Approximating Irrational Numbers Quotes

Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out, fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets - but the beast ~ Stanisław Lem

I understand and sympathize with the reasonable needs of a reasonable number of people on a finite continent. All life depends upon other life. But what is happening today, in North America, is not rational use but irrational massacre. Man the Pest, multiplied to the swarming stage, is attacking the remaining forests like a plague of locusts on a field of grain. ~ Edward Abbey

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly disimilar to, and I might even say in priciple the same as, my method described above of introducing trasfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite. ~ Georg Cantor

The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers. ~ Georg Cantor

The thing that struck me was his intensity. Whatever he was interested in he would generally carry to an irrational extreme." Jobs had honed his trick of using stares and silences to master other people. " One of his numbers was to stare at the person he was talking to. He would stare into their fucking eyeballs, ask some question, and would want a response without the other person averting their eyes. ~ Walter Isaacson

Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes called the Golden Proportion), are known as irrational numbers. They lie deep in the structure of the physical universe, and were seen by the Egyptians as the principles controlling creation, the principles by which matter is precipitated from the cosmic mind.

Today scientists recognize the Comma of Pythagoras, Pi and the Golden Proportion as well as the closely related Fibonacci sequence are universal constants that describe complex patterns in astronomy, music and physics. ...

To the Egyptians these numbers were also the secret harmonies of the cosmos and they incorporated them as rhythms and proportions in the construction of their pyramids and temples. ~ Jonathan Black

Just as the introduction of the irrational numbers ... is a convenient myth [which] simplifies the laws of arithmetic ... so physical objects are postulated entities which round out and simplify our account of the flux of existence ... The conceptional scheme of physical objects is [likewise] a convenient myth, simpler than the literal truth and yet containing that literal truth as a scattered part. ~ Willard Van Orman Quine

[T]he superstitions to be feared in the present day are much less religious than political; and of all the forms of idolatry I know none more irrational and ignoble than this blind worship of mere numbers. - William Lecky, Democracy and Liberty ~ Bryan Caplan

I chose the name Pi because it's an irrational number (one with no discernable pattern). Yet scientists use this irrational number to come to a "rational" understanding of the universe. To me, religion is a bit like that, "irrational" yet with it we come together we come to a sound understanding of the universe. ~ Yann Martel

Einstein's remark on the limitlessness of human stupidity is made even more disturbing by the discovery that infinity comes in different sizes. Answering 'How much stupider?' or trying to measure the minimal idiocy bounded by an IQ test are mysteries which are themselves infinitely less alarming than simply attempting to tally the anti-savant population. One can count all the natural idiots (they're the same as the even number of idiots – twice as many), but the number of real idiots continues forever: all the counting idiots (finger reckoners) plus all the fractional idiots (geniuses on a bad day) plus all the irrational idiots (they go on and on and on) add up to a world in which the approaching upper limit of our set of natural resources has its complement in the inexhaustible lower limit of our set of mental ones. ~ Bauvard

I believe in numbers. The ones you can see and the ones you can't. The real and the imaginary, the rational and the irrational, and every point on lines that go on forever. Numbers have never let me down. They don't waffle. They don't lie. They don't pretend to be what they're not. They're timeless. ~ Amy Harmon

What good your beautiful proof on the transcendence of Pi: Why investigate such problems, given that irrational numbers do not even exist? ~ Leopold Kronecker

It is shown that the golden ratio plays a prominent role in the dimensions of all objects which exhibit five-fold symmetry. It is also showed that among the irrational numbers, the golden ratio is the most irrational and, as a result, has unique applications in number theory, search algorithms, the minimization of functions, network theory, the atomic structure of certain materials and the growth of biological organisms. ~ Richard A. Dunlap

Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc ... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. ~ Abraham Robinson

You must strive to multiply bread so that it suffices for the tables of mankind, and not rather favor an artificial control of birth, which would be irrational, in order to diminish the number of guests at the banquet of life. ~ Pope Paul VI

predecessor of Isaac Newton at Cambridge University, maintained that irrational numbers have no meaning independent of geometric lengths. ~ Morris Kline

Do you know what the foundation of mathematics is?" I ask. "The foundation of
mathematics is numbers. If anyone asked me what makes me truly happy, I would
say: numbers. Snow and ice and numbers. And do you know why?"
He splits the claws with a nutcracker and pulls out the meat with curved tweezers.
"Because the number system is like human life. First you have the natural numbers.
The ones that are whole and positive. The numbers of a small child. But human
consciousness expands. The child discovers a sense of longing, and do you know
what the mathematical expression is for longing?"
He adds cream and several drops of orange juice to the soup.
"The negative numbers. The formalization of the feeling that you are missing
something. And human consciousness expands and grows even more, and the child
discovers the in between spaces. Between stones, between pieces of moss on the
stones, between people. And between numbers. And do you know what that leads
to? It leads to fractions. Whole numbers plus fractions produce rational numbers.
And human consciousness doesn't stop there. It wants to go beyond reason. It adds
an operation as absurd as the extraction of roots. And produces irrational numbers."
He warms French bread in the oven and fills the pepper mill.
"It's a form of madness.' Because the irrational numbers are infinite. They can't be
written down. They force human consciousness out bey ~ Peter Høeg

The same thing is happening with the State. Call to mind what the State was at the end of the XVIIIth Century in all European nations. Quite a small affair! Early capitalism and its industrial organisation, in which the new, rationalised technique triumphs for the first time, had brought about a commencement of increase in society. A new social class appeared, greater in numbers and power than the pre-existing: the middle class. This astute middle class possessed one thing, above and before all: talent, practical talent. It knew how to organise and discipline, how to give continuity and consistency to its efforts. In the midst of it, as in an ocean, the "ship of State" sailed its hazardous course. The ship of State is a metaphor re-invented by the bourgeoisie, which felt itself oceanic, omnipotent, pregnant with storms. That ship was, as we said, a very small affair: it had hardly any soldiers, bureaucrats, or money. It had been built in the Middle Ages by a class of men very different from the bourgeois - the nobles, a class admirable for their courage, their gifts of leadership, their sense of responsibility. Without them the nations of Europe would not now be in existence. But with all those virtues of the heart, the nobles were, and always have been, lacking in virtues of the head. Of limited intelligence, sentimental, instinctive, intuitive - in a word, "irrational." Hence they were unable to develop any technique, a thing which demands rationalisation. They did not ~ Jose Ortega Y Gasset

Our number one opponents are not Liverpool, Arsenal or Manchester United. It is ourselves at Chelsea. ~ Didier Drogba

When I was a young boy, I can remember in the community that I grew up in, seeing people in the community who had numbers that were on their arms. ~ Bernie Sanders

Pay attention to when the cart is getting before the horse. Notice when a painful initiation leads to irrational devotion, or when unsatisfying jobs start to seem worthwhile. Remind yourself pledges and promises have power, as do uniforms and parades. Remember in the absence of extrinsic rewards you will seek out or create intrinsic ones. Take into account [that] the higher the price you pay for your decisions the more you value them. See that ambivalence becomes certainty with time. Realize that lukewarm feelings become stronger once you commit to a group, club, or product. Be wary of the roles you play and the acts you put on, because you tend to fulfill the labels you accept. Above all, remember the more harm you cause, the more hate you feel. The more kindness you express, the more you come to love those you help. ~ Anonymous

Just as summer-killed meat draws flies, so the court draws spurious sages, philosophists, and acosmists who remain there as long as their purses and their wits will maintain them, in the hope (at first) of an appointment from the Autarch and (later) of obtaining a tutorial position in some exalted family. At sixteen or so, Thecla was attracted, as I think young women often are, to their lectures on theogony, thodicy, and the like, and I recall one particularly in which a phoebad put forward as an ultimate truth the ancient sophistry of the existence of three Adonai, that of the city (or of the people), that of the poets, and that of the philosophers. Her reasoning was that since the beginning of human consciousness (if such a beginning ever was) there have been vast numbers of persons in the three categories who have endeavored to pierce the secret of the divine. If it does not exist, they should have discovered that long before; if it does, it is not possible that Truth itself should mislead them. Yet the beliefs of the populace, the insights of the rhapsodists, and the theories of the metaphysicians have so far diverged that few of them can so much as comprehend what the others say, and someone who knew nothing of any of their ideas might well believe there was no connection at all between them.

May it not be, she asked (and even now I am not certain I can answer), that instead of traveling, as has always been supposed, down three roads to the same destination, t ~ Gene Wolfe

Observe any meetings of people, and you will always find their eagerness and impetuosity rise or fall in proportion to their numbers. ~ Lord Chesterfield

Today, we live in the Age of Envy.

"Envy" is not the emotion I have in mind, but it is the clearest manifestation of an emotion that has remained nameless; it is the only element of a complex emotional sum that men have permitted themselves to identify.

Envy is regarded by most people as a petty, superficial emotion and, therefore, it serves as a semihuman cover for so inhuman an emotion that those who feel it seldom dare admit it even to themselves. . . . That emotion is: hatred of the good for being the good.

This hatred is not resentment against some prescribed view of the good with which one does not agree. . . . Hatred of the good for being the good means hatred of that which one regards as good by one's own (conscious or subconscious) judgment. It means hatred of a person for possessing a value or virtue one regards as desirable.

If a child wants to get good grades in school, but is unable or unwilling to achieve them and begins to hate the children who do, that is hatred of the good. If a man regards intelligence as a value, but is troubled by self-doubt and begins to hate the men he judges to be intelligent, that is hatred of the good.

The nature of the particular values a man chooses to hold is not the primary factor in this issue (although irrational values may contribute a great deal to the formation of that emotion). The primary factor and distinguishing characteristic is an emotional mechanism set in reverse: ~ Ayn Rand

No one was interested in picking up a midlist series, even though I have a decent fanbase and respectable numbers. ~ J.A. Konrath

I was standing amid floor-to-ceiling shelves of books in wonder and awe when my view of stories suddenly and forever changed. There were enormous piles of books lying in corners. Books covered the walls. Books even lined the staircases as you went up from one floor to the next. It was as if this used bookstore was not just a place for selling used books; it was like the infrastructure itself was made up of books. There were books to hold more books, stories built out of stories.

I was standing in Daedalus Books in Charlottesville, Virginia, and I had recently read Mortimer J. Adler's How to Read a Book. I was alive with the desire to read. But at that particular moment, my glee turned to horror. For whatever reason, the truth of the numbers suddenly hit me. The year before, I had read about thirty books. For me, that was a new record. But then I started counting. I was in my early twenties, and with any luck I'd live at least fifty more years. At that rate, I'd have about 1,500 books in me, give or take.

There were more books than that on the single wall I was staring at.

That's when I had a realization of my mortality. My desire outpaced reality. I simply didn't have the life to read what I wanted to read.

Suddenly my choices in that bookstore became a profound act of deciding. The Latin root of the word decide - cise or cide - is to "cut off' or "kill." The idea is that to choose anything means to kill off other options you m ~ Justin Whitmel Earley

Bible literacy matters because it protects us from falling into error. Both the false teacher and the secular humanist rely on biblical ignorance for their messages to take root, and the modern church has proven fertile ground for those messages. Because we do not know our Bibles, we crumble at the most basic challenges to our worldview. Disillusionment and apathy eat away at our ranks. Women, in particular, are leaving the church in unprecedented numbers.1 ~ Jen Wilkin

Bealer has a number of reasons for thinking that a naturalistic epistemology is self-undermining. Let me focus on one of these. (I've tried to take on all of them in the first chapter of Knowledge and Its Place in Nature.) ~ Hilary Kornblith

Most of us are unaware of our deep-seated faith in numbers. ~ Carl Eckart

If we remembered everything, we should on most occasions be as ill off as if we remembered nothing. It would take us as long to recall a space of time as it took the original time to elapse, and we should never get ahead with our thinking. All recollected times undergo, accordingly, what M. Ribot calls foreshortening; and this foreshortening is due to the omission of an enormous number of facts which filled them. ~ William James

The conceptions of idle talk, of superfluities, and of vain ostentation, all designations of an irrational attitude without objective purpose, thus ~ Max Weber

So I forcibly shove aside my prickles of pissed-off, which is easier than it sounds when millions of little sequined caffeine dancers are doing their big Broadway number on your internal stage. (Page 173) ~ Deb Caletti

Writing is the only art form where a good number of the artists make a slice of their living criticizing one another in print, in public. ~ Christian Bauman

The year the police called Sherrena, Wisconsin saw more than one victim per week murdered by a current or former romantic partner or relative. 10 After the numbers were released, Milwaukee's chief of police appeared on the local news and puzzled over the fact that many victims had never contacted the police for help. A nightly news reporter summed up the chief's views: "He believes that if police were contacted more often, that victims would have the tools to prevent fatal situations from occurring in the future." What the chief failed to realize, or failed to reveal, was that his department's own rules presented battered women with a devil's bargain: keep quiet and face abuse or call the police and face eviction. ~ Matthew Desmond

I would rather be loved by millions and hated by the same number rather than moderately liked by everyone. ~ Josh Bernstein

I have a Sharpie. I love Sharpies. You know what they say on them? Not for letter writing. That sucks. Now I have to communicate with my dad using numbers. ~ Mitch Hedberg

The edge of a painting is its frontier ... where the artist negotiates his boundaries with the real world ... where art begins and ends and where the eye enters and leaves the image. It determines, in an infinitely subtle number of ways, how you read a painting - which, unlike a book or a piece of music, has no pre-determined beginning or end. ~ Andrew Graham-Dixon

We see, from almost every conceivable angle throughout the Scripture, that there is no doctrine more clearly taught than that it is God's will to heal all who have need of healing, and that they may fulfill the number of their days according to His promise. ~ F. F. Bosworth

The unluckiest of the Caribbean's sick came, in search of cures: a poor woman who, since childhood, had been counting the beats of her heart so long that she had run out of numbers to count; a Jamaican who, because of the tormenting sound the stars made, never slept; a sleepwalker who rose from bed at night, and in sleep undid all the things he had done in waking; and many other ailments too, less serious in nature. ~ Gabriel Garcia Marquez

One of the questions I've always hated answering is how do people make money in open source. And I think that Caldera and Red Hat - and there are a number of other Linux companies going public - basically show that yes, you can actually make money in the open-source area. ~ Linus Torvalds

As damaging as the obsessive emphasis on testing often proves to be for kids in general, I believe that the effects are still more harmful in those schools in which the resources available to help the children learn the skills that will be measured by these tests are fewest, the scores they get are predictably the lowest, and the strategies resorted to by principals in order to escape the odium attaching to a disappointing set of numbers tend to be the most severe. ~ Jonathan Kozol

If we smoke, there's no magic number of cigarettes, or number of years, that we can smoke before we know we'll get lung cancer. ~ Katharine Hayhoe