By which convention 04 (for example) is no valid 2 digit number?

Question

When a person asks: "What is the smallest number (natural numbers) with two digits?"

You answer: "10".

But by which convention 04 is no valid 2 digit number?

Thanks alot in advance

Answer 1

Simple answer would be $0$ has no value infront of a number.So,$04,004,0004,000000000004$ are all same.As,this $0$ has no meaning or significance in the representation of the number,$0$ infront of a number is not considered.

Answer 2

The question asks for a number, not a representation of a number.

The digit sequence 04 is a representation of the number $4$ (also known as $1+1+1+1$), but 4 is also a representation of that same number.

Every natural number has many such representations, with varying amounts of leading zeroes, but when the question asks about a "number with two digits" it mus implicitly assume that a number has a well-defined number of digits. The only reasonable way to make sense of that is it interpret the question as

What is the smallest number whose usual decimal representation has two digits?

(where "usual" means with no leading zeroes except that $0$ itself is represented as 0 because using the empty string of digits to represent something would be confusing).

$4$ does not qualify as an answer to this because its usual representation is 4 rather than 04.

Answer 3

It depends on the context of the problem. For example. A number $n$ is called "magical" if $n^2$ ends with the digits of $n$. Below is a list of the first six magical numbers whose units digit is $5$.

 5²      =           25   
 25²     =          625    
 625²    =       390625    
 0625²   =       390625    
 90625²  =   8212890625    
 890625² = 793212890625    

In this case, I'd want to say that $0625$ is a magical four-digit number because $0625^2$ end with the digits $0625$.

If I told someone I was making a six-digit yearly salary, it would probably be wrong to say that $\$005489$ was a six-digit income.