should be noted that most researchers believe that P is not equal to NP.

Most of his predecessors had considered the differential calculus as bound up with geometry, but Euler made the subject a formal theory of functions which had no need to revert to diagrams or geometrical conceptions.

Some of the greatest minds in the history of science, including Kepler, Halley, and Euler, had speculated as to the existence of a so-called “hollow Earth.” One day, it was hoped, the technique of intra-planetary “short-cutting” about to be exercised by the boys would become routine, as useful in its way as the Suez or the Panama Canal had proved to surface shipping. At the time we speak of, however, there still remained to our little crew occasion for stunned amazement, as the Inconvenience left the South Indian Ocean’s realm of sunlight, crossed the edge of the Antarctic continent, and began to traverse an immense sweep of whiteness broken by towering black ranges, toward the vast and tenebrous interior which breathed hugely miles ahead of them. Something did seem odd, however. “The navigation’s not as easy this time,” Randolph mused, bent over the chart table in some perplexity. “Noseworth, you can remember the old days. We knew for hours ahead of time.” Skyfarers here had been used to seeing flocks of the regional birds spilling away in long helical curves, as if to escape being drawn into some vortex inside the planet sensible only to themselves, as well as the withdrawal, before the advent of the more temperate climate within, of the eternal snows, to be replaced first by tundra, then grassland, trees, plantation, even at last a settlement or two, just at the Rim, like border towns, which in former times had been the sites of yearly markets, as dwellers in the interior came out to trade luminous fish, giant crystals with geomantic properties, unrefined ores of various useful metals, and mushrooms unknown to the fungologists of the surface world, who had once journeyed regularly hither in high expectation of discovering new species with new properties of visionary enhancement.

Materialists simply cannot conceive of immaterial, non-sensory things. Yet the irony is that mathematics is entirely immaterial and non-sensory ... and is the core of scientific materialism! Work that one out. It’s mathematics that defines “immaterial substance” and does so via the God Equation, and Fourier frequency singularities. Of course, Hobbes had the excuse that he didn’t know any of that ... but modern scientists have no such excuse. They have complete access to Euler’s Formula and Fourier mathematics. What they lack is intelligence, imagination and a proper ontology and epistemology, and any understanding whatsoever of what mathematics actually is.

General becomes a Statesman, but he must not cease to be the General. He takes into view all the relations of the State on the one hand; on the other, he must know exactly what he can do with the means at his disposal. As the diversity, and undefined limits, of all the circumstances bring a great number of factors into consideration in War, as the most of these factors can only be estimated according to probability, therefore, if the Chief of an Army does not bring to bear upon them a mind with an intuitive perception of the truth, a confusion of ideas and views must take place, in the midst of which the judgment will become bewildered. In this sense, Bonaparte was right when he said that many of the questions which come before a General for decision would make problems for a mathematical calculation not unworthy of the powers of Newton or Euler.

Humanitarian Arithmetic (Sonnet 1354) If it takes $300bn to end world hunger, and 7 trillion to fund the next AI wonder, how many people have to starve to death, to feed the appetite of the cyberworld? If Britain's NHS costs about $200bn, and US military costs 800 billion dollars, how many have to suffer from sickness, for the tribal chiefs to feel secure? If it takes $20bn to end homelessness in the US, and trillions to colonize Mars, how many have to sleep in cardboard boxes, for heirs of billionaires to breed on Mars? You don't need to be a Ramanujan or Euler, to solve this simple arithmetic equation. But you do need a living human heart, to take responsibility for the solution.