user18948: Instead of 'a1', why not '11'? First file, first rank.
Instead of 'g6', why not '76'? Seventh file, sixth rank.
And so on. Why not just use two numbers to represent the square, as opposed to 1 letter and 1 number? 
I think the notation with the letter is more confusing for beginners and children. For example, with the 'g6'-square, it's not immediately obvious to a complete beginner where the 'g'-file is. They usually have to look down on the letter on the chess board, or, if the board doesn't have letters, they will usually count in their head, from left to right: 'a ... b.. c.. d.. e ... f... g, aha there it is'. On the other hand, if the 'g'-file was simply the '7'-file, it's immediately obvious where it is, since it's much easier to simply find the 7th file by eye-sight.

user17411: One reason would be to avoid confusion with earlier methods - such as numbering each square from 1 to 64, starting on black's side of the board.
"28" could either be file 2, rank 8 (b8), or the 28th square from the upper left (d5).

user3872: I believe that it is more of a historic thing where chess players went from descriptive to algebraic notation, really because strong players started adopting algebraic notation. Admittedly, using 2 numbers would probably be just as easy to use once you get use to it.
Infact, ICCF(international correspondence chess federation) uses such coordinate systems in order not to confuse people from different countries who would use different alphabets for algebraic notation.  

user18809: Imagine White has two knights on b5 and e2 respectively (ie, 25 and 52) I think it would be confusing to write N243 rather than Nbd4 (we could still write N5d4, making a "mistake" but still not changing our move)
Also, there is no advantage to write everything in numbers, I'd say it loses legibility, as now our eyes can easily tell pieces (capital letters), files (lowercase letters) and ranks (numbers)

18948: <p>Instead of 'a1', why not '11'? First file, first rank.</p>

<p>Instead of 'g6', why not '76'? Seventh file, sixth rank.</p>

<p>And so on. Why not just use two numbers to represent the square, as opposed to 1 letter and 1 number? </p>

<p>I think the notation with the letter is more confusing for beginners and children. For example, with the 'g6'-square, it's not immediately obvious to a complete beginner where the 'g'-file is. They usually have to look down on the letter on the chess board, or, if the board doesn't have letters, they will usually count in their head, from left to right: 'a ... b.. c.. d.. e ... f... g, aha there it is'. On the other hand, if the 'g'-file was simply the '7'-file, it's immediately obvious where it is, since it's much easier to simply find the 7th file by eye-sight.</p>
 17411: <p>One reason would be to avoid confusion with earlier methods - such as numbering each square from 1 to 64, starting on black's side of the board.</p>

<p>"28" could either be file 2, rank 8 (b8), or the 28th square from the upper left (d5).</p>
 3872: <p>I believe that it is more of a historic thing where chess players went from descriptive to algebraic notation, really because strong players started adopting algebraic notation. Admittedly, using 2 numbers would probably be just as easy to use once you get use to it.
Infact, ICCF(international correspondence chess federation) uses such coordinate systems in order not to confuse people from different countries who would use different alphabets for algebraic notation.  </p>
 18809: <p>Imagine White has two knights on b5 and e2 respectively (ie, 25 and 52) I think it would be confusing to write N243 rather than Nbd4 (we could still write N5d4, making a "mistake" but still not changing our move)</p>

<p>Also, there is no advantage to write everything in numbers, I'd say it loses legibility, as now our eyes can easily tell pieces (capital letters), files (lowercase letters) and ranks (numbers)</p>
