The Magic Of 0.999… = 1
How a simple proof taught me many life lessons
I recently read a post on Medium about a mathematical problem “Argue In Your Head: 15 divides 2¹⁰⁰⁰⁰-1”. It reminded me of the mathematical fact noted above that 0.999…=1. I actually left a comment there relating the two although next time I’ll try to reduce the typos and mistakes. So I wanted to go over this relationship in more details to share the magic.
Eluding Me As A Child
I remember in 4th grade I was taught long division with recurring decimal fractions. I calculate 1/3 and got 0.333… which I found to be a really interesting relationship. The three from the bottom of the fraction magically becomes the recurring fraction, interesting… Let’s try a few more numbers. And do 1/9 got 0.111…, wow the 1 just started replicating itself. Try some more numbers 2/9, 3/9, 4/9. And then I started looking for the elusive 0.999…, and I couldn’t find it. That magical numerator and denominator eluded me. Despondent I immediately did what any 9 year old would have done, forget about it and play video games.
Finding Those Magical Numbers
Many years later in my Calc II class in college I learned the formula for summing an infinite series. Finally, I had the secret formula that would lead give me this magic numerator and denominator.
And so one sleepless night I decided to find these two elusive numbers. 0.999… can be written as 0.9 + 0.09 + 0.009+… or 9/10 + 9/10² + 9/10³ + … which the formula above gives you 9*(1/(1–1/10)-1). Which I did in my head. And I got 1. No, that didn’t sound right. Tried again. Got the same answer. That was surprising. And so I tried something else.
- 1/9 = 0.111…
- 2/9 = 0.222…
- 3/9 = 0.333…
- 4/9 = 0.444…
- 5/9 = 0.555…
- 6/9 = 0.666…
- 7/9 = 0.777…
- 8/9 = 0.888…
- 9/9, umm, well, I see the pattern but it’s supposed to be 1. Or is it just that the pattern is true. And 0.999…=1
We can try it another way. 1/9=0.111… but what happens if you multiply both sides by 9. On the left side 9/9=1. On the right side, we don’t need to go to start at the rightmost digit (that doesn’t exist for that matter) because each 1 is multiplied by 9 so there is never a remainder. And we get the magical 1 = 0.999… again. That night I slept having solved a mystery that had been there for half of the 18 years of my life. The next morning I used this new found information to annoy David Fajardo at lunch. No, he wasn’t annoyed because I was talking about such a lame topic. He was annoyed because I shattered his world view. He’s only slightly less lame than me.
Why I Cared So Much
0.999… is special. I wanted to find it because it was the largest number I could think of that was less than 1. And here, I was, confronted with the reality that it wasn’t less than 1, close to 1. It was the number 1 itself. A little bit of false hope that I said goodbye to at that time. You see now why David was annoyed at me. I robbed him of the same comfort.
Coming To Terms With Infinity
Later I learned more about infinity. Hilbert’s Hotel and the weirdness of how there are the same number of integers as there are natural numbers. That went against my idea of counting and comparing. But after a lot of consideration I accepted it. And then I accepted that there are as many rational numbers as natural numbers. I started to accept infinity is infinity. And then I got thrown Cantor’s theorem. And that shattered my world view again.
But then the 0.999… = 1 pulled me out of this state of disbelief. Because it had already taught me what I need to know about Cantor’s theorem. By accepting that 0.999… and 1 are not two distinct numbers, I accepted that the only number that was a candidate to be an adjacent number was disqualified. A fundamental aspect of Cantor’s theorem is there are an infinite number of real numbers between any two numbers.
And this goes back to the idea of limits in Calculus. You can get closer and closer to a number, and there are still an infinite number of numbers still close to it. 0.999… = 1 is a reminder on why we use this complicated notation. Because it is a way to express the reality that as close as we get, between any two distinct numbers there are an infinite number of rational numbers. And there no such thing as an adjacent number.
There is something truly beautiful about this relationship 0.999… = 1. It’s something that’s actually unnatural to accept but if you accept it, you can find multiple proofs for it that are extremely simple and elegant. And accepting this proof is accepting something that we don’t want to believe and we cling to. But when your worldview is getting shattered it reminds you that you’re learning what you always knew. And it makes it so much easier to accept reality. And when you accept that reality, you find there were other things you didn’t quite accept that you didn’t think were a problem. And so all of your life is improved. And that is the magic of 0.999 = 1. It is not a fun little theorem. It is a beautiful life lesson.
