A Most Elegant Equation Quotes

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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).”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics

“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.”
― David Stipp, A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics