Largest Number Smaller than One
Date: Fall 2012
If 0.99 (repeating) =1, then what is the largest number possible that is smaller than 1?
But actually, 0.99 repeating never becomes exactly equal to 1. It is always a bit short no matter how many 9's.
True, 0.9999... quickly approaches 1 for practical use. But think of asymptotes you ran into in graphing functions.
Or the definition of the derivative. Although delta X approaches zero, it never really gets there. Same with 0.9999...
Hope this helps
Oklahoma State University Institute of Technology
The real number system is continuous. If you are a certain “distance” less than 1, there is always a way to be between you and one. If there is such a thing as the largest number possible that is smaller than 1, let x equal it. Now take the average of the two: (1+x)/2. This is closer than x is to 1. It is larger than x but smaller than 1. 0.99.… is so close to 1 that you cannot tell the difference between them. 0.99…. is not really 1. It is so close to 1 that the difference between the two numbers is smaller than can be said. As a result, 0.99…. is exactly the same size as 1. Because it works the same as 1 in calculations, you might just as well call it 1.
Dr. Ken Mellendorf
Illinois Central College
Thanks for the question. 0.999 (repeating) does not equal 1. It is slightly less than 1. One can prove that there are an infinite number of real numbers between 0 and 1 inclusive. Thus, there can be no number which satisfies the hypotheses of your question.
I hope this helps.
The largest possible number (in the decimal system of counting) that is less than one is:
0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 . . .
In Binary system:
Such that when you add a 1 in the last decimal place out in the realm of infinity, all of the other digits convert to a binary 10, carrying the 1.
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Update: November 2011