Gentleman Gustaf here to talk to you about changing. I've had a lot of success getting very good at a lot of things, and I credit it to one fact: I'm not afraid to be wrong. In fact, when I look back on my life, I want to say 'man, I was wrong a lot'. The emphasis, of course, is on the 'was'. Rejecting wrong opinions in favor of less obviously wrong answers is the main step on the path to having right opinions. I'm not the most talented person I know, and as much as I'd like to emulate Will Smith, I'm not the hardest working person I know. But as an ex-philosophy major, I know what it means to prove or disprove a point, and I follow up on it. A problem facing a lot of League of Legends players is simple: they are unwilling to be proven wrong.Widespread belief that 'I'd be at least X Elo but for Elo Hell' seems a direct result of people's inability to figure out when they are wrong instead of others, for example. When I first played League of Legends, I got Manamune on Veigar (I didn't even know AP was a thing). Obviously a lot has changed since then. I would not want to have stubbornly stuck to that build, or now, I would be a much worse player. On the other hand, a splash of attack speed on Cho, once considered trolly (and one of my preferred styles on him at the time), was played by HotshotGG successfully and became so standard that it became his new recommended build in the shop. I've been jungling Cho'Gath since almost 2 years ago, and it only recently became considered a strong build. Many strong builds were once considered weak, and it's only with time and exposure they become accepted. So how do we differentiate these good but hidden builds from bad builds? And how should we respond?
Knowledge vs Intuition
There is an important distinction between knowledge and intuition. Often times, our intuition lies to us. Quantum Mechanics and Relativity are very unintuitive concepts, and yet they are very true. The problem comes when people are given 'knowledge' that violates their intuition. They have to choose one or the other. I argue that intuition should guide you to ideas, but that you need to test those ideas many ways. As such, in many ways, the road to getting better is the road to denying intuition when it gets in the way. As such, today, I am going to revisit one of the longest running internet debates. It is a long standing debate because to anybody who knows much math, it is obviously true, and yet it violates a lot of people's intuitions. The proofs for it are very easy to understand, but people's intuitions are so strongly against it that they will force themselves to find 'reasons' to dispute these simple proofs; reasons which have no mathematical backing.
3 Reasons why .999... = 1
Take the number represented by .999..., also represented as .999 repeating. Loosely, this refers to the number representation of an infinite number of 9s following a decimal, although a much more common representation for this number is indeed, 1. How?
Reason #1 - The Number Theory Argument
The first argument is very simple: given 2 distinct real numbers, there must be an infinite number of numbers in between them. In between .8 and .9, for example, we have .81, .82. .83, .84, .85, .86, .87, .88, and .89. But in between .88 and .89 we have .881, .882, .883 and so on... We can continue this process ad infinitum. So what number lies between .999... and 1? Well, each digit of .999 is a 9. Before, with .8, we could simply add a number to the end of it. But .999... has no 'end', that strategy won't work here. Anywhere in the number we insert a number, we have 2 options. We can insert a 9 (leaving it the same number), or we can insert a non-9 (leaving it a smaller number). This can be stated in another way by asking what 1-.999... equals. I can already hear somebody trying to suggest .000...1, but it's very easy to show why you can't add a number to the end of an infinitely repeating decimal. Take .000...1. Now half it. You should be left with .000...05, also known as .000...5. And so if we allow appending numbers to the end of repeating decimals, we also allow half of a number to equal 5 times that number. Uh-oh.
Algebraic (1/3*3 or 1/11+10/11)
1/3 = .333...
In calculus, we have what's called a geometric sum. The most common example is that presented in one of Zeno's paradoxes.
You have a starting number, a number you are multiplying it by repeatedly, and a sum of each result. So, for example, let's take 9/10 as our base number, and 1/100 as our multiplier. The first number in our sum will be .9 (9/10). The next number will be .009 (9/10 * 1/100). The next will be .00009 (9/10 * 1/100 * 1/100), and so on. If we add all these numbers together, we get .909090. According to calculus, geometric series' sum to a/(1-r), where A is the first number, and r is the multiplier. This gives us a sum of 9/10/(1-1/100), or 10/11. Luckily enough, .909090 = 10/11, so we can keep using math. So when we change the multiplier to 1/10, we instead get .9, .09, .009, and so on. This gives us the series .999... And what is a/(1-r)? 9/10/(1-1/10). Or 1.
This also resolves Zeno's paradox, because with an a of 1/2, and an r of 1/2, we get a sum of 1/2/(1-1/2), or also 1. In other words, the turtle finishes the race.
Upper and Lower Bound Proofs
By appeal to upper and lower bounds, we can come by a most interesting proof that .999... = 1 = 1.000...1
We can define 1 as follows: it is both the highest number in the set bounded by 0 and 1 [0,1], and it is the lowest number in the set bounded by 1 and 2 [1,2] We can continue to decrease the size of those sets by 1/10. So 1 is also the highest number in the set bounded by [.9,1], and [.99,1], and [.999,1]. In fact, for any finite number of 9s, .999... will be greater than the lower bound. That is, picking any number less than 1, .999... is greater than every number less than 1. .999... must either be larger than 1, equal to 1, or less than 1. However, since it is greater than every number less than 1, it cannot itself be less than 1 (or it would be greater than itself). It is obviously not larger than 1, and so the only remaining option is that it equal 1.
For some of you, there will inevitably be a voice in the back of your head that says 'but .999... can't equal 1. They're different numbers'. First of all, the same number can have different representations (1.5=1+1/2=2-.5=.75*2=3/2). But more important than that is the following advice. Crush that voice. When it says 'but I don't like that', shut it up, because reasons always trump intuitions. Now, there can be bad reasons, but that is just another reason for us to train our ability to reason.
To give a LoL related example, recently, I encountered the following AD/support synergy/counter chart. And while I liked a lot of what I saw, I thought things like 'what about X combination?' 'how can X and Y champions counter each other?' and eventually 'what was this chart-maker thinking on some of these?' In other words, I had an intuition. But once I discovered that the chart was made by Spellsy, a very talented support player who has personally beaten me in lane, I started thinking 'he must have some reason, I must figure out what it is'. This caused me to question my intuition. Luckily, those reasons are available here, so I didn't have to wonder for long. This allowed me to abandon my intuition, having good reason to do so.
But Gentleman Gustaf, every week, you propose something that isn't always backed up by high Elo play!
There is one reason I am confident with my AD Carry series. I have math to back it up. If my math is disputed, I have no legs on which to stand. Moreover, sometimes there are non-mathematical disputes which render the math non-representative, such as with Black Cleaver, where you can't necessarily count on stacking the debuff on one target.
Nobody gets bonus points for sticking to their opinion. Nobody gets retroactively moved up to high Elo for having picked a jungler or an item combo or a strategy before it became common. The way you get better is by finding better players, seeing what they do, understanding why they do it, and adopting it. Of course, the other option is to be one of those innovators, but they are such a small percentage of the population that odds are, if you've come up with a new idea, you're one of the crazies, not one of the innovators. So pop open a stream, and watch some high Elo games (Solo/Duo Queue, scrims, and tournaments), and don't be afraid to copy what you see, if it's different than what you do (unless you get the idea that they're trolling; I'm looking at you, Saintvicious jungling Veigar). There's no shame in copying the pros, because, after all, they are pros.
EDIT: There are plenty of historical examples of people sticking to their guns and being right, but they had compelling reasons to do so. The Discovery of Neptune is one of the best reasons. But intuition has a very low batting average when it comes to math and sciences. One of the greatest minds the world has ever seen, Albert Einstein, disbelieved in certain aspects of Quantum Mechanics on intuitive grounds, the most famous being 'God does not play dice with the universe'. However, the predictions of QM have become increasingly accurate. In this case, intuition may be strong, but reasons become hard to refute.
A video on this topic I made from a little while back. It may reveal how embarrassingly bad I was at the game 8 months ago, and who doesn't like that?