fractionmathematics
Main
Aspects of this topic are discussed in the following places at Britannica.
Assorted References
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major reference
( in arithmetic: Rational numbers )
From a less abstract point of view, the notion of division, or of fraction, may also be considered to arise as follows: if the duration of a given process is required to be known to an accuracy of better than one hour, the number of minutes may be specified; or, if the hour is to be retained as the fundamental unit, each minute may be represented by 1/60 or by ... .
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contribution by De Morgan
( in De Morgan, Augustus )
...the square root of minus one) that suggested the idea of quaternions. He made a useful contribution to mathematical symbolism by proposing the use of the solidus (oblique stroke) for the printing of fractions.
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use of term
( in term )
In mathematics, the terms of a fraction are the numerator and denominator. The terms of a proportion are the four numbers or expressions that enter into the proportion. Similarly, the terms of a sum are the numbers that are added together to constitute the sum or the numerical expressions denoting them. In this sense, an infinite series is thought of as a sum of an infinite number...
computations in
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Chinese mathematics
( in mathematics, East Asian: Arithmetic of fractions )
Division is a central operation in The Nine Chapters. Fractions are defined as a part of the result of a division, the remainder of the dividend being taken as the numerator and the divisor as the denominator. Thus, dividing 17 by 5, one obtains a quotient of 3 and a remainder of 2; this gives rise to the mixed quantity 3 + 2/5. The fractional parts are thus always less than one,...
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Egyptian mathematics
( in mathematics: The numeral system and arithmetic operations )
Computations involving fractions are carried out under the restriction to unit parts (that is, fractions that in modern notation are written with 1 as the numerator). To express the result of dividing 4 by 7, for instance, which in modern notation is simply 4/7, the scribe wrote 1/2 + 1/14. The procedure for finding quotients in this form merely extends the usual method for the division of...
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major reference arithmetic
From a less abstract point of view, the notion of division, or of fraction, may also be considered to arise as follows: if the duration of a given process is required to be known to an accuracy of better than one hour, the number of minutes may be specified; or, if the hour is to be retained as the fundamental unit, each minute may be represented by 1/60 or by ... .
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contribution by De Morgan De Morgan, Augustus
...the square root of minus one) that suggested the idea of quaternions. He made a useful contribution to mathematical symbolism by proposing the use of the solidus (oblique stroke) for the printing of fractions.
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use of term term
In mathematics, the terms of a fraction are the numerator and denominator. The terms of a proportion are the four numbers or expressions that enter into the proportion. Similarly, the terms of a sum are the numbers that are added together to constitute the...
computations in
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Chinese mathematics mathematics, East Asian
Division is a central operation in The Nine Chapters. Fractions are defined as a part of the result of a division, the remainder of the dividend being taken as the numerator and the divisor as the denominator. Thus, dividing 17 by 5, one obtains a quotient of 3 and a remainder of 2; this gives rise to the mixed quantity 3 + 2/5. The fractional parts are thus always less than one,...
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Egyptian mathematics mathematics
Computations involving fractions are carried out under the restriction to unit parts (that is, fractions that in modern notation are written with 1 as the numerator). To express the result of dividing 4 by 7, for instance, which in modern notation is simply 4/7, the scribe wrote 1/2 + 1/14. The procedure for finding quotients in this form merely extends the usual method for the division of...
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place in arithmetic arithmetic
...unit 1/d is defined by the property d × 1/d = 1. The number n × 1/d is written n/d and is called a common fraction. It may be considered as the quotient of n divided by d. The number d is called the denominator (it determines the fractional unit or denomination), and n...
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use of Euclidean algorithm Euclidean algorithm
The Euclidean algorithm is useful for reducing a common fraction to lowest terms. For example, the algorithm will show that the GCD of 765 and 714 is 51, and therefore 765/714 = 15/14. It also has a number of uses in more advanced mathematics. For example, it is the basic tool used to find integer solutions to linear equations...
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tests of kidney function renal system
Estimation of the GFR and RPF allows the proportion of available plasma perfusing the kidney that is filtered by the glomerulus to be calculated. This is called the filtration fraction and on average in healthy individuals is 125/600, or about 20 percent. Thus about one-fifth of plasma entering the glomeruli leaves as filtrate, the remaining four-fifths continuing into the efferent glomerular...
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heart failure cardiovascular disease
...This increase in size of the ventricular cavity (called ventricular dilation), however, also results in a reduction in the percentage of the left ventricular volume of blood that is ejected (called ejection fraction) and has significant functional consequences. Ejection fraction, therefore, is a benchmark for assessing ventricular function and failure on a chronic basis.
expression of a number as the sum of an integer and a quotient, the denominator of which is the sum of an integer and a quotient, and so on. In general,
where a0, a1, a2, … and b0, b1, b2, … are all integers.
In a simple continued fraction (SCF), all the bi are equal to 1 and all the ai are positive integers. An SCF is written, in the compact form, [a0; a1, a2, a3, …]. If the number of terms ai is finite, the SCF is said to terminate, and it represents a rational number; for example, 802/251 = [3; 5, 8, 6]. If the number of these terms is infinite, the SCF does not terminate, and it represents an irrational number; for example, √23 = [4; 1, 3, 1, 8], in which the bar spans a sequence of terms that repeats indefinitely. A nonterminating SCF in which a sequence of terms recurs represents an irrational number that is a root of a quadratic equation with rational coefficients. Nonterminating SCFs that represent numbers such as π or e can be evaluated after any given number of terms to obtain a rational approximation to the irrational quantity.
Student Encyclopædia Britannica articles specifically written for elementary and high school students.
- An Introduction to the Continued Fraction
- Essay on this type of numbers. Covers their relation to Euclid’s algorithm, Fibonacci numbers, and pi. Includes a puzzle on splitting rectangles into squares.
- Wolfram MathWorld - Continued Fraction
- Physics, Spaceflight and the Earth’s Magnetism - Continued Fraction
- MathPath - Continued Fraction
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