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## Rational number
In mathematics, a
*a*/*b*+*c*/*d*=(*ad*+*bc*)/*bd*- (
*a*/*b*)(*c*/*d*)=*ac*/*bd*
a/b and c/d are equal if and only if ad=bc.
The set of all rational numbers is denoted by
In mathematics, the term "rational XXX" means that the underlying field considered is the field of rational numbers. For example, rational polynomials.
## History## Egyptian fractionsAny positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance, 5/7 = 1/2 + 1/6 + 1/21. For any positive rational number, there are infinitely many different such representations. These representations are calledEgyptian fractions, because the ancient Egyptians used them. The hieroglyph used for this is the letter that looks like a mouth, which is transliterated R, so the above fraction would be written as R2R6R21. The Egyptians also had a different notation for dyadic fractions.## Formal Construction
Mathematically we may define them as an ordered pair of integers (
- (
*a*,*b*) + (*c*,*d*) = (*a*×*d*+*b*×*c*,*b*×*d*) - (
*a*,*b*) × (*c*,*d*) = (*a*×*c*,*b*×*d*)
- (
- (
*a*,*b*) ~ (*c*,*d*) if, and only if,*a*×*d*=*b*×*c*.
- (
Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.
We can also define a total order on - (
*a*,*b*) ≤ (*c*,*d*) if, and only if,*ad*≤*bc*.
- (
## PropertiesThe setQ, together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers Z.
The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of
The algebraic closure of The set of all rational numbers is countable. Since the set of all real numbers is uncountable we can say that almost all real numbers are irrational. The rationals are a densely ordered set: between any two rationals there sits another one, in fact infinitely many other ones. ## Real numbersThe rationals are a dense subset of the real numbers: every real number is arbitrarily close to rational numbers. A related property is that rational numbers are the only numbers with finite expressions of continued fraction.
By virtue of their order, the rationals carry an order topology. The rational numbers are a (dense) subset of the real numbers, and as such they also carry a subspace topology. The rational numbers form a metric space by using the metric
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