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What is the difference between infinitely many solutions and all real numbers
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The set of even integers is a set of infinitely many members but
is is not the set of all real numbers.
A subset of real numbers can also be an infinite set – without
including all real numbers. For example:* All integers
* All rational numbers (fractions)
* All multiples of pi (pi multiplied by an integer)
* All numbers of the type 1/n, for an integer n
* Etc.
Let me use an example.
y^2 = -x (where y^2 means y squared)
Then y = sq rt (-x). There is an infinite number of solutions,
some of which are imaginary numbers and some are real.
So when you say ‘infinitely many solutions’ this includes
imaginary numbers. All real numbers is a subset of that.