When asked to form the least number from the digits 4 5 0 1 and you form 0145 which invariably is 145 How have you not used the zero when you all know zero is nothing and the goal is to form the l?

Answer 1

Consider this: with 145 you have also not used two zeros - 00145 = 145 - but only one was supplied, so you have not used ONLY the supplied digits.

If the zero is being supplied, then it must be used with some meaning:

The zero is not nothing when it comes somewhere between a non-zero digit and the decimal point (which for a whole number is "hiding" after the last digit); for if it were, the two numbers 105 and 15 would be equal - after all only "nothing" has been removed from the first.

Our zero has developed over history from the Babylonian place holder which was used to show an empty place value column - otherwise when written down the numbers "1 5" and "15" could be mistaken for each other.

With 145, the thousands column is empty, BUT does not need to be shown as such because the number cannot be misread; but with 105, the tens columns is effectively empty, but without anything written there 105 becomes 1 5 which could be misread as 15.

Thus to answer the question, the zero can only be used somewhere between a non-zero digit and the decimal point - for then it has some meaning. (The correct answer I leave as an exercise for the reader.)