Why 0.9999 . . . Equals 1.0000 Part 1

King Zog

When I talked about open and closed intervals to my seventh-graders, I stumbled a bit when it came to talking about 0.99999 . . . as a number. Does that number with 9’s going on forever reside in the open interval (0,1)? That interval is the set of numbers between 0 and 1, not including 0 and 1 (that’s the “open” part). I told the students that for technical reasons we can't say .9999 . . . is in the interval, and in fact we say that 0.99999 . . . = 1.00000. I thought I'd better write up a handout to explain the weirdness. It's something I'd never heard of myself until I was in my 50's, when Professor Christopher Connell told me about it. The handout became this Substack article.

Let’s start with the proposition¹ at hand:

Theorem 1: 0.999 . . . = 1.

Proof. To define the number zog1, set zog1 = 0.3333 . . . Note that that zog1 = 1/3. Three times zog1 equals one, because 3 x (1/3) = 1.

Thus, using the 0.3333 . . . way to write zog1, three times zog1 equals 3 x 0.3333 . . . = 1.

But 3 x 0.3333... = 0.999 . . .

Therefore, it must be that 0.999 . . . = 1.

Q.E.D.²

I chose the name zog1 for the variable for 0.999 . . . in honor of Zog I, King of Albania from 1928 to 1939. Born Ahmet Muhtar Zogolli, he later changed his name to King Zog. He was a favorite of my college debating club, and once when we were close to electing as chairman a junior who was a tad lacking in humility, we instead elected King Zog and elected the junior as Acting Chairman.

The key step of the proof is the statement that 0. 333 . . . = 1/3. Given that, ordinary arithmetic gives us 0.999 . . . = 1. When 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 by 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 when it’s not. It’s built using the idea of infinity. We either have to say that 0.999 . . . is undefined, like 4/0, or let it equal 1, as is done in Theorem 1. What Theorem 1 does is to assume that we want to define and can define 0.999. . . in such a way that arithmetic works on it.

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.

Another way to think about it is that 0.999 . . . is as 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 get all mixed up because there's no way to make it its own number without wrecking arithmetic. Whichever path you take, we get 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:

The excellent 1662 long poem by Michael Wigglesworth 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.

And talk about .0000001.

And this is not something about using base-10. It works in base 2. 1/3 is .010101 . . .

.99999 is 0.11111110011001100110011 . . . in binary

See https://x.com/IsaacKing314/status/1986147865677406655

---

If you liked this Substack, you’d like "I Am the Very Model of a Modern Major-General," Part I” and “The Two-Envelope Paradox: Stated but Not Resolved”.

---

1. My high school classmate A. A., a computer science professor, pointed out something interesting about propositions. In math (and my field of economics too), a proposition is a true statement, usually with a proof attached. In logic, a proposition is any statement, even one known to be false.

2. But consider this objection:

My response to this objection is that the right-hand-side quantity is never changing. As we add a bit to .333, we subtract a bit from .0001. The number .000 . . . 001 makes no sense. If the zeroes go on forever, you never get to the 1.