Does 0.999... equal one?

The number 0.9 does not equal one. And the number 0.99 does not equal one. No matter how many times we add another nine, the resulting number still does not equal one.

The question "Does 0.999... equal one?" is confusing since the term "0.999..." implies that the nines repeat without end, making it impossible to get to the word "equal" when trying to make sense of the question.

In order to make sense of the question, it is necessary to define the term "0.999...".

Variable

If we define the term as a variable which approaches one as nines are added, then no it does not equal one since it does not refer to the same number at any time. Rather, it refers to a different number each time a nine is added.

Contradiction

One might attempt to define the term as a zero followed by a decimal point followed by an infinite sequence of nines, not a sequence which is continually growing, but a static sequence which has no end. Such an attempt to define would be contradictory since specifying the end of a static sequence is necessary in order to define the sequence.

In order to define a static sequence it is necessary to define both the beginning and the end of the sequence. For example, suppose that the first element of a sequence is nine, the second element is nine, and likewise the third element is nine. Has the sequence been defined? Not yet. If we additionally specify that the third element is the last element, then the sequence is defined. Once the sequence is defined we can interpret the sequence as a number, 999.

Class

A zero followed by a decimal point followed by an unlimited (in a sense infinite) number of nines may define a class, rather than refer to the contradiction above. The sequences 0.9, 0.99, and 0.999 are all members of this class, meaning that each of these sequences satisfy the constraints of the class. The class itself is a description, not a sequence or a group of sequences.

Limit

One might define the term to refer to the limit of the variable described above. Since the limit is one the term would refer to one. To avoid confusion it would be necessary to define a term referring to the variable.

Rather than define the term to refer directly to the limit, one might consider defining the term to evaluate to the limit. If the term refers to the variable described above then the term does not evaluate to the limit, because the variable refers to a different number each time a nine is added. If the term refers to the contradiction described above then the term does not evaluate to anything. If the term refers to the class described above then again the term does not evaluate since it is a description not a sequence. In order to pursue this course it would be necessary to specify what the term refers to while ensuring that what the term refers to can evaluate to one.

Last revised 25 September 2018

Copyright © 2018 Kevin Cook