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A Fraction of the Whole

A Fraction of the Whole is a 2008 novel by Steve Toltz. It follows three generations of the eccentric Dean family in Australia and the people who surround them.

Characters

Jasper Dean

Jasper Dean is Martin Dean's illegitimate son and Terry Dean's nephew. He narrates most of the novel, save some sections which are narrated by Martin. Jasper's difficult relationship with his father is the central subject of the book, and he leads a confused childhood due to Martin's constant bizarre lessons and diatribes. His conflict is a fear of turning into his father, and he often works to distance himself from Martin, though their bond is strangely loving in its own way.

Martin Dean

Martin Dean, the father of Jasper Dean, is paranoid, philosophical, and intelligent. As a child, he spent four years and four months in a coma; upon waking up, the resulting unfamiliarity with the world led to his later misanthropy. Martin is transfixed by his inevitable death and will stop at nothing to leave his mark on the world; this leads to a series of "immortality projects" which inevitably end up backfiring. He is also determined to indoctrinate his son Jasper with his beliefs. Throughout the book, his sanity is debatable.

Terry Dean

Terry Dean is Martin's younger half-brother. He is described as blond and handsome. He grew up as a natural athlete who loved sports with a religious fervor, hating all cheats. When Terry is injured and can no longer take part in any sport, he turns to a life of crime, going from juvenile delinquency to a vigilante crusade against every cheating athlete in Australia. He is eventually captured, and presumed dead when a brushfire burns down his prison.

Caroline Potts

A childhood love of both Martin and Terry and the daughter of the richest man in the town where they grew up. She leaves to travel the world, returning for her father when he is blinded in an explosion caused by Terry but eventually reappears, marrying Martin when they are both middle-aged.

Astrid

Jasper's mother, accidentally impregnated by Martin after a chance meeting on New Year's. Though Martin initially loves her, he becomes more and more exasperated with her as she sinks into depression during pregnancy. Jasper has never met her.

Eddie

A longtime friend of Martin's who met him in Paris. Jasper is suspicious of Eddie, as he takes long, unexplained vacations, constantly lends Martin money, and is always taking pictures of them. Eddie is trained as a doctor and dreams of returning to his home in Thailand to start a practice.

Anouk

A beautiful woman whom Martin hires as a housekeeper after catching her keying his car. Her attempts to point out Martin's flaws lead to his psychological breakdown, but she becomes a voice of sanity who stays on to help the Deans. She is constantly in the middle of a nasty breakup and is versed in meditation.

Harry West

A hardened criminal residing in the prison that overlooks Martin and Terry's hometown. Harry is hired by Martin to act as a mentor for Terry's fledgling criminal operation. As Harry's sanity deteriorates, Martin becomes his only confidante, and Harry asks him for help in publishing his vision, a textbook for criminals.

Martin's parents

Never named in the book. Martin and Terry's mother is a Jewish immigrant who arrived in Australia by way of Shanghai after Martin's father is shot. Their mother is devoted to both sons. Terry's father is an alcoholic who took his wife to a small town in New South Wales when he found work building a prison. He blames Martin for sending Terry down the wrong path in life, and has an uneasy relationship with Martin.

Plot

A Fraction of the Whole uses a multi-perspective narrative, often going back in time to show Martin's perspective on events before returning to Jasper's story in the present. The framing narrative of the novel is written from the perspective of Jasper, writing secretly from the prison cell he is incarcerated in for an initially undisclosed crime.

Martin enters

The story jumps back to when Jasper was five, and was pulled out of school by his father, Martin. Rather than using a typical school curriculum, Martin teaches his son his beliefs about how life is, how it should be, and how to survive it.

Martin's childhood

Martin gives Jasper a highly detailed account of his own childhood. He has dealt with many problems in his life, from Terry's criminal behavior, to Martin's own depression, to his four year coma, to his mother poisoning him while she went mad from fear of her terminal cancer. Martin clearly remembers telling his brother that the two criminal kids, whom Terry would later join, are cheating. These kids had been beating Martin up, and he knows that telling Terry this lie would make him go after them. Terry does go after the bullies, and they stab him in the leg. This injury cripples Terry for life and renders him incapable of playing sports.

Martin comes up with the idea of a suggestion box, where everyone in town is welcome to enter recommendations for town life. It starts off well, but soon everyone in town is criticized by someone else. Each slip is anonymous, making it impossible for anyone to get mad, except at the person who invented the suggestion box. No one ever finds out that it was Martin. Finally, there is the loss of Martin's true love, Caroline, to Terry. After Terry is imprisoned, she leaves the town and visits every now and then.

Martin's mother is diagnosed with Cancer, and Martin vows that he won't leave her, effectively trapping himself in the town he hates. Once the town burns down, killing his mother and stepfather, and burning down the prison, he leaves town for good, having wanted to for so many years. Before he leaves he collects what he believes to be his brother's ashes from the prison, and scatters them in a puddle.

Martin's young adulthood

Martin leaves his hometown in Australia for Paris. He has picked Paris because he figures that he may as well start where he believes Caroline Potts to be. He has traced the postcard he has received from her to the original address. Upon arrival, he learns that she has recently moved, and no one is quite sure where.

Martin decides to live in Paris, where he meets two important people. Eddie comes off as a very friendly Thai who loves to take pictures and constantly takes Martin's photograph. Eddie is not the type of person Martin likes and he decides never to see him again but unfortunately for Martin, Eddie becomes his dearest and longest friend. Eddie is always there for Martin, giving him jobs and money when he needs it.

Martin also meets Astrid (real name unknown) in a café. He finds her extremely attractive and assumes that his affair with her will be a one night stand, but in fact it becomes the exact opposite. Astrid and Martin move in together, and Astrid unexpectedly becomes pregnant. During her pregnancy, Astrid becomes crazy, repeatedly painting a violent and horrific face and trying to converse with God. She becomes angry when God does not respond, so Martin starts pretending to be him, answering her questions while hiding in the bathroom. He learns a great deal about her, and realizes that she is becoming suicidal. After giving birth to Jasper, Astrid commits suicide.

Middle

Eddie continues to help the Dean family financially. The Deans meet another central character, Anouk. Although they meet on undesirable grounds (Anouk vandalizes Martin's car), they become close family friends, as Martin hires Anouk to clean for them. Martin is deemed mentally unstable and is sent to mental institution. Jasper (Martin's child with Astrid) is sent to a foster home against his will.

When Martin is released, he buys a rotting, broken-down house in the middle of nowhere. Martin builds a house and labyrinth on the property to have maximum privacy. Jasper,

Partial fraction

In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.

In symbols, one can use partial fraction expansion to change a rational function in the form

\frac{f(x)}{g(x)}

where Æ’ and g are polynomials, into a function of the form

\sum_j \frac{f_j(x)}{g_j(x)}

where gj (x) are polynomials that are factors of g(x), and are in general of lower degree. Thus the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of fractions, that produces a single rational fraction with a numerator and denominator usually of high degree. The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that function as a sum of fractions, where:

  • the denominator of each term is a power of an irreducible (not factorable) polynomial and
  • the numerator is a polynomial of smaller degree than that irreducible polynomial.To decrease the degree of the numerator directly, the Euclidean algorithm can be used, but in fact if Æ’ already has lower degree than g this isn't helpful.

The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it. Therefore the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, linear functional transformations, of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition. See e.g. partial fractions in integration for an account of the use of the partial fractions in finding antiderivatives. Just which polynomials are irreducible depends on which field of scalars one adopts. Thus if one allows only real numbers, then irreducible polynomials are of degree either 1 or 2. If complex numbers are allowed, only 1st-degree polynomials can be irreducible. If one allows only rational numbers, or a finite field, then some higher-degree polynomials are irreducible.

Basic principles

The basic principles involved are quite simple; it is the algorithmic aspects that require attention in particular cases. On the other hand, the existence of a decomposition of a certain kind is an assumption in practical cases, and the principles should explain which assumptions are justified.

Assume a rational function R(x) = ƒ(x)/g(x) in one indeterminatex has a denominator that factors as

g(x) = P(x)Q(x)

over a fieldK (we can take this to be real numbers, or complex numbers). If P and Q have no common factor, then R may be written as

A/P + B/Q

for some polynomials A(x) and B(x) over K. The existence of such a decomposition is a consequence of the fact that the polynomial ring over K is a principal ideal domain, so that

CP + DQ = 1

for some polynomials C(x) and D(x) (see Bézout's identity).

Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomials. To take this further, if required, write

G(x)/F(x)n

as a sum with denominators powers of F and numerators of degree less than F, plus a possible extra polynomial. This can be done by the Euclidean algorithm, polynomial case. The result is the following theorem:

Therefore when the field K is the complex numbers, we can assume that each pi has degree 1 (by the fundamental theorem of algebra) the numerators will be constant. When K is the real numbers, some of the pi might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur. This therefore is a case that requires discussion, in the systematic theory of integration (for example in computer algebra).

Procedure

Given two polynomials P(x) and Q(x) = (x-\alpha_1)(x-\alpha_2) \cdots (x-\alpha_n), where the αi are distinct constants and deg&nbsp;P&nbsp;<&nbsp;n, partial fractions are generally obtained by supposing that

\frac{P(x)}{Q(x)} = \frac{c_1}{x-\alpha_1} + \frac{c_2}{x-\alpha_2} + \cdots + \frac{c_n}{x-\alpha_n}

and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise. (This is a variant of the method of undetermined coefficients.)

This approach does not account for several other cases, but can be modified accordingly:

  • If deg&nbsp;P&nbsp;>&nbsp;deg&nbsp;Q, then it is necessary to perform the division
P(x)

Ejection fraction

In cardiovascular physiology, ejection fraction (Ef) is the fraction ofblood pumped out of the right and left ventricles with each heart beat. The term ejection fraction applies to both the right and left ventricles; one can speak equally of the left ventricular ejection fraction (LVEF) and the right ventricular ejection fraction (RVEF). RVEF and LVEF may vary widely from one another incumbent upon physiologic state. Ventricular Dyssynchrony represents pathology in which the LVEF and RVEF combined may be less than 100%. Without a qualifier, the term ejection fraction refers specifically to that of the left ventricle. Its reverse operation is the injection fraction.

Overview

By definition, the volume of blood within a ventricle immediately before a contraction is known as the end-diastolic volume. Similarly, the volume of blood left in a ventricle at the end of contraction is end-systolic volume. The difference between end-diastolic and end-systolic volumes is the stroke volume, the volume of blood ejected with each beat. Ejection fraction (Ef) is the fraction of the end-diastolic volume that is ejected with each beat; that is, it is stroke volume (SV) divided by end-diastolic volume (EDV):

E_f = \frac{SV}{EDV} = \frac{EDV - ESV}{EDV}

Normal values

In a healthy 70-kg (154-lb) man, the SV is approximately 70 ml and the left ventricular EDV is 120 ml, giving an ejection fraction of 70/120, or 0.58 (58%).

Right ventricular volumes being roughly equal to those of the left ventricle, the ejection fraction of the right ventricle is normally equal to that of the left ventricle within narrow limits.

Healthy individuals typically have ejection fractions between 50% and 65%. However, normal values depend upon the modality being used to calculate the ejection fraction, and some sources consider an ejection fraction of 55-75% to be normal. Damage to the muscle of the heart (myocardium), such as that sustained during myocardial infarction or in cardiomyopathy, impairs the heart's ability to eject blood and therefore reduces ejection fraction. This reduction in the ejection fraction can manifest itself clinically as heart failure.

The ejection fraction is one of the most important predictors of prognosis; those with significantly reduced ejection fractions typically have poorer prognoses. However, recent studies have indicated that a preserved ejection fraction does not mean freedom from risk.

The QT interval as recorded on a standard Electrocardiogram is generally agreed to be an exemplary display of depolarization of the ventricles. Widening of the QT interval is a reliable and inexpensive method in determination of mismatched flow states between the RV and LV.

Measurement

Ejection fraction is commonly measured by echocardiography, in which the volumes of the heart's chambers are measured during the cardiac cycle. Ejection fraction can then be obtained by dividing stroke volume by end-diastolic volume as described above.

Accurate volumetric measurement of performance of the right and left ventricles of the heart is inexpensively and routinely echocardiographically interpreted worldwide as a ratio of Dimension between the ventricles in Systole and Diastole. For example, a ventricle in greatest dimension could measure 6cm while in least dimension 4cm. Measured and easily reproduced beat to beat for ten or more cycles, this ratio may represent a physiologically normal EF of 60%. Mathematical expression of this ratio can then be interpreted as the greater half as Cardiac Output and the lesser half as Cardiac Input.

Other methods of measuring ejection fraction include cardiac MRI, fast scan cardiac computed axial tomography (CT) imaging, ventriculography, Gated SPECT, and the MUGA scan. A MUGA scan involves the injection of a radioisotope into the blood and detecting its flow through the left ventricle. The historical gold standard for the measurement of ejection fraction is ventriculography.



From Yahoo Answers

Question:the questions on the test are 1. Which of the following pairs of numbers contain like fractions? 2. 7 1/5 6 2/5 = ? 3. What is the sum of 2/5 and 2/4? 4. What is the least common denominator of 3/4, 4/5, 2/3? 5. What is the product of 3 2/3 and 14 2/5? 6. Divide 6/13 by 6/12 . 7. Jane is making a suit which requires 2 5/8 yards for the jacket and 1 3/4 yards for the skirt. What is the total amount of material she needs? 8. Which of the following is an example of a proper fraction? 9. A bus on a regular schedule takes 3 1/4 hours to reach its destination. The express bus takes 2 1/2 hours to make the same trip. How much travel time can be saved by taking the express? 10. Ralph spends 15 1/3 hours per month playing tennis. How many hours does he play tennis in a year? (There are twelve months in a year.) 11. What is the reciprocal of 6/5? 12. A family spends 1/10 of its annual income for housing, 1/4 for food and clothing, 1/5 for general expenses, and 2/15 for entertainment. What fractional part of their income is spent on these items altogether? 13. What is the fraction 18/24 reduced to its lowest terms? 14. 7/8 = ?/48 15. A chef prepared five chocolate tortes for a dinner party. The guests consumed 2 5/16 tortes. How many tortes are left? 16. Write 10 5/12 as an equivalent improper fraction. 17. Simone has 5 employees in her flower shop. Each employee works 6 4/15 hours per day. How many hours, in total, do the 5 employees work per day? 18. 15 6 2/3 = ? 19. What is the difference between 126 1/4 and 78 2/3? 20. 4/15 of the 315 members of a book club are male. How many female members are there in the club? but i don't want anyone to answer these for me i want to learn how to do it myself but i have no idea what im doing T_T

Answers:It would probably be best to go and see your teacher for extra help. 2. An easy way to subtract fractions is to turn them into improper fractions (whole number multiplied by denominator. then plus numerator) 3. To add fractions, find the least common denominator, change the fractions and then add. 11. A reciprocal is when you flip the number, so ex. 3/4's reciprocal would be 4/3 or 1 1/3. the reciprocal of 3 (3/1) would be 1/3. I would highly recommend going to your teacher for help.

Question:Hi! I need to find the LCD for each pair of fractions and I don't know nor do I know how to find the LCD? Here is the questions: 1. 1/2 & 3/5= 2. 2/3 & 1/4= 3. 5/6 & 4/9= 4. 2/5 & 3/4= 5. 3/10 & 1/12= 6. 7/8 & 2/3= 7. 2/7 & 1/8= 8. 11/12 & 3/8= Please & Thank you for you're help! Chelsei!

Answers:1. 10 2. 12 3. 18 4. 20 5. 60 7. 56 8. 24 i gave u the denomenators

Question:Describe some practical applications for fractions in your daily life. What are the challenges you have experienced regarding the use of fractions?

Answers:You have a sheet cake and 12 guests. Therefore each would get 1/12 of the cake if it were divided equally. A recipe calls for 2/3 cup of sugar at one point and then later another 1-2/3 cup. You only have 2 cups of sugar to begin with, so how do you know if you have to buy (or borrow!) more sugar before you start baking? Your weedeater runs on a mixture of oil/gas. The ratio listed on the oil container is 1pint to 5 gallons of gas. You only have a 1 gallon gas can. How do you know how much oil to put in it? You're watching "This Old House" on Saturday afternoon and you see this great bookcase that they are making. When they are finished, it's 8 feet tall and 4 feet wide. Your ceilings are only 6 feet tall and the biggest wall space you have is 2-1/2 feet wide. How will you proportion the bookcase so that it still looks like the one on TV but as a miniature version? Your car breaks down and requires standard size wrenches (not metric). The 1/2" wrench won't fit, but the 3/4" and 5/8" are too big. What's the next smallest wrench size that is still larger than the 1/2"? Personally, I have never had a problem with fractions, but my wife has a heck of time with them. Finding the lowest common denominator is one of her pet peeves. Converting mixed fractions to pure fractions is another. Multiplying fractions is easier to her than dividing them because you don't invert one of them. But I think the absolute hardest for her is to compare fractions without finding the LCD. It's hard for her to visualize say 15/32 as compared to 1/2.

Question:find two fractions equivalent to each fraction 5/6, 15/30, 45/60 Rewrite each pair of fractions with a common denominator 5/8 and 3/4, 2/5 and 1/2, 9/9 and 5/7 Rewrite each fraction in simplest form 9/54, 20/40, 100/110 Are the fractions 5/1, 5/5, and 1/5 equivalent In what situation can you use only multiplication to find equivalent fractions to a given fraction? give one example

Answers:Two fractions equivalent to each: Just divide or multiply both top AND bottom by the same number. 5/6: 10/12 OR 15/18 15/30: 5/10 OR 1/2 45/60: 8/12 OR 4/6 Rewrite each pair or fractions with common denominator: Find the difference between the two bottom numbers, and multiply top and bottom number. 5/8 and 3/4: 4X2=8, 3X2=6. So, 5/8 and 6/8. 2/5 and 1/2: 2/5 and 2.5/5 9/9 and 5/7: 9/9 and ~5.7/9 Rewrite each in simple form: Find greatest common factor and divide. 9/54: 1/6 20/40: 1/2 100/110: 10/11 Are these fractions equivalent? No. 5/1 and 5/5 are, because they are both 5 wholes. 1/5 is not because it is a fifth of a whole. In what situation can you use multiplication to find equivalent fractions? I'm sorry but I do not understand this question.