I have questions about the number systems which confused me.
As all we know we are using decimal system and also we can talk about natural numbers, integers and so on. Could we talk about natural numbers in for example in quinary system?
additional question : in quinary system we say the least number with two digits is $(10)_5$, but why we didn't write down as $-(44)_5$ ? i don't know if it is $-(44)_5$ ? or $(-44)_5$ ? sorry for bad langugage , thanks in advance :)
On
The naturals, integers, etc. are types of numbers that are independent of the system used to represent them. We can certainly talk of the naturals in base $5$. They are the same as the naturals in base $10$, though most of them have a different representation.
For your second question, this does not depend on the base. In base $10$, we have $10$ is the smallest positive two digit number. When people talk of the smallest two digit number they usually do not bother with positive but do mean it. If you count negative numbers, the smallest two digit number is $-99$. I would normally not write base $5$ numerals with the parentheses, so if we allow negative numbers the smallest two digit base $5$ number is $-44_5$
Remember, the natural numbers in base ten are non-negative numbers. (Some also exclude $0$ from the set of natural numbers, so that $\mathbb N = \{1, 2, 3, \ldots\}$.)
So the least two-digit natural number in base 10 is $10 = $1\times 10 + 0\times 1= 10_{10}$.
Likewise, the natural numbers in base $5$ must be non-negative, (perhaps excluding 0, i.e. $\{1, 2, 3, 4, 10, 11, 12, 13,14, 20\ldots\}$.
The least two-digit number in base 5 is given by $10_5 = 1\times 5 + 0 \times 5^0 = 5_{10}.$
If you want to include negative numbers, as well, in considering the least 2-digit number in base $5$, then I would suggest you write it as $$-(44)_5 = -(4\times 5^0 + 4\times 5) = -(24)_{10}$$