fraction

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fraction

[Lat.,=breaking], in arithmetic, an expression representing a part, or several equal parts, of a unit.

Notation for Fractions

In writing a fraction, e.g., 2-5 or 2/5, the number after or below the bar represents the total number of parts into which the unit has been divided. This number is called the denominator. The number before or above the bar, the numerator, denotes how many of the equal parts of the unit have been taken. The expression 2-5, then, represents the fact that two of the five parts of the unit or quantity have been taken. The present notation for fractions is of Hindu origin, but some types of fractions were used by the Egyptians before 1600 B.C. Another way of representing fractions is by decimal notation (see decimal systemdecimal system
[Lat.,=of tenths], numeration system based on powers of 10. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10.
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).

Characteristics of Fractions

When the numerator is less than the denominator, the fraction is proper, i.e., less than unity. When the reverse is true, e.g., 5-2, the fraction is improper, i.e., greater than unity. When a fraction is written with a whole number, e.g., 3 1-2, the expression is called a mixed number. This may also be written as an improper fraction, as 7-2, since three is equal to six halves, and by adding the one half, the total becomes seven halves, or 7-2. A fraction has been reduced to its lowest terms when the numerator and denominator are not divisible by any common divisor except 1, e.g., when 4-6 is reduced to 2-3.

Arithmetic Operations Involving Fractions

When fractions having the same denominator, as 3-10 and 4-10, are added, only the numerators are added, and their sum is then written over the common denominator: 3-10+ 4-10= 7-10. Fractions having unlike denominators, e.g., 1-4 and 1-6, must first be converted into fractions having a common denominator, a denominator into which each denominator may be divided, before addition may be performed. In the case of 1-4 and 1-6, for example, the lowest number into which both 4 and 6 are divisible is 12. When both fractions are converted into fractions having this number as a denominator, then 1-4 becomes 3-12, and 1-6 becomes 2-12. The change is accomplished in the same way in both cases—the denominator is divided into the 12 and the numerator is multiplied by the result of this division. The addition then is performed as in the case of fractions having the same denominator: 1-4+ 1-6= 3-12+ 2-12= 5-12. In subtraction, the numerator and the denominator are subjected to the same preliminary procedure, but then the numerators of the converted fractions are subtracted: 1-4− 1-6= 3-12− 2-12= 1-12.

In multiplication the numerators of the fractions are multiplied together as are the denominators without needing change: 2-3× 3-5= 6-15. It should be noted that the result, here 6-15, may be reduced to 2-5 by dividing both numerator and denominator by 3. The division of one fraction by another, e.g., 3-5÷ 1-2, is performed by inverting the divisor and multiplying: 3-5÷ 1-2= 3-5× 2-1= 6-5. The same rules apply to the addition, subtraction, multiplication, and division of fractions in which the numerators and denominators are algebraic expressions.

Fraction

 

in arithmetic, a quantity consisting of an integral number of parts of a unit. A fraction is represented by the symbol m/n, where n, the denominator of the fraction, indicates the number of parts into which the unit is to be divided and m, the numerator of the fraction, indicates the number of such parts taken. A fraction may be viewed as the quotient obtained by dividing one integer (m) by another (n). If m is divisible by n without a remainder, then the quotient m\n denotes an integer (for example, 6/3 = 2, 33/11 = 3). The numerator and denominator of a fraction may be simultaneously multiplied or divided by the same number without changing the value of the fraction. Any fraction can be represented in reduced form, that is, as a fraction whose numerator and denominator do not have common factors; for example, 16/72 is not in reduced form [16/72 = 2x8/9x 8 = 2/9], but 27/64 is. To add fractions with the same denominator, we add their numerators and take the same denominator: (a/b) + (c/b) + (d/b) = (a + c + d)/b. To add several fractions with different denominators, it is necessary to bring them to a common denominator. Subtraction of fractions is done in the same way. To multiply several fractions, we divide the product of their numerators by the product of their denominators: (a/b) x (c/d) = ac/bd. Defining division as an inverse operation of multiplication implies the following rule for division: (a/b) ÷ (c/d) = ad/bc. If the numerator of a fraction is less than the denominator, the fraction is called a proper fraction; if the opposite is true, it is called an improper fraction. An improper fraction may be shown to be the sum of an integer and a proper fraction (a mixed number). For this it is necessary to divide the numerator (with remainder) by the denominator; for example,

This proposition of elementary arithmetic can be extended to all real numbers: a real number x can be represented uniquely as x = n + d, where n is an integer and 0 ≤ d < 1. The integer n is called the integral part of x and is denoted by [x]. The number d = x - [x] is called the fractional part of x.

Decimal fractions are fractions whose denominator is a power of 10. Such fractions are written without denominators; for example, 5,481,475/10,000 = 548.1475 and 23/1,000 = 0.023.

Operations with fractions are encountered in the ancient Egyptian Ahmes papyrus (c. 2000 B.C.) where the only admissible fractions are fractions of the type l/n (aliquot fractions). Hence the distinctive “Egyptian” problem of representing any fraction as the sum of unequal fractions of the type l/n (in addition to aliquot fractions, the Egyptians had a special symbol for the fraction 2/3); for example, 7/29 = (1/5) + (1/29) + (1/145).

In ancient Babylonian manuscripts we encounter so-called sexagesimal fractions, that is, fractions having a denominator that is a power of 60. The number 60 played a significant role in classical arithmetic; the division of a unit into 60 and 3,600 = 602 parts has been preserved to the present day in the division of an hour or a degree into 60 minutes (1/60) and of a minute into 60 seconds. The ancient Hindus, apparently, were the first to conceive the modern symbol for a fraction.

REFERENCES

Entsiklopediia elementarnoi matematiki, book 1: Arifmetika. Moscow-Leningrad, 1951.
Depman, I. la. Istoriia arifmetiki, 2nd ed. Moscow, 1965.

Fraction

 

a portion of a granular or lumpy solid (such as crushed rock, sand, or powder) or of a liquid mixture (such as petroleum) isolated according to a specific criterion. In sieve analysis, fractions are isolated by particle or grain size; in gravity concentration, by density; and in petroleum distillation, by boiling point.

fraction

[′frak·shən]
(chemistry)
One of the portions of a volatile liquid within certain boiling point ranges, such as petroleum naphtha fractions or gas-oil fractions.
(mathematics)
An expression which is the product of a real number or complex number with the multiplicative inverse of a real or complex number.
(metallurgy)
In powder metallurgy, that portion of sample that lies between two stated particle sizes. Also known as cut.
(science and technology)
A portion of a mixture which represents a discrete unit and can be isolated from the whole system.

fraction

1. Maths
a. a ratio of two expressions or numbers other than zero
b. any rational number that is not an integer
2. Chem a component of a mixture separated by a fractional process, such as fractional distillation
3. Christianity the formal breaking of the bread in Communion
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"The result is the ability to see variability in not only patient status--for example, if the FiO2 data point increases, it signals a worsening oxygenation status that is a huge red flag - but also in clinical practice, such as weaning.
Abbreviations AVM: Arteriovenous malformations CT: Computerized tomography ECMO: Extracorporeal membrane oxygenation CXR: Chest X-ray NO: Inhaled Nitric Oxide FiO2: Fractional concentration of inspired oxygen PaO2: Partial pressure of arterial oxygen PFO: Patent foramen ovale HHT: Hemorrhagic telangiectasia MRI: Magnetic Resonance Imaging.
Indice de shunt: usa la relacion entre la PaO2 / FiO2,
Expiratory positive airway pressure was set at sufficient gas exchange maintenance level and FiO2 was set according to a value of SpO2 of greater than 90%, as measured by pulse oximetry.
Once ROSC is achieved, adjust the FiO2 to maintain oxygen saturation between 94% and 99% (Table 1 Class IIb, LOE C), as hyperoxia can increase reperfusion injury after an ischemic insult.
In children who received FiO2 > 60% for a period of > 6 days, mortality was significantly higher compared to those who received FiO2 > 60% for [less than or equal to] 3 days (100% vs.
Variables Oxygen therapy Sp[O.sub.2] (%) Hospital admission Room air HFNC 90 ICU admission 40 L/min, FiO2 0.40 HFNC 94 ICU day 2 40 L/min, FiO2 0.40 HFNC 95 ICU day 3 40 L/min, FiO2 0.40 HFNC 95 ICU day 4 40 L/min, FiO2 0.40 96 Variables Pa[O.sub.2] Respiratory rate Systolic blood (mmHg) ([min.sup.-1]) pressure (mmHg) Hospital admission 61.7 18 88 ICU admission 64.5 28 111 ICU day 2 73.5 19 94 ICU day 3 73.3 17 124 ICU day 4 89.3 17 122 ICU: intensive care unit; Sp[O.sub.2]: oxygen saturation of peripheral artery; Pa[O.sub.2]: partial pressure of arterial oxygen; HFNC: high-flow nasal cannula; Fi[O.sub.2]: fraction of inspiratory oxygen.
Internal oxygen concentrator: Together with the Oxygen Direct system, delivers the equivalent of 6 L/min of oxygen or 40% FiO2. They system is up to two times more energy efficient than other portable oxygen concentrators.
The mean or median values for FEV1, FVC, MIP, MEP and the ratio between partial pressure of oxygen in arterial blood (PaO2) and fraction of inspired oxygen (FiO2) were 101+-17, 102+-17, 66 (IQR: 59-74), 114 (IQR: 100-138) and 379+-49, respectively, before operation compared to 78+-18, 76+-18, 53 (IQR: 48-59), 85 (IQR: 73-95) and 331+-49 after operation (p<0.001) (Table-1).
The concentrations of NO and FiO2 were reduced every 6 to 12 hour if disease conditions of patients improved.
Al decimo dia paciente con mejoria clinica, disminucion de los requerimientos de vasopresor, mejoria de la oxigenacion PO2/ FIO2 205, normalizacion de tiempos de coagulacion, conteo plaquetario, azoados y transaminasas.