Name: David
Who is asking: Teacher
Level: Secondary
Question:
I have read your answers to the questions on rational numbers, esp.
6.9999... = ? and still have a question: The simple algebraic stunt
of converting
repeating decimals to rational numbers seems to work for all numbers
except X.999999.... where X is any integer. The fact that the method
yields the
integer X+1 in each case seems to violate the completeness axiom
of the real numbers, 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. In the case
of 3.999... for
example, it seems that both the number 4 and the number 3.9999....
occupy the same point on the number line. How is this possible???
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 non-zero digits after the decimal? For example I know what 2.345 means
2.345 = 2 + 3/10 + 4/100 + 5/1000
but what about 2.343443444344443...?
One way to make sense of this is to consider the increasing sequence of numbers
2.3, 2.34, 2.343, 2.3434. 2.34344, 2.343443,...
Each number in this sequence has a finite number of non-zero 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