Quandaries
and Queries 

Name: David
Question: 

Hi David, You have asked exactly the right question. The argument I gave in 6.9999... = ? seems to be a trick. It's harder than that. How can 3.9999... and 4 occupy the same point on the number line, and if they don't, what number is between them? What do we mean by a number represented with an infinite number of nonzero digits after the decimal? For example I know what 2.345 means
but what about 2.343443444344443...? One way to make sense of this is to consider the increasing sequence of numbers
Each number in this sequence has a finite number of nonzero digits after the decimal so they all make sense to me. I then define 2.343443444344443... to be the smallest number that is larger than all of the numbers 2.3, 2.34, 2.343, 2.3434. 2.34344, 2.343443,... (The proof that such a number exists and that there is only one such number take some work and requires some of the principles of calculus.) Defining the decimal numbers this way gives the nice properties that you list in your question, "namely that there is no space on the number line which does not have an number and conversely that every geometric point on the number line is associated with a unique real number". When you do this with 3.9999... however you are faced with a problem. The smallest number that is larger than each number in the sequence 3.9, 3.99, 3.999, 3.9999,... is 4. At first it seems counterintuitive, but it is common for us to represent the same number in different ways. For example 0.8, ^{8}/_{10} ,^{4}/_{5} and 80% all represent the same number. The same is true for 3.9999... and 4, they are two ways of representing the same number. Penny 

