What is a counting number?
The definition of natural number is given as The counting numbers {1, 2, 3, ...}, are called natural numbers. They include all the counting numbers i.e. from 1 to infinity. at the link .
What does counting number mean, are there any numbers which are not countable?
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$\begingroup$Well, yes. Consider 0, or -1, or $\frac{1}{2}$, or $\sqrt{2}$, or $\pi$, or... None of these are natural numbers, but they are real numbers.
$\endgroup$ 6 $\begingroup$The counting numbers are the positive integers. The positive numbers have always been considered suitable for counting. For example, a shepherd might have $47$ sheep; he probably would never say that he has $\frac{189}{4}$ sheep or $2 + \sqrt{-43}$ sheep (I can barely begin to imagine what such a complicated number means).
But what about $0$? Is that a counting number? Traditionally, no. In the earliest days of shepherding, $0$ was not even accepted as a valid number. Say our shepherd sold all his sheep. Then he would say he has no sheep, but he wouldn't say he has $0$ sheep. And if he had $47$ sheep, the idea that he could sell $50$ sheep would probably have been unthinkable.
$\endgroup$ 1 $\begingroup$The counting numbers are $\mathbb{Z}^{+} = \big\{1, 2,3....,∞\big\} $ also known as the natural numbers.
Even though the counting numbers go on on forever, they can be counted.
Given a set $X$,
$X$ is denumerable or enumerate if there is a bijection $\mathbb{Z}^{+} → X$.
A set is countable if either it is finite or it is denumerable.
Thus the set $\mathbb{Z}^{+} = \big\{1, 2,3....,∞\big\} $, is countable since there is a bijection $\mathbb{Z}^{+} → \mathbb{Z}^{+}$.
This gives the listing $\mathbb{Z}^{+} = \big\{1, 2,3....,∞\big\} $
The set of real $\mathbb{R}$ numbers is uncountable (not countable) since there is no bijection such that $\mathbb{R}^{} → \mathbb{Z}^{+}$.
For example, $∞$ cannot be map to 1, since infinity is not a number.
Thus there has to be a one to one and surjective relationship between the set of the natural numbers with the given set of numbers.
$\endgroup$ 1 $\begingroup$Counting numbers are numbers you can use to count whole items. You can use the positive integers to count marbles, for example. Whether 0 is a counting number or not, that's a big debate you may or may not be interested in. As another answerer so amusingly noted, fractional and complex numbers are unsuitable for counting live animals.
As to whether numbers of a particular kind are themselves countable, that's a different issue that requires different words: you need to ask about countable numbers, not counting numbers. For example, there are infinitely many rational numbers between 0 and 1 and you can count them in a theoretical sense even though you can't count them in a practical sense. I would start with $\frac{1}{2}$, continue with $\frac{1}{3}$, $\frac{2}{3}$, then $\frac{1}{4}$, $\frac{3}{4}$, $\frac{1}{5}$, $\frac{2}{5}$, etc. In short, I can give you a concrete algorithm that could theoretically count all these numbers.
There are also infinitely many irrational numbers between 0 and 1. I would start with $-1 + \sqrt{2}$. But what about $\pi - 3$? What about all those transcendental numbers we don't have symbols for? Even if I was a genius (which I am not), I couldn't give you an algorithm to count those numbers.
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