My reason for questioning this relates to the Hilbert space-filling curve. can anyone convince me that there is a single unambiguous correct answer to this question? Are there more rational numbers than integers, or not?Īlthough I've already accepted an answer, I'll just add some extra context. You can map finite range of integers of any size to each rational this way and, since there is no finite upper bound to the stepping amount, you could argue there are potentially infinite integers for every single rational. You can use 1.10 for the first rational and 11.20 for the second etc. Since you can count the rationals, you can equally count stepping by any amount for each rational. To me, this seems a much more reasonable approach, implying that there are infinite rational numbers for every integer.īut even then, this is just one of many alternative ways to map between ranges of rationals and ranges of integers. Between zero and one there are infinitely many rational numbers, between one and two there are infinitely many rational numbers, and so on. The second reason is that it's very easy to construct alternative mappings. If it were possible to count to infinity, it would be possible to count one step less and stop at count infinity-1 which must be different to infinity.
#ALL INTEGERS ARE RATIONAL NUMBERS CODE#
Infinity is code for "no matter how far you count, you have never counted enough". You can't even count all positive integers. You can count to and number any rational, but you cannot number all rationals. First, this logic seems to assume that infinity is a finite number. The trouble is, I find this very hard to accept.
So far we have counted out 6 rationals, and if we continue long enough, we will eventually count to any specific rational you care to mention. If the constant is 4 there's 1/3, 2/2 and 3/1. If the constant is 3, there's 1/2 and 2/1. To count the rationals, consider sets of rationals where the denominator and numerator are positive and sum to some constant. I've ignored sign-related issues, but these are easily handled. The set of rationals is countably infinite, therefore every rational can be associated with a positive integer, therefore there are the same number of rationals as integers. Whole numbers are a subset of the set of rational numbers and can be written as a ratio of the whole number to 1.I've been told that there are precisely the same number of rationals as there are of integers. Integers such as -1 and -6 are not whole numbers. Explain your choice.Įvery integer is included in the set of rational numbers. Integers are a subset of the set of rational numbers.Ī whole number can be written as a fraction with a denominator of 1, so every whole number is included in the set of rational numbers. The whole numbers are a subset of the rational numbers.Įvery integer is a rational number, but not every rational number is an integer.įor example, rational numbers such as 3/5 and -5/2 are not integers.Įvery whole number is an integer, but it is not true that every integer is a whole number. Tell whether the given statement is true or false.
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