Q:

regular expression for a number

It is a regular expression for a number. I have several questions about it.

(0U1U2U3U4U5U6U7U8U9)*   Does it means a set containing a number from 0 to 9 and then concatenate itself n times, or a set containing all of those 10 numbers and then concatenate itself n times. 
Is the first (0U1U2U3U4U5U6U7U8U9) necessary if (0U1U2U3U4U5U6U7U8U9)* means a set containing all of those 10 numbers and then concatenate itself n times. 
Is this set represent the natural number or only part of the natural number.

Thank you.

A:

The regular expression
$$0\cup 1\cup 2\cup 3\cup 4\cup 5\cup 6\cup 7\cup 8\cup 9$$
represents any single digit, including $0$. Call this expression $\rho$. Then $\rho^*$ represents any finite sequence of things matching $\rho$, which in this case means any finite string of digits. Note that finite includes empty, the string consisting of no digits at all. Thus, $\rho\rho^*$, which is what you have, represents any string consisting of a single digit ($\rho$) followed by any finite (possibly empty) string of digits ($\rho^*$). In other words, $\rho\rho^*$ represents any non-empty finite string of digits; the purpose of the initial $\rho$ is to ensure that there is at least one digit.
In words this definition of number amounts to saying that a number is any non-empty finite string of digits, where the digits are $0,1,2,3,4,5,6,7,8$, and $9$. Note that this definition allows leading zeroes: it allows $00123$ as a number, for instance. It does, however, include all natural numbers, including $0$; indeed, it includes each of them infinitely many times, since it allows any number of leading zeroes.

