Integer Quotes

Is a woman a thinking unit at all, or a fraction always wanting its integer?

Innocent droplets of rain Make almost all events Quite natural. (from "A Rainy Day")

Only with a leaf can I talk of the forest,

It rains And rains And rains. But there is a sky above the rain, Nothing can rot the sky. Earth has turned to mud. What of it? The heart of the planet is made of fire, of ardent sun. (from "A Rainy Day")

He was shaken by an unwelcome insight. Lives did not add as integers. They added as infinities.

God made the integers; all the rest is the work of Man.

Let them shoot us in the head, My blood will grow roots and will blossom.

ABYSS Our country lives Among the dead And dies among the living Sometimes.

Handcuffs weigh much more than gravestones. (from "Gratitude")

Increasing pressure on students to subject themselves to ever more tests, whittling themselves down to rows and rows of tight black integers upon a transcript, all ready to goose-step straight into a computer.

SECOND SUN So much blood Has been spent in this world, But we have not yet built a sun of blood. Listen, my friend, To these trembling words: A second sun will be born of our blood in the form of a heart.

SOWING LIGHTNING Seize Bolts of lightning from the sky And plant them in fields of life. They will grow like tender sprouts of fire. Charge somber thoughts With unexpected flash, You, my lightning in the soil!

IN OUR CELLS They keep us in our cells For a long time... And, if we get out, We lug them with us on our shoulders, Like a porter with a chest of goods.

What did we lose, what was lost in us? To whom do these distances belong that separated us and that now bind us? Are we still one or have we both broken into pieces? How gentle this dust is- Its body now, and mine, at this very minute are one and the same

The forest has shrunk And fear has expanded, The forests have dwindled, There are less animals now, less courage and less lightning, less beauty and the moon lies bare, deflowered by force and then abandoned.

TIME Time And how it slips through my fingers Without putting its ring on them, And I remain simply its lover

He was shaken by an unwelcome insight. Lives did not add as integers. They added as infinities. I

Seven is my favorite number," he said. "Why?" He nuzzled gently at her stomach. "There are seven colors in a rainbow, seven days of the week, and..." His voice lowered seductively, "...seven is the lowest natural number that can't be represented as the sum of the squares of three integers." "Mathematics," she exclaimed, laughing breathlessly. "How stirring.

Between one and one Between integer and integer Is itself’s nothing The abstract zero. Between I and I Between self and self Is itself’s everything The abstract Hero That self may equate to Or keep ever as two.

THE CURSE May they never Return home at night... May you have no part of eventide, May you have no room of your own, Nor road, nor return. May your days be all exactly the same, Five Fridays in a row, Always an unlucky Tuesday, No Sunday, May you have no more little worries, Tears or inspiration, For you yourself are the greatest worry on earth: Prisoner!

Base yourself on what you feel, even when you alone feel it.

The nineteenth-century mathematician Leopold Kronecker said, “God made the integers; all the rest is the work of man.

Only learn with reservations. An entire life is not enough to unlearn what you naively, submissively, have allowed to be placed in your head---innocent one---without imagiging the consequences.

Gud i himlen, kjærligheten er et flygtig stof!

Always there where children die stone and star and so many dreams become homeless.

Kurt Gödel was the first person to realize and exploit the fact that the positive integers, though they might superficially seem to be very austere and isolated, in fact constitute a profoundly rich representational medium. They can mimic or mirror any kind of pattern.

Could it be that the planets are castaway heads.

When will the death Of Death ever come? from "The Siege

forty-one was a “very special number, the initial integer in the longest continuous string of quadratic primes.

In they'd come, integers; out they came, squared.

Om integer te zijn en alleen te kunnen wezen, moet je iets ontdekken dat het de moeite waard maakt ervoor te lijden.

So a)To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers? Plus and minus, self- evidently; sometimes multiplication, and yes. division. But these signs are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically insoluble?

Augie. You're a good girl," said Dad. This was more hope than truth. "You're nine and a half." "NINE," she corrected. She hated how adults never rounded down to the nearest integer when talking to children.

You remember the footprint All that is forgotten you remember from eternity You remember the footprint which filled with death As the myrmidon approached. You remember the child's trembling lips As they had to learn their farewell to their mother. You remember the mother's hands which scooped out a grave For the child which had starved at her breast. You remember the mindless words That a bride spoke into the air to her dead bridegroom.

I love you as if all hearts were a mirror of mine

The root of the word “integrity” is “integer.” It’s a math term - and it refers to whole numbers. The word itself implies “wholeness.” These are the questions we must ask ourselves frequently. “Am I whole?” “Are there parts of my character that are lacking?

Since most sexual abuse begins well before puberty, preventive education, if it is to have any effect at all, should begin early in grade school. Ideally, information on sexual abuse should be integrated into a general curriculum of sex education. In those communities where the experiment has been tried, it has been shown conclusively that children can learn what they most need to know about sexual abuse, without becoming unduly frightened or developing generally negative sexual attitudes. In Minneapolis, Minnesota, for example, the Hennepin County Attorney's office developed an education program on sexual assault for elementary school children. The program was presented to all age groups in four different schools, some eight hundred children in all. The presentation opened with a performance by a children’s theater group, illustrating the difference between affectionate touching, and exploitative touching. The children’s responses to the skits indicated that they understood the distinction very well indeed. Following the presentation, about one child in six disclosed a sexual experience with an adult, ranging from an encounter with an exhibitionist to involvement in incest. Most of the children, both boys and girls, had not told anyone prior to the classroom discussion. In addition to basic information on sexual relations and sexual assault, children need to know that they have the right to their own bodily integity.

The Woman Who Forgot Everything But in old age all drifts in blurred immensities. The little things fly off and up like bees. You forgot all the words and forgot the object too; And reached your enemy a hand where roses and nettles grew.

an overexertion to which one is driven by inner content is easy to bear.

Men kvinden hun var som alle vise visste før: uendelig ringe i ævner, men rik i uansvarlighet, i forfængelighet, i letfærdighet. Hun har meget av barnet, men intet av dets uskyld.

With energy there is this difference, that there are no blocks, so far as we can tell. Also, unlike the case of the blocks, for energy the numbers that come out are not integers. I

The integers of death.

My days are her name The dreams, when the sky is sleepless over my sorrow, are her name The obsession is her name and the wedding, when slayer and sacrifice embrace is her name. Once I sang: every rose as it tires, is her name as it journeys, is her name. Did the road end, has her name changed?

Vi er som brever som er sendt ut: vi befiner os ikke længer under befordringen, vi er kommet frem. Så er det da om vi har hvirvlet glæder og sorger op ved vårt indhold eller vi intet indtryk har efterlat.

MSE has that property—and it is the only definition of overall error that has it. In figure 6, we have computed the value of MSE in the set of five measurements for ten possible integer values of the line’s true length.

With each integer on the Richter scale, there is a tenfold increase in the number of earthquakes that occur annually. On average, there is one magnitude 8 event, ten magnitude 7 events, a hundred magnitude 6 events, and so on, each year. If we consider this from an energy standpoint, the smaller earthquakes account for a significant fraction of the total seismic energy released each year. The one million magnitude 2 events (which are too small to be felt except instrumentally) collectively release as much energy as does one magnitude 6 earthquake. Although the larger events are certainly more devastating from a human perspective, they are geologically no more important than the myriad less newsworthy small ones.

In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just thoroughly existent as the finite positive integers.

360 degrees of the circle can be divided evenly by all integers from 1 through 10 except 7; thus, 7 was considered by pre-talismanic witches and sages to “break out of the circle," because breaking the circle by 7 produces an irrational number, whose nonrepeating decimal sequence runs to infinity.]  

For one who sets himself to look at all earnestly, at all in purpose toward truth, into the living eyes of a human life: what is it he there beholds that so freezes and abashes his ambitious heart? What is it, profound behind the outward windows of each one of you, beneath touch even of your own suspecting, drawn tightly back at bay against the backward wall and blackness of its prison cave, so that the eyes alone shine of their own angry glory, but the eyes of a trapped wild animal, or of a furious angel nailed to the ground by his wings, or however else one may faintly designate the human 'soul,' that which is angry, that which is wild, that which is untamable, that which is healthful and holy, that which is competent of all advantaging within hope of human dream, that which most marvelous and most precious to our knowledge and most extremely advanced upon futurity of all flowerings within the scope of creation is of all these the least destructible, the least corruptible, the most defenseless, the most easily and multitudinously wounded, frustrated, prisoned, and nailed into a cheating of itself: so situated in the universe that those three hours upon the cross are but a noble and too trivial an emblem how in each individual among most of the two billion now alive and in each successive instant of the existence of each existence not only human being but in him the tallest and most sanguine hope of godhead is in a billionate choiring and drone of pain of generations upon generations unceasingly crucified and is bringing forth crucifixions into their necessities and is each in the most casual of his life so measurelessly discredited, harmed, insulted, poisoned, cheated, as not all the wrath, compassion, intelligence, power of rectification in all the reach of the future shall in the least expiate or make one ounce more light: how, looking thus into your eyes and seeing thus, how each of you is a creature which has never in all time existed before and which shall never in all time exist again and which is not quite like any other and which has the grand stature and natural warmth of every other and whose existence is all measured upon a still mad and incurable time; how am I to speak of you as 'tenant' 'farmers,' as 'representatives' of your 'class,' as social integers in a criminal economy, or as individuals, fathers, wives, sons, daughters, and as my friends and as I 'know' you?

BLOODY LIPS The bloody wound Of the gladiator Gurgles out life's end. The cries of acclimations from the stands Fill the sky with raging tigers. Waving their arms about to incite the masses The aging notables add an air of dignity to the arena. Making their separate entries they K N E E L over the still-warm corpses Of the young. Their withered lips they pose Upon the fresh flowing wounds And, to prolong their lives – so they believe, Suck, ravenously suck out the blood, blood, blood. Fresh blood from the sun Flowing into filthy veins As into sewage pipes, And thus the Heart of the Nation is abandoned.

As part of "moral philosophy," the concept of "natural liberty" clicks easily into place. Man, as an ethical integer, is either free to choose between good and bad courses within the limits of his circumstances, or he is not. If he is not free, if he can only accept what is handed to him from above (by fate, or by decree of the human agents of fate), then there is not much use in talking about morality or ethics. To make any sense of the idea of morality, it must be presumed that the human being is responsible for his actions-and responsibility cannot be understood apart from the presumption of freedom of choice.

Illam meae si partem animae tulit Maturior vis, quid moror altera? Nec carus aeque, nec superstes Integer. Ille dies ultramque Ducet ruinam. [Wenn meinen besten Teil der Seele die Parzen vor der Zeit abrissen, was zaudert der andere, der mir nicht lieber, nicht überlebender ist! Ein Tag stürzt uns beide ins Grab.]

In a way, in a profound way, I mean, Christ was never pushed off the dead end. At the moment when he was tottering and swaying as if by a great recoil, this negative backwash rolled up and stayed his death. The whole negative impulse of humanity seemed to coil up into a monstrous inert mass to create the human integer, the figure one, one and indivisible. There was a resurrection which is inexplicable unless we accept the fact that men have always been willing and ready to deny their own destiny. The earth rolls on, the stars roll on, but men: the great body of men which makes up the world, are caught in the image of the one and only one.

Now they thanked everyone and laughed, and papers were signed, and congratulations offered, and all stood for a moment unwilling to leave this genteel living room where there was such softness. The newlyweds thanked everyone again shyly and went out the door into the cool morning. They laughed, rosy. In they'd come, integers; out they came, squared.

how much is 2+2? Suppose Joseph says: 2+2 = purple, while Maxwell says: 2+2 = 17. Both are wrong but isn't it fair to say that Joseph is wronger than Maxwell? Suppose you said: 2+2=an integer. You'd be right, wouldn't you? Or suppose you said: 2+2=an even integer. You'd be rather righter. Or suppose you said:2+2=3.999. Wouldn't you be nearly right?

Det er ingen herlighet til som suset i skogen, det er som å gynge, det er som galskap; Uganda, Tananarivo, Honolulu, Atacama, Venezuela -

Tak for livet, det var morsomt å leve!

Det er for øvrig ikke tvil om at det skal en viss grad av hjærnetomhet til for å kunne gå og være varig tilfreds med sig selv og alt.

The remaining integer is either 6, 7, 8 or 9. It is more likely to be 8 or 9. The number of continents was either 68 or 76 and a half. Can you guess the equation? What is the formula for calculating the last integer, the continent one? I bet you don’t know unless you are skilled in maths. Fine. Here’s the answer. 68 divided by 8.5 is 8, and 76.5 divided by 8.5 is 9.

5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one. 5.5 So a) To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers?Plus and minus, self-evidently; sometimes multiplication, and yes, division. But these sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically improbable and mathematically insoluble? 5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v? B = s - v (*/+) a (to the first) Or a (to the second) + v + a (to the first) x s = b 5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility? 5.8 Or is "link" a false metaphor? 5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for such breaking? On the links immediately on the other side, or on the whole chain? But what do you mean by "the whole chain"? How far do the limits of responsibility extend? 6.0 Or we might try to draw the responsibility more narrowly and apportion it more exactly. And not use equations and integers but instead express matters in the traditional narrative terminology. So, for instance, if...." - Adrian Finn

The primary concern of mathematics is number, and this means the positive integers…. Mathematics belongs to man, not God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer exists, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

Harriet grinned at Betty Armstrong, hearing the familiar academic wrangle begin. Before ten minutes had passed, somebody had introduced the word "values." An hour later they were still at it. Finally the Bursar was heard to quote: "God made the integers; all else is the work of man." "Oh, bother!" cried the Dean. "Do let's keep mathematics out of it. And physics. I cannot cope with them.

It as mathematical, marriage, not, as one might expect, additional; it was exponential. This one man, nervous in a suite a size too small for his long, lean self, this woman, in a green lace dress cut to the upper thigh, with a white rose behind her ear. Christ, so young. The woman before them was a unitarian minister, and on her buzzed scalp, the grey hairs shone in a swab of sun through the lace in the window. Outside, Poughkeepsie was waking. Behind them, a man in a custodian's uniform cried softly beside a man in pajamas with a Dachshund, their witnesses, a shine in everyone's eye. One could taste the love on the air, or maybe that was sex, or maybe that was all the same then. 'I do,' she said. 'I do,' he said. They did. They would. Our children will be so fucking beautiful, he thought, looking at her. Home, she thought, looking at him. 'You may kiss,' said the officiant. They did, would. Now they thanked everyone and laughed, and papers were signed and congratulations offered, and all stood for a moment, unwilling to leave this gentile living room where there was such softness. The newlyweds thanked everyone again, shyly, and went out the door into the cool morning. They laughed, rosy. In they'd come integers, out they came, squared. Her life, in the window, the parakeet, scrap of blue midday in the London dusk, ages away from what had been most deeply lived. Day on a rocky beach, creatures in the tide pool. All those ordinary afternoons, listening to footsteps in the beams of the house, and knowing the feeling behind them. Because it was so true, more than the highlights and the bright events, it was in the daily where she'd found life. The hundreds of time she'd dug in her garden, each time the satisfying chew of spade through soil, so often that this action, the pressure and release and rich dirt smell delineated the warmth she'd felt in the cherry orchard. Or this, each day they woke in the same place, her husband waking her with a cup of coffee, the cream still swirling into the black. Almost unremarked upon this kindness, he would kiss her on the crown of her head before leaving, and she'd feel something in her rising in her body to meet him. These silent intimacies made their marriage, not the ceremonies or parties or opening nights or occasions, or spectacular fucks. Anyway, that part was finished. A pity...

But what G. Cantor posits as the defining formal property of an infinite set is that such a set can be put in a 1-1C with at least one of its proper subsets. Which is to say that an infinite set can have the same cardinal number as its proper subset, as in Galileo's infinite set of all positive integers and that set's proper subset of all perfect squares, which latter is itself an infinite set.

He had never indulged in the search for the True Substance, the One, the Absolute, the Diamond suspended from the Christmas tree of the Cosmos. He had always felt the faint ridicule of a finite mind peering at iridescence of the invisible through the prison bars of integers . And even if the Thing could be caught, why would he, or anyone else for that matter, wish the phenomenon to lose its curls, its mask, its mirror, and become the bald noumenon?

Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write, “Two plus two equals four,” “2 + 2 = 4,” or “Dos más dos es igual a cuatro.” The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures—we discover them, and invent only the notation for describing them.

In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

The danger of speaking about life exclusively in terms of problem and solution is that we are thus tempted to overlook the limitations of this detective game and the very existence of the initial arbitrary rules that makes the playing of it possible. The rule is to exclude from the terms of the problem everything that the solution cannot solve. It is diverting and useful to know that, for the chemist, a man is made up of a few pennyworth of salt, sugar, iron, and what not, together with an intolerable deal of water. But we must not assert that ‘man is, in fact, nothing but’ these things, or suppose that the solution of the pennyworths in the water will produce a complete and final solution of man. . . . It was said by Kronecker, the mathematician: ‘God made the integers; all else is the work of man.’ Man can table the integers and arrange them into problems that he can solve in the terms in which they are set. But before the inscrutable mystery of the integers themselves he is helpless, unless he calls upon that tri-unity in himself that is made in the image of God, and can include and create the integers.

Here are the basic principles of Constructivism as practiced by Kronecker and codified by J.H. Poincare and L.E.J. Brouwer and other major figures in Intuitionism: (1) Any mathematical statement or theorem that is more complicated or abstract than plain old integer-style arithmetic must be explicitly derived (i.e. 'constructed') from integer arithmetic via a finite number of purely deductive steps. (2) The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., 'constructing') whatever mathematical entities it's concerned with.

A good example of the archetypal ideas which the archetypes produce are natural numbers or integers. With the aid of the integers the shaping and ordering of our experiences becomes exact. Another example is mathematical group theory. ...important applications of group theory are symmetries which can be found in most different connections both in nature and among the 'artifacts' produced by human beings. Group theory also has important applications in mathematics and mathematical physics. For example, the theory of elementary particles and their interactions can in essential respects be reduced to abstract symmetries. [The Message of the Atoms: Essays on Wolfgang Pauli and the Unspeakable]

I don't feel very brave,” I said. Without my planning it, our eyes met. He looked, for that moment, like an eighteen-year-old who had been sealed in his bedroom for so long that he'd forgotten how to properly talk to humans. “It doesn't matter how you feel,” he said. “What matters is truth, and the truth is, you're afraid of other things you don't need to fear.

Argumentum Ornithologicum I close my eyes and see a flock of birds. The vision lasts a second, or perhaps less; I am not sure how many birds I saw. Was the number of birds definite or indefinite? The problem involves the existence of God. If God exists, the number is definite, because God knows how many birds I saw. If God does not exist, the number is indefinite, because no one can have counted. In this case I saw fewer than ten birds (let us say) and more than one, but did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, which was not nine, eight, seven, six, five, etc. That integer—not-nine, not-eight, not-seven, not-six, not-five, etc.—is incon-ceivable. Ergo, God exists.

We are so stricken We are so stricken that we think we're dying when the street casts an evil word at us. The street does not know it, but it cannot stand such a weight; it is not used to seeing a Vesuvius of pain break out. Its memories of primeval times are obliterated, since the light became artificial and angels only play with birds and flowers or smile in a child's dream

If you object (as some of us did to Dr. Goris) that Cantor's transfinite numbers aren't really numbers at all but rather sets, then be apprised that what, say, 'P(Infinity to the Infinity +n), really is is a symbol for the number of members in a given set, the same way '3' is a symbol for the number of members in the set {1,2,3}. And since the transfinites are provably distinct and compose an infinite ordered sequence just like the integers,they really are numbers, symbolizable (for now) by Cantor's well-known system of alephs or '(Aleph symbol's). And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers-although, just as with 0, the rules for these operations are very different in the case of (Alephs) and have to be independently established and proved.

Chorus of Comforters We are gardeners who have no flowers, No herb may be transplanted From yesterday to tomorrow. The sage has faded in the cradles-- Rosemary lost its scent facing the new dead-- Even wormwood was only bitter yesterday. The blossoms of comfort are too small Not enough for the torment of a child's tear. New seed may perhaps be gathered In the heart of a nocturnal singer. Which of us may comfort? In the depth of the defile Between yesterday and tomorrow The cherub stands Grinding the lightnings of sorrow with his wings But his hands hold apart the rocks Of yesterday and tomorrow Like the edges of a wound Which must remain open That may not yet heal. The lightnings of sorrow do not allow The field of forgetting to fall asleep. Which of us may comfort? We are gardeners who have no flowers And stand upon a shining star And weep.

Chorus of Clouds We are full of sighs, full of glances, We are full of laughter And sometimes we wear your faces. We are not far from you. Who knows how much of your blood rose And stained us? Who knows how many tears you have shed Because of our weeping? How much longing formed us? We play at dying, Accustom you gently to death. You, the inexperienced, who learn nothing in the nights. Many angels are given you But you do not see them.

There are numerous brain rhythms, from approximately 0.02 to 600 cycles per second (Hz), covering more than four order of temporal magnitude. Many of these discrete brain rhythms have been known for decades, but it was only recently recognized that these oscillation bands form a geometric progression on a linear frequency scale or a linear progression on a natural logarithmic scale. leading to a natural separation of at least ten frequency bands. The neighbouring bands have a roughly constant ratio of e = 2,718 - the base for the natural logarithm. Because of this non-integer relationship among the various brain rhythms, the different frequencies can never perfectly entrain each other. Instead, the interference they produce gives rise to metastability, a perpetual fluctuation between unstable and transiently stable states, like waves in the ocean. The constantly interfering network rhythms can never settle to a stable attractor, using the parlance of nonlinear dynamics. This explains the ever-changing landscape of the EEG.

Boston and Chicago are two great seats of mathematical research located in major American cities. Until they won in 2004, if you asked a baseball fan in Boston what they most hoped to see in their lifetime, they would have answered a World Series win for the Boston Red Sox. Chicago Cubs fans are still waiting. Ask a mathematician in either of those cities or anywhere else in the world what they would most hope to see in their lifetime, and they would most likely answer: "A proof o the Riemann hypothesis!" Perhaps mathematicians, like Red Sox fans, will have their prayers answered in our lifetimes, or at least before the Cubs win the World Series.

The properties that define a group are: 1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer (e.g., 3 + 5 = 8). 2. Associativity. The operation must be associative-when combining (by the operation) three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed. Addition, for instance, is associative: (5 + 7) + 13 = 25 and 5 + (7 + 13) = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first. 3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3. 4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + (-4) = 0 and (-4) + 4 = 0. The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.

What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)

What Cantor's Diagonal Proof does is generate just such a number, which let's call R. The proof is both ingenious and beautiful-a total confirmation of art's compresence in pure math. First, have another look at the above table. We can let the integral value of R be whatever X we want; it doesn't matter. But now look at the table's very first row. We're going to make sure R's first post-decimal digit, a, is a different number from the table's a1. It's easy to do this even though we don't know what particular number a1 is: let's specify that a=(a1-1) unless a1 happens to be 0, in which case a=9. Now look at the table's second row, because we're going to do the same thing for R's second digit b: b=(b2-1), or b=9 if b2=0. This is how it works. We use the same procedure for R's third digit c and the table's c3, for d and d4, for e and e5, and so on, ad inf. Even though we can't really construct the whole R (just as we can't really finish the whole infinite table), we can still see that this real number R=X.abcdefhi... is going to be demonstrably different from every real number in the table. It will differ from the table's 1st Real in its first post-decimal digit, from the 2nd Real in its second digit, from the 3rd Real in its third digit,...and will, given the Diagonal Method here, differ from the table's Nth Real in its nth digit. Ergo R is not-cannot be-included in the above infinite table; ergo the infinite table is not exhaustive of all the real numbers; ergo (by the rules of reductio) the initial assumption is contradicted and the set of all real numbers is not denumerable, i.e. it's not 1-1 C-able with the set of integers. And since the set of all rational numbers is 1-1C-able with the integers, the set of all reals' cardinality has got to be greater than the set of all rationals' cardinality. Q.E.D.*

Many of the really great, famous proofs in the history of math have been reduction proofs. Here's an example. It is Euclid's proof of Proposition 20 in Book IX of the Elements. Prop. 20 concerns the primes, which-as you probably remember from school-are those integers that can't be divided into smaller integers w/o remainder. Prop. 20 basically states that there is no largest prime number. (What this means of course is that the number of prime numbers is really infinite, but Euclid dances all around this; he sure never says 'infinite'.) Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2,3,5,7,11,...,Pn) is exhaustive and finite: (2,3,5,7,11,...,Pn) is all the primes there are. Now think of the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2,3,5,...,Pn), because dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2,3,5,...,Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2,3,5,...,Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction. And since the assumption that there's a largest prime entails a contradiction, modus tollens dictates that the assumption is necessarily false, which by LEM means that the denial of the assumption is necessarily true, meaning there is no largest prime. Q.E.D.

Unlike classically spinning bodies, such as tops, however, where the spin rate can assume any value fast or slow, electrons always have only one fixed spin. In the units in which this spin is measured quantum mechanically (called Planck's constant) the electrons have half a unit, or they are "spin-1/2" particles. In fact, all the matter particles in the standard model-electrons, quarks, neutrinos, and two other types called muons and taus-all have "spin 1/2." Particles with half-integer spin are known collectively as fermions (after the Italian physicist Enrico Fermi). On the other hand, the force carriers-the photon, W, Z, and gluons-all have one unit of spin, or they are "spin-1" particles in the physics lingo. The carrier of gravity-the graviton-has "spin 2," and this was precisely the identifying property that one of the vibrating strings was found to possess. All the particles with integer units of spin are called bosons (after the Indian physicist Satyendra Bose). Just as ordinary spacetime is associated with a supersymmetry that is based on spin. The predictions of supersymmetry, if it is truly obeyed, are far-reaching. In a universe based on supersymmetry, every known particle in the universe must have an as-yet undiscovered partner (or "superparrtner"). The matter particles with spin 1/2, such as electrons and quarks, should have spin 0 superpartners. the photon and gluons (that are spin 1) should have spin-1/2 superpartners called photinos and gluinos respectively. Most importantly, however, already in the 1970s physicists realized that the only way for string theory to include fermionic patterns of vibration at all (and therefore to be able to explain the constituents of matter) is for the theory to be supersymmetric. In the supersymmetric version of the theory, the bosonic and fermionic vibrational patters come inevitably in pairs. Moreover, supersymmetric string theory managed to avoid another major headache that had been associated with the original (nonsupersymmetric) formulation-particles with imaginary mass. Recall that the square roots of negative numbers are called imaginary numbers. Before supersymmetry, string theory produced a strange vibration pattern (called a tachyon) whose mass was imaginary. Physicists heaved a sigh of relief when supersymmetry eliminated these undesirable beasts.

For example, consider a stack (which is a first-in, last-out list). You might have a program that requires three different types of stacks. One stack is used for integer values, one for floating-point values, and one for characters. In this case, the algorithm that implements each stack is the same, even though the data being stored differs. In a non-object-oriented language, you would be required to create three different sets of stack routines, with each set using different names. However, because of polymorphism, in Java you can create one general set of stack routines that works for all three specific situations. This way, once you know how to use one stack, you can use them all. More generally, the concept of polymorphism is often expressed by the phrase “one interface, multiple methods.” This means that it is possible to design a generic interface to a group of related activities. Polymorphism helps reduce complexity by allowing the same interface to be used to specify a general class of action.

Let’s zoom in on a particular form of synesthesia as an example. For most of us, February and Wednesday do not have any particular place in space. But some synesthetes experience precise locations in relation to their bodies for numbers, time units, and other concepts involving sequence or ordinality. They can point to the spot where the number 32 is, where December floats, or where the year 1966 lies.8 These objectified three-dimensional sequences are commonly called number forms, although more precisely the phenomenon is called spatial sequence synesthesia.9 The most common types of spatial sequence synesthesia involve days of the week, months of the year, the counting integers, or years grouped by decade. In addition to these common types, researchers have encountered spatial configurations for shoe and clothing sizes, baseball statistics, historical eras, salaries, TV channels, temperature, and more.

I've actually got a new sorta-boyfriend and I suppose maybe I'll have shake-the-rafters, rattle-the-windows sex with him since I'll finally be legal and all." "Eponine," Phil hissed. Maube I'd only said it for the reaction, because there was absolutely no way that would be happening withing the month. I could count the numbers of boys I'd merely kissed on one hand. I didn't even need a hand at all to number the guys I'd slept with. I figured the integer wasn't about to change any time soon, either. "Obviously I'm kidding, Phil." His posture relaxed, only slightly. "We'll be quiet.

for example, to see whether in the developing subject, i.e. the child, integers are directly constructed starting from class logic by biunivocal correspondence and the construction of a “class of equivalent classes” as Frege and B. Russell thought, or whether the construction is more complex and presupposes the concept of order.

int at the beginning means that main will return an integer to the operating system

return 0; will return zero (which is the integer referred to on line 3) to the operating system. When a program runs successfully its return value is zero

An easy direct calculation shows that d dt exp[!g(t)][$(t) ! a] = 0, so, indeed, exp[!g(t)][$(t) ! a] = $(0) ! a. Taking t = 1, exp[!2"in($; a)][$(1) ! a] = $(0) ! a, or, exp[!2"in($; a)] = 1, which implies that n($; a) is indeed integral. The function is obviously continuous, and being integer, it is constant on connected components. Clearly also, it tends to 0 as a tends to +, so it is identically 0 on the unbounded component. ! The next result is an immediate corollary of the invariance of path integrals of analytic functions under homotopy.Proposition. If $0 and $1 are paths which are homotopic in C \ {a} for some point a, then n($0; a) = n($1; a). Cauchy’s Integral Formula. Let $ a piecewise smooth curve in a region G which is null homotopic there, and let f be an analytic function on G. Then n($; a)

When we combine the set of whole numbers with their opposites, including zero, we obtain a set of numbers we call integers. That

First we define a simple list of integer values, then we use the standard functions filter(), map() and reduce() to do various things with that list.

There is a somewhat surprising source of cyclic groups: if p is prime, the group ((Z/pZ) ! , ·) is cyclic. We will prove a more general statement when we have accumulated more machinery (Theorem IV.6.10), but the adventurous reader can already enjoy a proof by working out Exercise 4.11. This is a relatively deep fact; note that, for example, (Z/12Z) ! is not cyclic (cf. Exercise 2.19, and Exercise 4.10). The fact that (Z/pZ) ! is cyclic for p prime means that there must be integers a such that every non-multiple of p is congruent to a power of a; the usual proofs of this fact are not constructive, that is, they do not explicitly produce an integer with this property. There is a very pretty connection between the order of an element of the cyclic group (Z/pZ) ! and the so-called ‘cyclotomic polynomials’; but that will have to wait for a little field theory (cf. Exercise VII.5.15

An algebraic integer of degree two is simply a root of a quadratic polynomial of the form X2 + aX + b with a, b ordinary integers.

The reason special names are given to these quadratic irrationalities is that any quadratic algebraic integer is a linear combination (with ordinary integers as coefficients) of 1 and one of these fundamental quadratic algebraic integers.

The collection of all real or complex numbers that are integral linear combinations of 1 and τd is closed under addition, subtraction, and multiplication, and is therefore a ring, which we denote by Rd. That is, Rd is the set of all numbers of the form a + bτd where a and b are ordinary integers. These rings Rd are our first, basic, examples of rings of algebraic integers beyond that prototype, , and they are the most important rings that are receptacles for quadratic irrationalities. Every quadratic irrational algebraic integer is contained in exactly one Rd.

there are only four units in the ring R-1 of Gaussian integers, namely ±1 and ±i; multiplication by any of these units effects a symmetry of the infinite square tiling

Numbers JavaScript has a single number type. Internally, it is represented as 64-bit floating point, the same as Java’s double. Unlike most other programming languages, there is no separate integer type, so 1 and 1.0 are the same value. This is a significant convenience because problems of overflow in short integers are completely avoided, and all you need to know about a number is that it is a number. A large class of numeric type errors is avoided.

Every positive integer is one of Ramanujan’s personal friends. John E. Littlewood

Example 11-1. Measuring memory usage of 100,000,000 of the same integer in a list In [1]: %load_ext memory_profiler  # load the %memit magic function In [2]: %memit [0]*int(1e8) peak memory: 790.64 MiB, increment: 762.91 MiB