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Topic: Rational number

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 Irrational number - Wikipedia, the free encyclopedia
Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange.
The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2.
For the nineteenth century it remained to complete the theory of complex numbers, to separate irrationals into algebraic and transcendental, to prove the existence of transcendental numbers, and to make a scientific study of a subject which had remained almost dormant since Euclid, the theory of irrationals.
en.wikipedia.org /wiki/Irrational_number   (1831 words)

 Rational number - Simple English Wikipedia
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers.
The simplest form is when a and b have no common divisors, and every non-zero rational number has exactly one simplest form of this type with positive denominator (it is the number, which stands under the fraction-line).
The set of all rational numbers is countable.
simple.wikipedia.org /wiki/Rational_number   (379 words)

 PlanetMath: rational number
This is version 10 of rational number, born on 2001-10-19, modified 2005-02-14.
The set of rational numbers is dense when considered as a subset of the real numbers.
Under this ordering relation, the rational numbers form a topological space under the order topology.
planetmath.org /encyclopedia/RationalNumber.html   (174 words)

 Teaching Rational Number and Decimal Concepts
That rational numbers of the form a/b are complex entities consisting of at least three separate but yet related concepts—the numerator a, the denominator b, and multiplicative relationship between a and b, which defines the entity a/b.
Rational numbers are the first set of numbers children experience that are not based on a counting algorithm of some type.
Rational numbers can he interrupted in at least six ways: a part to whole comparison, a decimal, a ratio, an indicated division (quotient), an operator, and as a measure.
education.umn.edu /rationalnumberproject/92_2.html   (12736 words)

 Rational Numbers from Repeating Fractions
is the decimal expansion of the rational number 315/990.
All rational numbers less than 1 (that is, those for which p is less than q) can be expanded into a decimal fraction, but this expansion may require repetition of some number of trailing digits.
A rational number is any which can be written in the form p/q, where p and q are integers.
acm.uva.es /p/v3/332.html   (408 words)

 rational number - Hutchinson encyclopedia article about rational number
Numbers such as π are called irrational numbers.
are all rational numbers, whereas π (which represents the constant 3.141592...
In mathematics, any number that can be expressed as an exact fraction (with a denominator not equal to 0), that is, as a ÷ b where a and b are integers; or an exact decimal.
encyclopedia.farlex.com /rational+number   (115 words)

 The Real Number System
This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.
Note that the negative sign in front of a number is part of the symbol for that number: The symbol “–3” is one object—it stands for “negative three,” the name of the number that is three units less than zero.
The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns.
www.jamesbrennan.org /algebra/numbers/real_number_system.htm   (1567 words)

 11: Number theory
The sets of solutions in rational numbers to algebraic equations may be viewed as algebraic varieties, and thus studied with tools of 14: Algebraic Geometry.
Number theory is one of the oldest branches of pure mathematics, and one of the largest.
Questions in algebraic number theory often require tools of Galois theory; that material is mostly a part of 12: Field theory (particularly the subject of field extensions).
www.math.niu.edu /~rusin/known-math/index/11-XX.html   (2572 words)

 Rational Numbers
A rational number is a number that can be expressed as a fraction or ratio (rational).
a rational number (given in decimal form) as a fraction, "say" the decimal using place value to help form the fraction.
Rational numbers can be ordered on a number line.
regentsprep.org /Regents/math/rational/Lrat.htm   (154 words)

 Math Forum: Ask Dr. Math FAQ: Integers, Rational Numbers, Irrational Numbers
A real number is a number that is somewhere on a number line, so any number on a number line that isn't a rational number is irrational.
The square root of 2 is an irrational number because it can't be written as a ratio of two integers.
In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers.
mathforum.org /dr.math/faq/faq.integers.html   (721 words)

 Rational Number Library
To be precise, rational numbers are exact as long as the numerator and denominator (which are always held in normalized form, with no common factors) are within the range of the underlying integer type.
Internally, rational numbers are stored as a pair (numerator, denominator) of integers (whose type is specified as the template parameter for the rational type).
The mathematical concept of a rational number is what is commonly thought of as a fraction - that is, a number which can be represented as the ratio of two integers.
www.boost.org /libs/rational/rational.html   (3088 words)

That is, rationalizing a float by either method and then converting it back to a float of the same format produces the original number.
If number is a float, rational returns a rational that is mathematically equal in value to the float.
rationalize returns a rational that approximates the float to the accuracy of the underlying floating-point representation.
www.lispworks.com /documentation/HyperSpec/Body/f_ration.htm   (131 words)

 What's a number?
To be rational a number ought to have at least one fractional representation.
However, rational numbers form a countable set whereas irrational form a set which is not countable.
2, being the length of the diagonal in a unit square, was the first number proved not to be rational.
www.cut-the-knot.org /do_you_know/numbers.shtml   (3531 words)

 MathSteps: Grade 7: Rationals: What Is It?
Rational numbers are simply numbers that can be written as fractions or ratios (this tells you where the term rational comes from).
When a decimal or fractional approximation for an irrational number is used to compute (as in finding the area of a circle), the answer is always approximate and should clearly indicate this.
Rational numbers sound like they should be very sensible numbers.
www.eduplace.com /math/mathsteps/7/a   (369 words)

 Rational number - Wikipedia, the free encyclopedia
The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part).
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero.
The rational numbers are an important example of a space which is not locally compact.
en.wikipedia.org /wiki/Rational_number   (369 words)

 Number: What Is "This Many?" -- Platonic Realms MiniText
What it says is, the set of rational numbers is the set consisting of all numbers of the form p divided by q, where p and q are elements of the set of integers and q is not zero.
Historically, the rational numbers are nearly as old as the natural numbers.
For this reason, much of the work we do in mathematics will require us to add to the rational numbers a new set – a set of numbers which is the topic of our next section.
www.mathacademy.com /pr/minitext/number   (3230 words)

 Rational Spiral
Rational numbers can be expressed as the division of two integers.
For each line of input, print out the rational number in this form: numerator, followed by a space, followed by a forward slash, followed by a space, followed by the denominator.
Note: if the rational number is negative, the minus sign must be printed with the numerator.
acm.uva.es /p/v4/493.html   (285 words)

 The Prime Glossary: rational number
The rational numbers are those which have repeating decimal expansions (for example 1/11=0.09090909..., and 1=1.000000...=0.999999...).
In mathematics rational means "ratio like." So a rational number is one that can be written as the ratio of two integers.
Most real numbers (points on the number-line) are irrational (not rational).
primes.utm.edu /glossary/page.php?sort=RationalNumber   (85 words)

 Rational number - Wikipedia, the free encyclopedia
Each rational number can be written in many forms, for example 3 / 6 = 2 / 4 = 1 / 2.
For any positive rational number, there are infinitely many different such representations.
The set of all rational numbers is countable.
en.wikipedia.org /wiki/Rational_number   (85 words)

 Rational and irrational numbers
When a rational number is expressed as a decimal, then either the decimal will terminate or there will be a predictable pattern of digits.
There is no decimal -- no rational number -- whose square is exactly 2.000000000000000.
A rational number, then, can always be expressed as such a decimal.
www.themathpage.com /aPreCalc/rational-irrational-numbers.htm   (728 words)

 The ``Pi Is Rational'' Page :)
Another belief I hold is that either irrational numbers can not be expressed as the sum of an infinite number of rationals, or that irrational numbers are truly rational.
The problem with my argument, is that it is true that a rational plus a rational is a rational for any finite number of iterations.
We will prove that pi is, in fact, a rational number, by induction on the number of decimal places, N, to which it is approximated.
dse.webonastick.com /pi   (1089 words)

 Rational Triangles
The rational number associated with one acute angle has numerator and denominator both odd, but for the other angle they are of different parity.
Conversely, any rational number between zero and one is the tangent of half an acute angle of a rational right triangle.
It follows that the tangent of any bisected angle of the triangle is a rational number and therefore the original angle is a Heronian angle.
grail.cba.csuohio.edu /~somos/rattri.html   (831 words)

 Rational Number Tutorial
Rational number systems on computers have the following property: When you ask for an arithmetic expression to be evaluated, you will either get an exact answer or an overflow error.
This error is called an overflow error, and it occurred because the value of the expression was too large to represent in the current system: 4 is not a valid rational number when the minimum integer is -2 and the maximum integer is 2.
With a minimum integer of -2 and a maximum integer of 2, the simulator reveals that there are only seven valid rational numbers.
www.cs.utah.edu /~zachary/isp/applets/Rational/Rational.html   (812 words)

 Rational Number System
Note: Since irrational numbers cannot be expressed as a fraction they form decimals that are neither repeating nor terminating.
The real number system is made up of rational and irrational numbers.
Any number that cannot be written as a fraction where the numerator and denominator are integers.
argyll.epsb.ca /jreed/math9/strand1/1101.htm   (289 words)

 rational number. The American Heritage® Dictionary of the English Language: Fourth Edition. 2000.
A number capable of being expressed as an integer or a quotient of integers, excluding zero as a denominator.
www.bartleby.com /61/42/R0054200.html   (83 words)

 The Rational Number Project
The Rational Number Project (RNP) was the longest lasting cooperative multi-university research project in the history of mathematics education.
The vast majority of these are concerned with the learning and teaching of rational number concepts including fraction, decimal, ratio, indicated division, measure and operator.
We have also produced two curriculum texts for teachers that reflect our suggestions as to how rational number concepts should be taught to children.
education.umn.edu /rationalnumberproject   (1394 words)

 Rational and Irrational Numbers
All numbers that are not rational are considered irrational.
The number 8 is a rational number because it can be written as the fraction 8/1.
A rational number is a number that can be written as a ratio.
www.factmonster.com /ipka/A0876704.html   (229 words)

 Field extension - Wikipedia, the free encyclopedia
For example, C (the field of complex numbers) is an extension of R (the field of real numbers), and R is itself an extension of Q (the field of rational numbers).
see algebraic number and transcendental number.) If every element of L is algebraic over K, then the extension L/K is said to be algebraic, otherwise it is said to be transcendental.
Given a field extension L/K, L can be considered as a vector space over K, with vector addition being the field addition on L, and scalar multiplication being a restriction of the field multiplication on L.
en.wikipedia.org /wiki/Field_extension   (229 words)

 PlanetMath: theory of rational and irrational numbers
Such rational numbers, which are not integers, may be expressed as sum of partial fractions (the denominators being powers of distinct prime numbers).
The number e is irrational (this is not as difficult to prove as it is to show that e is transcendental).
Rational and irrational: the sum, difference, product and quotient of two non-zero real numbers, from which one is rational and the other irrational, is irrational.
planetmath.org /encyclopedia/TheoryOfRationalAndIrrationalNumbers.html   (229 words)

 Field (mathematics)
Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or complex numbers.
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers.
For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
www.brainyencyclopedia.com /encyclopedia/f/fi/field__mathematics_.html   (229 words)

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