Taking Down A Basic Proof

The first time I heard the idea that .999 = 1, it was explained to me like this:

1/3 = .333…

(1/3)3 = (.333…)3

1 = .999…

One divided by 3 equals point 3 repeating. If you multiply both sides by 3, you get 1 equals point 9 repeating.

When I first saw this proof, my mind was blown. I had heard that .999 repeating was exactly equal to 1 rather than almost, but I had never seen it proven. This was proof. I shared this with several people, and I was so proud to be “in the know”. But upon further examination, I see that there are a couple of problems with this seemingly easy math problem as I will demonstrate in this post.

The first is that 1 divided by 3 does not equal point 3 repeating. It’s almost equal. The explanation is long but educational. Bear with me on this one.

When we were kids first learning about division, we were taught a mathematical model that I’ll refer to as the Remainder Model. When we divided 10 by 3, we got “3 with a remainder of 1” (written as 3R1). If you wanted to check the math, you’d do it in reverse by multiplying by the number you divided by and then adding the remainder, so 3(3) + 1 gives you 10. This allowed us to divide uneven numbers exactly.

When we got older we replaced the Remainder Model with the Decimal Model. Instead of stopping at 3R1, we’d drop a zero from behind our original 3 (because it’s actually 3.000…) and then divide our remainder of 1 with a 0 (10) by 3 again, giving us 3.3. Now while we didn’t do this in school (at least I know *I* didn’t), we could have used both models to get an exact answer… all you’d have to do is carry the remainder as many times as you drop zeroes from behind the 3. Dividing 10 by 3 just a couple of times gave us 3.3R.1 (checked by multiplying 3(3.3) to get 9.9 and then adding the remainder of .1 to get 10), and doing it three times gave us 3.33R.01 and doing it ten times gave us 3.333333333R.000000001. Through inductive reasoning, we can see that if you divided 10 by 3 an infinite number of times you’d get 3.333…R.000…01. That remainder may get infinitely small, but it never disappears — no matter how many times you do the division, it will always carry a remainder.

Thus, 10 divided by 3 does not equal 3.333… but rather almost equals 3.333… because there’s a remainder. It’s easy to see why this is also true of dividing 1 by 3. In our proof above, we don’t actually get 1 = .999… but rather 1 = .999… with a remainder of .000…01.

I said above that there were “a couple of problems”, and this rounding error is just the first. The second one is with multiplying .333… by 3. “Why would that be a problem?” you may ask. Well, I don’t see it as a personal problem, but rather one for my opponents. There are those who would “reify” .333… (see my last post) and suggest that this problem is unsolvable because multiplication requires that both numbers have a finite end. After all, how else would you even start such a math problem? The answer is easy — through inductive reasoning. We know that any finite string of 3s multiplied by 3 will give us an equally long string of 9s, and there’s no reason to think this will change just because the string of 3s is infinitely long. But this is a problem for some people, and I’ve seen this proof less and less over time because mathematicians have realized that allowing this multiplication creates an obvious double standard. While this may seem a trivial problem compared to the first one, it is major enough to prevent many mathematicians from even offering this proof any more.

One final note: While I welcome criticism, please don’t tell me that my math is wrong unless you can suggest how it could have been done right. Don’t tell me that there is no remainder when you divide 1/3 unless you can demonstrate how it is done evenly. If you can’t offer an alternative explanation, there’s a good chance that there isn’t one.

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I’d like to answer an objection here, one that I hear commonly and that I’m sure I’ll hear again (even despite this post): people say that a number like 1.999…98 is impossible. If you aren’t familiar with that number, it is 1 point 9 repeating followed by a single 8, found by multiplying point 9 repeating by 2. I’ve explained how I came to this conclusion (inductive reasoning) and why it is consistent with what we know about math, but people like to think I’m being inconsistent with the definition of infinity.

You may have noticed the title of this post and thought, “What is reification?” Reification is a logical fallacy in which an abstract idea is treated like a physical object. For example, many people talk about feelings being “bottled up”, and suggest that letting out anger through venting will reduce the anger that you have. Studies have proven this wrong, because anger is not a physical object that can be reduced by “getting rid of it” through expressing it.

The problem with this objection to my method is that mathematicians who think that it’s impossible to “bookend” an infinite series of 9’s with a decimal point and an 8 will bookend other infinities without batting an eye. Consider the number of real numbers between 0 and 1. The obvious answer is “they’re infinite“, despite the existence of a floor and a ceiling to this series of numbers. One might argue that I’m arguing apples and oranges — the series between 0 and 1 is made up of numbers, not digits. So let’s convert that to digits. If you took all the real numbers between 0 and 1 and listed them from least to greatest, then drew a line crossing the first digit of each number, your result would be 01234567891, where every number except that final 1 would be repeated infinitely. Would this new number be consider a “real number”? “Sure!”, say proponents of Cantor’s diagonal argument, which creates a real number through almost identical means.

Do I agree that the number created is a real number? No, but that’s not the point. It’s not important to agree that a real number is made here, but instead to focus on the fact that the digits in this number are indisputably infinite, even though they end abruptly with other infinite series of digits and finally with one finite digit. This is because we’re not talking about a physical reality but a thought experiment. Numbers are just abstract ideas that we use to describe the relationships between mathematical models such as graphed lines, sets of objects, and sometimes just other numbers. While it may offend your sensibilities to try to grasp a concept like .000….01 (point zero repeating ending with a single one), there is no logical reason that such a number can’t exist.

One final note: While I welcome comments, please don’t tell me I’m wrong unless you can explain how I could have been right. If you can’t come up with an alternative explanation, there’s a good chance that there isn’t one.

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A Simple Proof

It’s almost too easy to explain why .999… does not equal 1. Even fools can see the reason right away — it appears to take a college education to get it wrong, as counter-intuitive as that may be.

If .999… = 1, and X – X = 0, then 1 – .999… should equal 0. It doesn’t. You get .000…01. How do I know this? Inductive reasoning.

1.00 – .99 = .01

1.00000 – .99999 = .00001

1.0000000000 – .9999999999 = .0000000001

1 minus point 9 repeating for any length equals point 0 repeating for one digit shorter than the string of 9s followed by a single 1.

There’s no reason to suppose that this pattern suddenly changes just because the string of 9s is infinitely long. For any finite length this is true, so for an infinite length it should still be true.

But let’s not stop there. If .999… = 1 then X squared = X squared. Does it? No.

.999 squared = .998001

.99999 squared = .9999800001

.999… squared = .999…980….0001

None of these equals 1 squared (which is 1). Again, I’ve demonstrated that any finite length of 9’s squared does not even come very close to being 1. It’s true that it gets closer as the string gets longer, but it will never change to become 1.000… No matter how long the number is, you’ll still be multiplying a 9 by a 9, and that will give you a 1, not a zero.

It’s even more revealing when you multiply both sides of the equation by 2.

.999…(2) = 1.999…98

1(2) = 2

Here you see that the distance between both sides doubles, which is consistent with any unbalanced equation. Take any two numbers, multiply both of them by 2, and you’ll always find that the difference between them also doubles (because if X-Y=Z, then 2X-2Y=2Z, where X and Y are any two numbers and Z is their difference).

There is literally nothing you can do to both sides of the equation .999… = 1 and have them still coming out equal. Many people assume that 9.999… = 10 (by multiplying both sides by 10) or that 3.999… = 4 (by adding 3 to both sides), but this is a logical fallacy known as “begging the question“… the conclusion is derived from prior belief that .999… = 1, not because it’s otherwise provable that these equations are true.

One final note: While I welcome criticism, please don’t tell me that I’m doing these problems wrong unless you can explain how they are done right. Don’t tell me that .999… multiplied by 2 does not equal 1.999…98 unless you can explain how those numbers are multiplied correctly and come out to a figure that supports your conclusion. If you can’t offer an alternative explanation, there’s a good chance that there isn’t one.

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Inductive Reasoning

Throughout this blog, I’ll refer many times to getting my answers through inductive reasoning, and it will probably benefit you to know what that is. Inductive Reasoning is, along with deductive reasoning, one of the two methods of figuring things out with logic. While deduction makes a more airtight case than induction, it is still an extremely useful tool that would be foolish to ignore. It is frequently used in mathematics, and may perhaps be necessary for any problem involving infinite numbers.

A common example of inductive reasoning is used in figuring out whether the sun will rise tomorrow. It rose this morning, and yesterday morning, and every morning of my life, and every recorded instance of mornings in history. Because the pattern fits so many intervals uninterrupted and without a single pattern break, it is said to be a strong inductive argument.

Mathematicians also use inductive reasoning to figure out, for example, what you get when you divide 1 by 3. We all know it’s (roughly) .333… , which is read as “point 3 repeating”. How does one reach this conclusion? We can’t keep dividing 10 by 3, dropping the remainder of 1, adding a 0, and dividing again by 3 over and over again. There isn’t enough time to do the whole problem, and it would be pointless to do it infinitely anyway because we all spot the pattern rather quickly. Because we know that this pattern will repeat every time without fail, we inductively reason that, no matter how many digits we add to the problem, we’ll get nothing but threes.

It’s important to note that this pattern doesn’t have a reason to break or interrupt — what is true about it at the beginning is true at the end, even if we keep it up into infinity. Strangely, some mathematicians would prefer to believe that the pattern does suddenly change when a number becomes infinitely long. Why? We’ll explore that in further blog posts, but the important thing to note here is that inductive reasoning will tell us what a pattern will do for any length of math problem, even an infinitely long one, and they are all strong arguments because there are an infinite number of examples in which each one works.

One final note: While I welcome criticism, please do not claim that I am wrong unless you can explain how I could have been right. If you don’t believe that inductive reasoning is useful for figuring out a problem like 1/3, please tell me how one does figure out such a problem. If you don’t have an alternative, there’s a good chance that there isn’t one.

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