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Tina
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When is x rational and irrational?
Also When is r positive and negative?
Also When is r positive and negative?
Tina said:When is x rational and irrational?
Also When is r positive and negative?
A = not A?HallsofIvy said:An irrational number is any number that is NOT irrational.
gravenewworld said:Aren't all irrational numbers just sequences of rational numbers?
A sequence of rational numbers would be rational.
The real numbers have the property that they are dense
Is there a proof of this somewhere?Hurkyl said:Any irrational number is equal to the limit of some sequence of rational numbers.
Is there a proof of this somewhere?
Ooops! I will edit that. Thanks.Alkatran said:A = not A?
... contrary to the rational numbers, the real numbers are dense in themselves :laughing:Hurkyl said:The rational numbers are a dense subset of the real numbers.
... the real numbers are not compact in themselvesHurkyl said:Or more compactly,
gravenewworld said:Not all the time. The real numbers have the property that they are dense, i.e. for any real number a there is a sequence of rational numbers {r_n} so r_n--->a. So say for the irrational number pi
r_1=3.0
r_2=3.1
r_3=3.14
r_4=3.141
and so on you get the idea. If a is irrational you can just choose r_n to be the rational numbers of the first n terms of the decimal expansion of a followed by zeroes. If r has its decimal expansion that agrees with the expansion of a to the mth place then the number differs from a less than 10^-m. So obviously the sequence of rationals {r_n} converges to a.
Alkatran said:A = not A?
Yes.gravenewworld said:Aren't all irrational numbers just sequences of rational numbers?
No, the limit of a sequence of rational numbers does not have to be rational.Alkatran said:A sequence of rational numbers would be rational.
1 then 2 -> 1.2
One way of defining the real numbers, in terms of rational numbers, is to define them to be equivalence classes of certain kinds of kinds of sequences of rational numbers.Ethereal said:Is there a proof of this somewhere?
A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not equal to 0. It can be written in the form of a/b, where a and b are integers.
An irrational number is a number that cannot be expressed as a ratio of two integers. These numbers cannot be written in the form of a/b, where a and b are integers. They are non-terminating and non-repeating decimals.
You can determine if a number is rational or irrational by looking at its decimal representation. If the decimal is non-terminating and non-repeating, then the number is irrational. If the decimal is either terminating or repeating, then the number is rational.
Yes, all integers are rational numbers. This is because they can be written in the form of a/b, where b=1. For example, 5 can be written as 5/1, making it a rational number.
No, a number cannot be both rational and irrational. It can only be one or the other. This is because the definition of a rational number excludes numbers that cannot be expressed as a ratio of two integers.