Unit 2 Numeric Fractions Assignment-Btechnd

Formal construction

A diagram showing a representation of the equivalent classes of pairs of integers

Mathematically we may construct the rational numbers as equivalence classes of ordered pairs of integers (m,n), with n ≠ 0. This space of equivalence classes is the quotient space (Z × (Z \ {0})) / ~, where (m1,n1) ~ (m2,n2) if, and only if, m1n2m2n1 = 0. We can define addition and multiplication of these pairs with the following rules:

{\displaystyle \left(m_{1},n_{1}\right)+\left(m_{2},n_{2}\right)\equiv \left(m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}\right)}\left(m_{1},n_{1}\right)+\left(m_{2},n_{2}\right)\equiv \left(m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}\right)
{\displaystyle \left(m_{1},n_{1}\right)\times \left(m_{2},n_{2}\right)\equiv \left(m_{1}m_{2},n_{1}n_{2}\right)}\left(m_{1},n_{1}\right)\times \left(m_{2},n_{2}\right)\equiv \left(m_{1}m_{2},n_{1}n_{2}\right)

and, if m2 ≠ 0, division by

{\displaystyle {\frac {\left(m_{1},n_{1}\right)}{\left(m_{2},n_{2}\right)}}\equiv \left(m_{1}n_{2},n_{1}m_{2}\right).}{\frac {\left(m_{1},n_{1}\right)}{\left(m_{2},n_{2}\right)}}\equiv \left(m_{1}n_{2},n_{1}m_{2}\right).

The equivalence relation (m1,n1) ~ (m2,n2) if, and only if, m1n2m2n1 = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set(Z × (Z \ {0})) / ~, i.e. we identify two pairs (m1,n1) and (m2,n2) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) We denote by [(m1,n1)] the equivalence class containing (m1,n1). If (m1,n1) ~ (m2,n2) then, by definition, (m1,n1) belongs to [(m2,n2)] and (m2,n2) belongs to [(m1,n1)]; in this case we can write [(m1,n1)] = [(m2,n2)]. Given any equivalence class [(m,n)] there are a countably infinite number of representation, since

{\displaystyle \cdots =[(-2m,-2n)]=[(-m,-n)]=[(m,n)]=[(2m,2n)]=\cdots .}\cdots =[(-2m,-2n)]=[(-m,-n)]=[(m,n)]=[(2m,2n)]=\cdots .

The canonical choice for [(m,n)] is chosen so that n is positive and gcd(m,n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write [(1,2)] instead of [(2,4)] or [(−12,−24)], even though [(1,2)] = [(2,4)] = [(−12,−24)].

We can also define a total order on Q. Let ∧ be the and-symbol and ∨ be the or-symbol. We say that [(m1,n1)] ≤ [(m2,n2)] if:

{\displaystyle (n_{1}n_{2}>0\ \land \ m_{1}n_{2}\leq n_{1}m_{2})\ \lor \ (n_{1}n_{2}<0\ \land \ m_{1}n_{2}\geq n_{1}m_{2}).}(n_{1}n_{2}/>0\ \land \ m_{1}n_{2}\leq n_{1}m_{2})\ \lor \ (n_{1}n_{2}<0\ \land \ m_{1}n_{2}\geq n_{1}m_{2}).”></dd></dl><p style=The integers may be considered to be rational numbers by the embedding that maps m to [(m,1)].

Properties

A diagram illustrating the countability of the rationals

The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers Z.

The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime field for characteristic zero.

The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the algebraic numbers.

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 densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that

{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}{\frac {a}{b}}<{\frac {c}{d}}

(where {\displaystyle b,d}b,d are positive), we have

{\displaystyle {\frac {a}{b}}<{\frac {ad+bc}{2bd}}<{\frac {c}{d}}.}{\frac {a}{b}}<{\frac {ad+bc}{2bd}}<{\frac {c}{d}}.

Any totally ordered 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 and topological properties

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.

By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric d(x,y) = |xy|, and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric d(x,y) = |xy|, above.

p-adic numbers

See also: p-adic Number

In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field:

Let p be a prime number and for any non-zero integer a, let |a|p = pn, where pn is the highest power of p dividing a.

In addition set |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p / |b|p.

Then dp(x,y) = |xy|p defines a metric on Q.

The metric space (Q,dp) is not complete, and its completion is the p-adic number field Qp. Ostrowski’s theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.

References

  1. ^ Jump up to:a b Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. ISBN 978-0-07-288008-3.
  2. Jump up^ Rouse, Margaret. “Mathematical Symbols”. Retrieved 1 April 2015.
  3. Jump up^ Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN 0-534-40264-X.
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