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A Natural infinite set refers to one whose members can be put into 1-to-1 correspondence with the natural numbers, while a real infinite set is one whose members can be put into 1-to-1 correspondence with the real numbers. Although both sets are infinite, they are not of the same cardinality (size).
The cardinality of the natural infinite set is denoted by À0 or Aleph-null. The cardinality of the real infinite set is 2 to the power À0, which is denoted by C. (Actually Aleph looks like an N with wriggly lines but this browser is incapable of displaying it.)
For more on the cardinality of infinite sets, see the related links. Georg Cantor's diagonal argument is exquisite - simple but immensely powerful. If you want to get a feel for transfinite arithmetic - read about Hilbert's Hotel paradox.
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What is the difference between natural infinite numbers and real infinite numbers?
A natural number is a counting number, such as 1, 2, 3. There are also known as whole numbers and integers. They can be infinitely large. A real number is a number, possibly a natural number, but more possibly not, because there are an infinite number of real numbers that lie between any two natural numbers, such as 1, 1.1, 1.11, 1.111, 111112, etc, ad infinitum. Real numbers can also be infinitely large.
What number lies between a square and a cube?
There are no natural limits to the exponent a number can be raised to, and there are an infinite number of real numbers between "2.0" and 3.0".
How many more real numbers are there than natural numbers?
Since there is an infinite number of real numbers and an infinite number of natural numbers, there is not more of one kind than of another.
What is the difference between real solutions and real roots when dealing with discriminant?
There is no difference between real solutions and real roots.
Is there a real number between 0.123 repeated and 0.124?
Yes, an infinite number of them.
