Difference between revisions of "Rational number"
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==Countability and density== | ==Countability and density== | ||
| − | An important theoretical property of the rationals is that they are countable. That is, the entire set of rational numbers can be put into a one-to-one correspondence with the integers. This is surprising at | + | An important theoretical property of the rationals is that they are countable. That is, the entire set of rational numbers can be put into a one-to-one correspondence with the integers. This is surprising at first glance, since the integers are "sparse" while the rationals seem to fill out the real line. To see this correspondence, we need to list all rational numbers in some order. List the rationals (that is, the rationals in which the fraction has been fully reduced) that have the sum of their numerators and denominators equal to one. 0/1 is the only such. Then follow those with the rationals that have the sum of their numerators and denominators equal to two. 1/1 is the only such, since 0/2 is not reduced. Follow those with the rationals that have the sum of their numerators and denominators equal to three. They are 1/2 and 2/1. Continue without end. |
A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology. | A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology. | ||
Revision as of 23:37, October 18, 2009
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This article/section deals with mathematical concepts appropriate for a student in mid to late high school. |
The rational numbers are the numbers representable as ratios of integers, that is, fractions. Mathematicians denote the set of rational numbers with an ornate capital letter: 
. They are the 3rd item in this hierarchy of types of numbers:
- The "natural numbers", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
- The "integers"—positive, negative, and zero
- The "rational numbers", or fractions, like 355/113
- The "real numbers", including irrational numbers
- The "complex numbers, which give solutions to polynomial equations
Contents
Rational numbers are actually defined as equivalence classes of ratios of integers, so that 2/3 and 4/6 are the same number.
If a rational number were to be represented as a decimal to infinite precision, that decimal would either terminate at some point (e.g. 1.25) or would eventually get into an endless repeating pattern (e.q. 1.250909090909..., which is 1376/1100).
Zero in the denominator of a rational number is not allowed. All rational numbers are finite.
Countability and density
An important theoretical property of the rationals is that they are countable. That is, the entire set of rational numbers can be put into a one-to-one correspondence with the integers. This is surprising at first glance, since the integers are "sparse" while the rationals seem to fill out the real line. To see this correspondence, we need to list all rational numbers in some order. List the rationals (that is, the rationals in which the fraction has been fully reduced) that have the sum of their numerators and denominators equal to one. 0/1 is the only such. Then follow those with the rationals that have the sum of their numerators and denominators equal to two. 1/1 is the only such, since 0/2 is not reduced. Follow those with the rationals that have the sum of their numerators and denominators equal to three. They are 1/2 and 2/1. Continue without end.
A more subtle theoretical property is that the rationals comprise a countable dense subset of the reals. This is used in some advanced theorems of topology.
