Rational Number

In Mathematics , a rational number is a number which can be expressed as a Ratio of two Integer s. Non-integer rational numbers (commonly called Fractions ) are usually written as the Vulgar Fraction $a/b$ , where ''b'' is not Zero . Each rational number can be written in infinitely many forms, such as $3/6=2/4=1/2$ , but it is said to be in simplest form when ''a'' and ''b'' have no common Divisor s except 1 (i.e., they are Coprime ). Every non-zero rational number has exactly one simplest form of this type with a positive denominator. A fraction in this simplest form is said to be an Irreducible Fraction , or a fraction in ''reduced form''. 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). The same is true for any other integral base above one, and is also true when rational numbers are considered to be P-adic Number s rather than Real Number s. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A Real Number that is not a rational number is called an Irrational Number . The Set of all rational numbers, which constitutes a Field , is denoted $\mathbb{Q}$ . Using the Set-builder Notation , $\mathbb{Q}$ is defined as : $\mathbb{Q} = \left\{rac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n e 0 ight\},$ where $\mathbb{Z}$ denotes the set of integers. THE TERM ''RATIONAL'' In the mathematical world, the adjective ''rational'' often means that the underlying Field considered is the field $\mathbb{Q}$ of rational numbers. For example, a Rational Integer is an Algebraic Integer which is also a rational number, which is to say, an ordinary integer, and a Rational Matrix is a matrix whose coefficients are rational numbers. Rational Polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a “polynomial over the rationals”. However, Rational Function does not mean the underlying field is the rational numbers, and a Rational Algebraic Curve is not an algebraic curve with rational coefficients. ARITHMETIC See Also: Fraction (mathematics)#Arithmetic with fractions Two rational numbers $a/b$ and $c/d$ are equal If And Only If $ad = bc$ . Two fractions are added as follows : $rac{a}{b} + rac{c}{d} = rac{ad+bc}{bd}.$ The rule for multiplication is : $rac{a}{b} \cdot rac{c}{d} = rac{ac}{bd}.$ Additive and Multiplicative Inverse s exist in the rational numbers : $- \left( rac{a}{b} ight) = rac{-a}{b} = rac{a}{-b} \quad\mbox{and}\quad \left(rac{a}{b} ight)^{-1} = rac{b}{a} \mbox{ if } a eq 0.$ It follows that the quotient of two fractions is given by : $rac{a}{b} \div rac{c}{d} = rac{ad}{bc}.$ EGYPTIAN FRACTIONS See Also: Egyptian fraction Any positive rational number can be expressed as a sum of distinct Reciprocal s of positive integers, such as : $rac{5}{7} = rac{1}{2} + rac{1}{6} + rac{1}{21}.$ For any positive rational number, there are infinitely many different such representations, called '' Egyptian Fraction s'', as they were used by the ancient Egyptians . The Egyptians also had a different notation for Dyadic Fraction s. FORMAL CONSTRUCTION Mathematically we may construct the rational numbers as Equivalence Class es of Ordered Pair s of Integer s $\left(a, b ight)$ , with $b$ not equal to zero. We can define addition and multiplication of these pairs with the following rules: : $\left(a, b ight) + \left(c, d ight) = \left(ad + bc, bd ight)$ : $\left(a, b ight) imes \left(c, d ight) = \left(ac, bd ight)$ and if c ≠ 0, division by : $rac{\left(a, b ight)} {\left(c, d ight)} = \left(ad, bc ight).$ The intuition is that $\left(a, b ight)$ stands for the number denoted by the fraction $frac{a}{b}.$ To conform to our expectation that $frac{2}{4}$ and $frac{1}{2}$ denote the same number, we define an Equivalence Relation $\sim$ on these pairs with the following rule: : $\left(a, b ight) \sim \left(c, d ight) \mbox{ if and only if } ad = bc$ This equivalence relation is a .) We can also define a Total Order on Q by writing : $\left(a, b ight) \le \left(c, d ight) \mbox{ if } (bd>0\mbox{ and } ad \le bc)\mbox{ or }(bd<0\mbox{ and } ad \ge bc)$ The integers may be considered to be rational numbers by the Embedding that maps $p\,$ to $1),\,$ where $[(a,b)]\,$ denotes the equivalence class having $(a, b)\,$ as a member. PROPERTIES The set $\mathbb{Q}$ , together with the addition and multiplication operations shown above, forms a Field , the Field Of Fractions of the Integer s $\mathbb{Z}$ . The rationals are the smallest field with for characteristic zero. The Algebraic Closure of $\mathbb{Q}$ , i.e. the field of roots of rational polynomials, is the Algebraic Number s. The set of all rational numbers is Countable . Since the set of all real numbers is uncountable, we say that Almost All real numbers are irrational, in the sense of Lebesgue Measure , i.e. the set of rational numbers is a Null Set . The rationals are a set which is countable, dense (in the above sense), and has no least or greatest element is Order Isomorphic to the rational numbers. REAL NUMBERS The rationals are a expansions as Regular Continued Fractions .
	Let <math>p</math> Be A	"http://wwwinformationdelightinfo/information/entry/prime_number" class="copylinks">Prime Number and for any non-zero integer <math>a</math> let <math>a_p = p^{-n}</math>, where <math>p^n</math> is the highest power of <math>p</math> Dividing <math>a</math>
	In Addition Write <math>0 P	0</math> For any rational number <math>rac{a}{b}</math>, we set <math>\leftrac{a}{b} ight_p = rac{a_p}{b_p}</math>

	CATEGORIES ABOUT RATIONAL NUMBER

	elementary mathematics

	field theory

	fractions

	real numbers

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Rational Number