Rational Number

Each rational number can be written in infinitely many forms, for example $3/6 = 2/4 = 1/2$ . The simplest form is when $a$ and $b$ have no common Divisor s, and every non-zero rational number has exactly one simplest form of this type with positive denominator. 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 1. 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 rational is called an Irrational Number . In mathematics, the term "rational ''something''" means that the underlying Field considered is the field $\mathbb{Q}$ of rational numbers. For example, rational Polynomial s or rational Prime Ideal s. The Set of all rational numbers is denoted by Q, or in Blackboard Bold $\mathbb{Q}$ . Using the Set-builder Notation $\mathbb{Q}$ is defined as such: : $\mathbb{Q} = \left\{rac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n e 0 ight\}$ ARITHMETIC : $rac{a}{b} + rac{c}{d} = rac{ad+bc}{bd}$ : $rac{a}{b} \cdot rac{c}{d} = rac{ac}{bd}$ Two rational numbers $rac{a}{b}$ and $rac{c}{d}$ are equal If And Only If $ad = bc$ Additive and multiplicative inverses exist in the rational numbers. : $- \left( rac{a}{b} ight) = rac{-a}{b}$ : $\left(rac{a}{b} ight)^{-1} = rac{b}{a} \mbox{ if } a eq 0$ HISTORY Egyptian fractions Any positive rational number can be expressed as a sum of distinct Reciprocal s of positive integers. For instance, $rac{5}{7} = rac{1}{2} + rac{1}{6} + rac{1}{21}$ For any positive rational number, there are infinitely many different such representations. These representations are called '' Egyptian Fraction s'', because the ancient Egyptian s used them. The Egyptians also had a different notation for Dyadic Fraction s. See also Egyptian Numerals . FORMAL CONSTRUCTION Mathematically we may define them as an Ordered Pair 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: : $\left(a, b ight) + \left(c, d ight) = \left(ad + bc, bd ight)$ : $\left(a, b ight) imes \left(c, d ight) = \left(ac, bd ight)$ To conform to our expectation that $2/4 = 1/2$ , we define an Equivalence Relation $\sim$ upon these pairs with the following rule: : $\left(a, b ight) \sim \left(c, d ight) \mbox{ iff } ad = bc$ This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the .) We can also define a Total Order on Q by writing : $\left(a, b ight) \le \left(c, d ight) \mbox{ iff } (bd>0\mbox{ and } ad \le bc)\mbox{ or }(bd<0\mbox{ and } ad \ge bc)$ 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 Characteristic 0: every other field of characteristic 0 contains a copy of $\mathbb{Q}$ . 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, the rationals are uniquely characterized by being countable, dense (in the above sense), and having no least or greatest element. REAL NUMBERS The rationals are a expressions of Continued Fraction .
	Let <math>p</math> Be A	"http://wwwinformationdelightinfo/encyclopedia/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