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  1. #11
    YAMATO2201 is offline Senior Member
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    Re: Irrational numbers

    Quote Originally Posted by probus View Post
    I think there must also be more and "larger" infinities because my intuition says there must be fewer real numbers than there are complex numbers.
    ℝ and ℂ have the same cardinality.
    https://math.stackexchange.com/quest...me-cardinality
    I am not a teacher.

  2. #12
    probus's Avatar
    probus is offline Moderator
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    Re: Irrational numbers

    Quote Originally Posted by jutfrank View Post
    You have in mind two number lines side by side, and see that one is longer than the other, and so you conclude that one infinity is bigger than the other because each line has a finite length. In my opinion, all you're saying with this is something about how the human mind likes to make sense of abstract ideas. This handy 'trick' of viewing infinity as complete is also evident when mathematicians talk of 'infinite sets' and when they attempt to express infinity as a number. Philosophers, being much more sensible types, tend to think of infinity as incomplete (not least because it's a much more difficult and interesting view!). With this view, there are no 'bigger' or 'smaller' infinities because, by definition, there is no concept of
    No. Although I put in the disclaimer that I am not a mathematician, I did take a lot of math courses as an undergraduate, and worked for a time as an actuary. I know what I'm talking about.

    The real numbers can indeed be represented by points on a line, and mathematicians do that. The technical term in mathematics for such correspondences is isomorphism. The real numbers and the points on a line can be regarded as identical for many purposes

    The "second line" lies not parallel to the first but across it at right angles. That is intuitively why the complex numbers are more "numerous" than the real numbers (C > aleph-null) in a sense. Just as mathematicians speak of the real number line, they also speak of the complex number plane.

    Since we are supposed to be discussing English, not mathematics, I am closing this thread.
    Last edited by emsr2d2; 16-Apr-2021 at 16:08. Reason: Fixed typo

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