Why do we have to say 一百 一千 but not 一十? This has always stumped me as all other base 10 numbers (>10) in Chinese you declare 1 through 9 for the quantity [1] of these units but you only have to declare 2 through 9 for 十 as 1 is implied [2]

So is there reason behind this or just a quirk in the numerical system? 

1:


  一百
  
  一千 
  
  一万


2:


  十六
  
  二十六




Note:
As @congusbongus has pointed out this is only applicable when talking about 十 through 十九
It's just conventional. Even common Chinese don't know why.

You do can say 16, for example,  一十六 or 十六.  When we emphasizing something, we say 一十六. While in common usage, we say 十六, because it's tired to add a prefix 一. It's redundant.