It seems intuitive that there should be more rational numbers than integers, but this is where intuition is wrong. The set of integers and the set of rational numbers have the same size in the same way that the set of even integers has the same size as the set of integers. However, the set of real numbers is larger (and is the same size as the set of complex numbers).H2X wrote: ↑Mon Feb 18, 2019 1:29 pmI am not an expert either, but clearly there are kinds of infinities which cannot be easily compared. While there is an infinite number of integer values, it seems logical that the also infinite number of rational numbers is larger, and real numbers larger still.
Little Problem with Complex Mathematics on Calculators
Re: Little Problem with Complex Mathematics on Calculators
Re: Little Problem with Complex Mathematics on Calculators
I could not begin to disagree, but could you please explain that, as neither of these things you note as having the same size feel intuitive to me, and that Reals and Complex are larger (but equal) is just as unclear. When you have time is fine, this is only a curiosity for me, and if I knew how to just google this, I'd do that.ijabbott wrote: ↑Tue Feb 19, 2019 6:00 pmIt seems intuitive that there should be more rational numbers than integers, but this is where intuition is wrong. The set of integers and the set of rational numbers have the same size in the same way that the set of even integers has the same size as the set of integers. However, the set of real numbers is larger (and is the same size as the set of complex numbers).
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Re: Little Problem with Complex Mathematics on Calculators
My intuition won't admit it's wrong...ijabbott wrote: ↑Tue Feb 19, 2019 6:00 pmIt seems intuitive that there should be more rational numbers than integers, but this is where intuition is wrong. The set of integers and the set of rational numbers have the same size in the same way that the set of even integers has the same size as the set of integers. However, the set of real numbers is larger (and is the same size as the set of complex numbers).H2X wrote: ↑Mon Feb 18, 2019 1:29 pmI am not an expert either, but clearly there are kinds of infinities which cannot be easily compared. While there is an infinite number of integer values, it seems logical that the also infinite number of rational numbers is larger, and real numbers larger still.
Granted, infinities are infinite, but that roughly translates to moving targets in my mind. They have a dynamic nature.
How are the sets of integers and rational numbers the same size, and ditto the sets of real and complex numbers? I see targets moving faster than me, forever out of my reach, but at different speeds from each other. Surely?
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Re: Little Problem with Complex Mathematics on Calculators
Take an integer, and consider it as two numbers, with their digits interleaved. Say, 123456: this can be considered as 135 combined with 246.
Any pair of integers can be represented as a single integer in this manner. In fact, with this kind of mapping, integers and pairs of integers correspond onetoone.
Now consider that a rational number is the quotient of two integers. Every rational number can be represented as such a quotient in infinitely many ways, only the form where numerator and denominator have no common factors being unique. On the other hand, each numerator/denominator pair corresponds to exactly one rational number.
Thus, there are no more rational numbers than there are pairs of integers.
Combine this with the fact that the number of pairs of integers is equal to the number of integers, and you can see that the number of rational numbers can be no greater than the number of integers. And finally, since every integer is a rational number, the number of rational numbers also cannot be less than the number of integers, and so, the cardinalities of both sets are the same.
https://en.m.wikipedia.org/wiki/Aleph_number
Update: I was a bit sloppy, ignoring negative numbers. Replace all mentions of "integer" with "natural number," and all mentions of "rational number" with "nonnegative rational number" if you prefer, and then the argument is solid. Extending it to cover negative numbers is left as an exercise for the reader.
Any pair of integers can be represented as a single integer in this manner. In fact, with this kind of mapping, integers and pairs of integers correspond onetoone.
Now consider that a rational number is the quotient of two integers. Every rational number can be represented as such a quotient in infinitely many ways, only the form where numerator and denominator have no common factors being unique. On the other hand, each numerator/denominator pair corresponds to exactly one rational number.
Thus, there are no more rational numbers than there are pairs of integers.
Combine this with the fact that the number of pairs of integers is equal to the number of integers, and you can see that the number of rational numbers can be no greater than the number of integers. And finally, since every integer is a rational number, the number of rational numbers also cannot be less than the number of integers, and so, the cardinalities of both sets are the same.
https://en.m.wikipedia.org/wiki/Aleph_number
Update: I was a bit sloppy, ignoring negative numbers. Replace all mentions of "integer" with "natural number," and all mentions of "rational number" with "nonnegative rational number" if you prefer, and then the argument is solid. Extending it to cover negative numbers is left as an exercise for the reader.
Re: Little Problem with Complex Mathematics on Calculators
Thomas, I agree and disagree.
Is the set of even numbers as big as the set of odd numbers? And is not the set of integers the union of these? Which set is bigger?
My intuition is screaming.
What I see when I try to envision these infinities is different sets of numbers growing at different rates, thus tending towards infinity at different slopes, diverging  not converging.
Thinking, or intuiting, about infinity is probably futile, but it is fun  in a weird sense of the word...
Is the set of even numbers as big as the set of odd numbers? And is not the set of integers the union of these? Which set is bigger?
My intuition is screaming.
What I see when I try to envision these infinities is different sets of numbers growing at different rates, thus tending towards infinity at different slopes, diverging  not converging.
Thinking, or intuiting, about infinity is probably futile, but it is fun  in a weird sense of the word...
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Re: Little Problem with Complex Mathematics on Calculators
Thanks for walking some of us through this stuff Thomas.
What about the initial 10 integers (0,1,2,3..9)? How can these be represented as a pair of integers? Does 1 map to 01? etc.Thomas Okken wrote: ↑Tue Feb 19, 2019 10:17 pmTake an integer, and consider it as two numbers, with their digits interleaved. Say, 123456: this can be considered as 135 combined with 246.
Any pair of integers can be represented as a single integer in this manner. In fact, with this kind of mapping, integers and pairs of integers correspond onetoone.
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Re: Little Problem with Complex Mathematics on Calculators
Yes, the set of even numbers is the same size as the set of odd numbers (obvious onetoone correspondence between 2k and 2k+1, for example), AND each of those is the same size as the set of all integers (correspondence between k and 2k). Intuition is a bad guide to dealing with infinities!
No problem there, you're just looking at the evennumbered vs. oddnumbered digits, that is, the digits representing 1, 100, 10000, vs. the digits representing 10, 1000, 100000, etc. So 0 maps to the pair 0, 0; 1 maps to the pair 0, 1; 10 maps to the pair 1, 0; etc.
Re: Little Problem with Complex Mathematics on Calculators
It isn't logical. There are the same number of rational numbers as there are integers, this we proved by Cantor. There are more real numbers (proved using Cantor's diagonalisation).
Infinity is not a number and shouldn't be considered one. It is far more subtle and elusive. Hilbert's hotel is a nice demonstration.
Pauli
Re: Little Problem with Complex Mathematics on Calculators
This could the key for the initial issue reported by Walter. While the distributivity law is perfectly valid in ℂ, this may not be valid anymore if you try to apply it to infinity.
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Re: Little Problem with Complex Mathematics on Calculators
I know. But.pauli wrote: ↑Wed Feb 20, 2019 1:02 amIt isn't logical. There are the same number of rational numbers as there are integers, this we proved by Cantor. There are more real numbers (proved using Cantor's diagonalisation).
Infinity is not a number and shouldn't be considered one. It is far more subtle and elusive. Hilbert's hotel is a nice demonstration.
Pauli
Interestingly, the same Hilbert's hotel Wikipedia article mentions "layers" and "levels" of infinity.
My intuition demands not that infinity be a number or have limits, but it insists that there is structure. It would be surprised if there is a finite "number" of such structures.
I can live with it. It gives interesting thoughts.
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