# Why 0.9999... Equals 1.0000...

In the 7th-grade math class I teach, when we discussed closed and open intervals I stumbled a bit when it came to talking about 0.99999... possibly being the edge of an open interval. I told the students that for technical reasons we can't say that, and 0.99999... = 1, but to explain why would be too hard. Job Currell found out why from his brother Isaac, and we talked about it in class, which was good, but I thought I'd better write up a handout to explain this weirdness. It's something I'd never heard of myself before I was in my 50's, when Professor Christopher Connell told me about it. That handout became this Substack article.

Theorem 1:0.999... = 1.

Proof.Define the numberzog1aszog1= 1/3 = 0.3333... Three timeszog1equals one, right?--- because 3 x (1/3) = 1.Thus, using the 0.3333... way to write

zog1, three timeszog1equals 3 x 0.3333... = 1.But 3 x 0.3333... = 0.999...

Therefore, it must be that 0.999... =1. Q.E.D.

I chose the name *zog1* for the variable for 0.999... in honor of Zog I, King of Albania from 1928 to 1939, when Italy conquered his country in World War II. He was born Ahmet Muhtar Zogolli, but changed his name to Zog. He was a favorite of my college debating club, and, in fact, when we were close to electing a chairman whose was a fine fellow but lacked humility, we instead elected King Zog and elected him as merely the Acting Chairman.

Mathematicians and math teachers have thought a lot about .999..., as you can see from the Wikipedia article. That article seems to me to make it unnecessarily complicated. I don't like the Archimidean Principle proofs there that are based on the number line and finding a least upper bound, since I don't see why making the least upper bound 1 matters to 0.999..., which it seems to me could be *less* than the least upper bound, whereas the argument assumes that it *is* the least upper bound. I think that proof is assuming its answer, by defining 0.999... in a certain way.

I don't see what's nonrigorous about the proof I gave of Theorem 1. It's the only way to make our ordinary rules of arithmetic work for 0.999..., a pretty convincing reason to accept that 0.999... = 1. Whenever we work with infinity in math, we have to be careful, because it's not a genuine number--- it's shorthand for "and then keep going with whatever you're doing forever". We don't say that 4/0 equals infinity, we just leave 4/0 undefined, because if we say that 4/0 = infinity, and 5/0 = infinity too, we end up having to say that 4 = 5 if we follow the ordinary rules of arithmetic. We can save arithmetic by saying that 0.999... = 1. It's weird, but that's because we think of 0.999... as a genuine number, but it really is not, since it's built using the idea of infinity. We either have to say that 0.999...is undefined, like 4/0, or define it to equal 1, as is done in Theorem 1. What Theorem 1 is doing is assuming implicitly-- that is, assuming without coming out and saying it is making that assumption-- that we want to define and can define 0.999... in such a way that arithmetic works on it.

Another way to think about it is that 0.999... is just another way to write "1", just as I defined *zog1* earlier as another way to write .333... If you think about it as a separate number, you will get all mixed up because there's no way to make it its own number without wrecking arithmetic.

Also, if you just think of it as another way to write "one", you can make a good joke:

Question: How many mathematicians does it take to screw in a lightbulb?

Answer: 0.999999...

[This article continues off of Substack at https://rasmusen.org/special/Cedars_School/04.05_3.99999_equals_4.pdf since I need to use a lot of mathematical notation, for which Substack is not well suited but LaTex and pdf are.]

p.s.: I ought to add this in somehow:

See the excellent and underrated 1662 poem by Michael Wigglesworth, which contains a useful stanza on infinity:

Naught join’d to naught can ne’er make aught,

nor Cyphers make a Sum;

Nor things finite, to infinite

by multiplying come:

A Cockle-shell may serve as well

to lade the Ocean dry

As finite things and reckonings

to bound Eternity.