Yet another failed attempt at showing 0.999…≠1
I’ve discussed before how mathematics can sometimes lead to very counterintuitive results. One of the most common, and famous, of these counterintuitive properties of math is that the number 0.999… (that is, zero point nine, nine, nine, repeating) is equal to 1. This one is so well known that it is fairly often taught even to Elementary and High School students. If you are unfamiliar with this discussion, I highly recommend that you watch this video from Vi Hart, in which she discusses 10 different reasons to accept this concept. Additionally, you may have fun watching this video, in which she lampoons the common objections to the concept.
Despite the fact that it is fairly simple to prove that 0.999…=1, the concept is so counterintuitive that I find people try to struggle against it– even when they know and accept the reasoning behind the equality. One such attempt comes from Presh Talwalkar. In the following video, Mr. Talwalkar attempts to demonstrate that on the Surreal number system, 0.999…≠1.
Unfortunately for Mr. Talwalkar, he is wrong. Even on the Surreals, it is still true that 0.999…=1.
In the video, Mr. Talwalkar acknowledges that it is absolutely true that 0.999…=1 on the Real numbers. However, he then asserts that it is not true that 0.999…=1 on the Surreal numbers. Right away, this should look fairly suspect to anyone familiar with the Surreals. The reason for that is that the Surreals are a superset which contains the Real numbers. Anything which is true for a number which exists within the Real numbers will similarly be true for that number in the Surreals. It is therefore entirely incoherent to claim that 0.999… is a different number in the Surreals than it is in the Reals.
Most of Mr. Talwalkar’s video is fairly accurate, though I would prefer a more rigorous treatment of its subject matter. The point where it goes wrong, however, comes when he attempts to discuss some “weird numbers” which can be constructed on the surreals. He begins by discussing , which he erroneously claims to be “1 divided by infinity” and “point zero repeating, with a one at the end.” Neither of these descriptions is even coherent. Infinity is not a number. You cannot divide 1 by infinity any more than you can divide 1 by Blue, or by Sweet, or by Alexander Hamilton. The latter description “point zero repeating, with a one at the end” is quite obviously self-contradictory. If we have “point zero repeating,” then there is no “end” at which to write a “one.” That said, the Surreal number which Mr. Talwalkar defines here, , is an actual number and can be utilized in mathematics. As defined, represents a number which is greater than zero, but smaller than all of the Real numbers. That is to say, for any Real number , it is true that .
He continues by defining another "weird number," . Again, this number is correctly defined, and it is an actual number. It is therefore possible to deduce certain properties which are possessed by . For example, since we know that , we can know with certainty that . However, Mr. Talwalkar oversteps the bounds of logic when he baldly asserts that "you can think about it as point nine repeating." He gives absolutely no justification for asserting that , and it is actually quite simple to prove that this assertion is, in fact, entirely untrue. I shall do so, now, by Reductio ad Absurdum.
Let’s start by assuming Mr. Talwalkar’s assertion to be true. Then, by exploring the properties of the numbers, I’ll show that this assertion leads to a logical contradiction, and that it therefore cannot be true.
- Given (1) and (2), and therefore
- Given (4) and (6),
- Given (5) and (7),
- Given (3) and (8),
This, of course, is nonsensical. A number must always equal itself. Therefore, our premise (1) cannot be true.
It doesn’t matter whether we are talking about the Rational numbers or the Real numbers or the Hyperreal numbers or the Surreal numbers. The simple fact of the matter is that 0.999… is equal to 1. For some reason, many people find this very difficult to accept, but it is absolutely true. Presh Talwalkar’s attempt to show otherwise fails just as surely as all those which came before it. To quote Vi Hart on the issue:
If you’re having Math problems, I feel bad for you, son.
I got 98.999… problems, but 0.999… equals 1.