Right, that's why we can't count with them. And multiplication by i is equivalent to a rotation by 90 degrees.
Infinity
- Thread starter Lithp
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That is what I thought, Dexter, and that is why I think complex numbers is Aleph 2.
Another way of looking at it is that in a set belonging to a higher order of infinity, there is ∞ to 1 correspondence with a set of lower order of infinity.
Thus, set 1, of integers is Aleph Null; set 2, of real numbers is Aleph One. Here there is 1 to ∞ correspondence. For every number in set 1 (integer) there are infinite real numbers, in set 2.
Similarly, if you compare set 2, real numbers with set 3, complex numbers, for each real number, there are an infinite number of complex numbers. here again, there is 1 to ∞ correspondence. That is why I think complex numbers is a higher order of infinity than real numbers. That would make it Aleph 2.
What do you think?
Dexter, we can look at it geometrically. A set of integers can be represented by zero dimensions. It is really a set of points, nothing more. That is Aleph Null.
The set of real numbers can be represented by one dimension, by a straight line. Real numbers correspond to different positions on the line. The set can be represented in one dimension, that is Aleph One.
The set of complex numbers can be represented by two dimensions. Complex number is a+ib. the number is represented on two axes. ‘a’ is on X axis, ‘b’ is on Y axis. The number a+ib is represented by the coordinates (a,b).
Thus a set of complex numbers can be represented by two dimensions, that is Aleph Two.
The set of real numbers can be represented by one dimension, by a straight line. Real numbers correspond to different positions on the line. The set can be represented in one dimension, that is Aleph One.
The set of complex numbers can be represented by two dimensions. Complex number is a+ib. the number is represented on two axes. ‘a’ is on X axis, ‘b’ is on Y axis. The number a+ib is represented by the coordinates (a,b).
Thus a set of complex numbers can be represented by two dimensions, that is Aleph Two.
Integers ARE real (whole) numbers- who cares if they are dots, points on straight line or points on a circumferance?
lol I am happy if I can add, do some calculus, and do some trig, etc. The nature of numbers is interesting, but has little bearing on how many fingers and toes I have.
I think you need to see the proof. Informally, it goes like this. Complex numbers are just pairs of real numbers, and so can be represented as coordinates in a plane. The question becomes, then, can we put the points in a plane into one-to-one correspondence with the points on a line? The answer is yes. Suppose we have a line one unit long, each point of which is labeled with a decimal fraction containing Aleph Null digits. You'll agree that's Aleph One points on the line? And further suppose we have a square one unit on a side. For each point on the line, form a pair of numbers by selecting alternating digits from its label. For the point 0.49586345245..., for instance, we'd get the pair (.456425..., .98354...) Doing that will get you every point in the square eventually, the line and the square contain the same number of points. The infinity of pairs of real numbers is the same as the infinity of real numbers, which means the infinity of complex numbers is Aleph One. The argument readily generalizes to 3 dimensions (pick triplets of every third digit from the line), and 4, and so on. There are some complications due to things like .500000... being the same as .49999999.... if they have Aleph Null digits which require some rather abstruse reasoning in set theory terms to deal with, but they don't falsify the argument.
Suppose we have a line one unit long, each point of which is labeled with a decimal fraction containing Aleph Null digits. You'll agree that's Aleph One points on the line?
I am with you here. Real numbers can be represented on a straight, one dimensional line.
And further suppose we have a square one unit on a side. For each point on the line, form a pair of numbers by selecting alternating digits from its label.
Huh? That is not cellar to me. So for instance, for a number 0.4, what number would you select? Could you give an example?
Doing that will get you every point in the square eventually, the line and the square contain the same number of points.
Now, I don’t see how that can be. Taking your number .456425, assume the square is one unit long (values go from 0 to 1). Then for the number .456425, why pick only .98254? for .456425, you will have to pick numbers from 0 to 1, which are infinite numbers.
I don’t see how all the numbers in a square can be represented on a straight line.
But then I am not a mathematician, maybe it can be done. Do you have a reference to it? Because really, if numbers in a square can be represented on a straight line, then you are right, set of complex numbers will not be a higher order of infinity. But I don’t see how that can be done. So do you have a reference to it?
I am with you here. Real numbers can be represented on a straight, one dimensional line.
And further suppose we have a square one unit on a side. For each point on the line, form a pair of numbers by selecting alternating digits from its label.
Huh? That is not cellar to me. So for instance, for a number 0.4, what number would you select? Could you give an example?
Doing that will get you every point in the square eventually, the line and the square contain the same number of points.
Now, I don’t see how that can be. Taking your number .456425, assume the square is one unit long (values go from 0 to 1). Then for the number .456425, why pick only .98254? for .456425, you will have to pick numbers from 0 to 1, which are infinite numbers.
I don’t see how all the numbers in a square can be represented on a straight line.
But then I am not a mathematician, maybe it can be done. Do you have a reference to it? Because really, if numbers in a square can be represented on a straight line, then you are right, set of complex numbers will not be a higher order of infinity. But I don’t see how that can be done. So do you have a reference to it?
Haven't been able to find one online that makes it clear, but you can find it informally discussed in Martin Gardner's Wheels, Life, and Other Mathematical Amusements, ISBN 0-7167-1588-0, and with more formal mathematics in Rudy Rucker's Infinity and the Mind, ISBN 3-7643-3034-1.
Maybe try thinking of it the other way. Start with the little square one unit on a side and select any point in it. From its coordinates, which will be a pair of numbers between 0 and 1, form a new number by interleaving their digits alternately. Suppose you selected the point (0.325465..., 0.87325...) Form the new number by taking 3 as the first digit, 8 as the second one, 2 as the third one, and so on, and you'll get .382753...etc. That number is on the real line between 0 and 1, and there's no location in the square for which that is not true. The number of points in a plane and the number of points on a line are the same.
Levity is all very well, but you might be surprised to know that you've actually raised some serious issues in topology and trans-finite mathematics with those questions, that have real world implications in terms of our understanding of the limits of what's possible and impossible. I commend a little book to your attention, and to anyone else interested in such questions, John D. Barrow's Impossibility: The Limits of Science and the Science of Limits, ISBN 0-09-977211-6.
And just to tie off any possible loose ends, so to speak, no you can't tie a granny knot in an infinitely long string. You have to fold the ends over and under each other to make a granny knot, and an infinite string has no ends. But you can tie other kinds of knots in it.
but you can find it informally discussed in Martin Gardner's Wheels, Life, and Other Mathematical Amusements, ISBN 0-7167-1588-0, and with more formal mathematics in Rudy Rucker's Infinity and the Mind, ISBN 3-7643-3034-1.
Dexter, that is where I read about orders of infinity, in a column by Gardner in Scientific American, many years ago. I am really writing from memory.
As to rest of it, at least now it is clear to me what you are trying to say. I will think about it.
Dexter, that is where I read about orders of infinity, in a column by Gardner in Scientific American, many years ago. I am really writing from memory.
As to rest of it, at least now it is clear to me what you are trying to say. I will think about it.
Having some second thoughts. I think I might be wrong about that, as long the string is just a conceptual object, not a real physical object. We can't, of course, have such a real physical object, it would fill the universe. If the positive real numbers, for instance, can have a beginning but no end, then it seems an infinite string could have one free end you can get hold of, and with that you *can* make a granny knot in it.
Worries!
1. There is no smallest positive real number.
2. If the string had one free end, it would be a string ray; they're dangerous.
3. If a Koch snowflake which has an infinite perimeter but a finite area were to melt near Fargo, would the meltwater breach the levees? More levee tea.