Numeric Fractions Assignment-Btechnd

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as “the rationals“, is usually denoted by a boldface Q (or blackboard bold {\displaystyle \mathbb {Q} }\mathbb {Q} , Unicode ℚ);[2] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for “quotient”.

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).

A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1]

The rational numbers can be formally defined as the equivalence classes of the quotient set (Z × (Z \ {0})) / ~, where the cartesian product Z × (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not 0 (n ≠ 0), and “~” is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2m2n1 = 0.

In abstract algebra, the rational numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals.

Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undefined)

Terminology

The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, “rational” is often used as a noun abbreviating “rational number”. The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (that is a point whose coordinates are rational numbers; a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term “polynomial over the rationals” is generally preferred, for avoiding confusion with “rational expression” and “rational function” (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.

Arithmetic

See also: Fraction (mathematics) § Arithmetic with fractions

Embedding of integers

Any integer n can be expressed as the rational number n/1.

Equality
{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}{\frac {a}{b}}={\frac {c}{d}} if and only if {\displaystyle ad=bc.}ad=bc.

Ordering

Where both denominators are positive:

{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}{\frac {a}{b}}<{\frac {c}{d}} if and only if {\displaystyle ad<bc.}ad<bc.

If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations:

{\displaystyle {\frac {-a}{-b}}={\frac {a}{b}}}{\frac {-a}{-b}}={\frac {a}{b}}

and

{\displaystyle {\frac {a}{-b}}={\frac {-a}{b}}.}{\frac {a}{-b}}={\frac {-a}{b}}.
Addition

Two fractions are added as follows:

{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}{\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.
Subtraction
{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}{\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.
Multiplication

The rule for multiplication is:

{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}{\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.
Division

Where c ≠ 0:

{\displaystyle {\frac {a}{b}}\div {\frac {c}{d}}={\frac {ad}{bc}}.}{\frac {a}{b}}\div {\frac {c}{d}}={\frac {ad}{bc}}.

Note that division is equivalent to multiplying by the reciprocal of the divisor fraction:

{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\times {\frac {d}{c}}.}{\frac {ad}{bc}}={\frac {a}{b}}\times {\frac {d}{c}}.
Inverse

Additive and multiplicative inverses exist in the rational numbers:

{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}={\frac {a}{-b}}\quad {\mbox{and}}\quad \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}{\mbox{ if }}a\neq 0.}-\left({\frac {a}{b}}\right)={\frac {-a}{b}}={\frac {a}{-b}}\quad {\mbox{and}}\quad \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}{\mbox{ if }}a\neq 0.
Exponentiation to integer power

If n is a non-negative integer, then

{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}}\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}

and (if a ≠ 0):

{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}\left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.

Continued fraction representation
Main article: Continued fraction

A finite continued fraction is an expression such as

{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},

where an are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a,b).

Other representations
  • common fraction: {\displaystyle {\frac {8}{3}}}{\frac {8}{3}}
  • mixed numeral: {\displaystyle 2{\tfrac {2}{3}}}2\tfrac{2}{3}
  • repeating decimal using a vinculum: {\displaystyle 2.{\overline {6}}}2.{\overline {6}}
  • repeating decimal using parentheses: {\displaystyle 2.(6)}2.(6)
  • continued fraction using traditional typography: {\displaystyle 2+{\cfrac {1}{1+{\cfrac {1}{2}}}}}2+{\cfrac {1}{1+{\cfrac {1}{2}}}}
  • continued fraction in abbreviated notation: [2; 1, 2]
  • egyptian fraction: {\displaystyle 2+{\frac {1}{2}}+{\frac {1}{6}}}{\displaystyle 2+{\frac {1}{2}}+{\frac {1}{6}}}
  • prime power decomposition: {\displaystyle {\frac {117}{1000}}=2^{-3}\times 3^{2}\times 5^{-3}\times 13}{\displaystyle {\frac {117}{1000}}=2^{-3}\times 3^{2}\times 5^{-3}\times 13}
  • quote notation: 3!6

are different ways to represent the same rational value.Order Now

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