關於 math question ，我們在網路上蒐集到這些相關的討論、資訊與評價

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📜 [專欄新文章] Uniswap v3 Features Explained in Depth

✍️ 田少谷 Shao

📥 歡迎投稿： https://medium.com/taipei-ethereum-meetup #徵技術分享文 #使用心得 #教學文 #medium

Once again the game-changing DEX 🦄 👑

Image source: https://uniswap.org/blog/uniswap-v3/

Outline

0. Intro1. Uniswap & AMM recap2. Ticks 3. Concentrated liquidity4. Range orders: reversible limit orders5. Impacts of v36. Conclusion

0. Intro

The announcement of Uniswap v3 is no doubt one of the most exciting news in the DeFi place recently 🔥🔥🔥

While most have talked about the impact v3 can potentially bring on the market, seldom explain the delicate implementation techniques to realize all those amazing features, such as concentrated liquidity, limit-order-like range orders, etc.

Since I’ve covered Uniswap v1 & v2 (if you happen to know Mandarin, here are v1 & v2), there’s no reason for me to not cover v3 as well ✅

Thus, this article aims to guide readers through Uniswap v3, based on their official whitepaper and examples made on the announcement page. However, one needs not to be an engineer, as not many codes are involved, nor a math major, as the math involved is definitely taught in your high school, to fully understand the following content 😊😊😊

If you really make it through but still don’t get shxt, feedbacks are welcomed! 🙏

There should be another article focusing on the codebase, so stay tuned and let’s get started with some background noise!

1. Uniswap & AMM recap

Before diving in, we have to first recap the uniqueness of Uniswap and compare it to traditional order book exchanges.

Uniswap v1 & v2 are a kind of AMMs (automated market marker) that follow the constant product equation x * y = k, with x & y stand for the amount of two tokens X and Y in a pool and k as a constant.

Comparing to order book exchanges, AMMs, such as the previous versions of Uniswap, offer quite a distinct user experience:

AMMs have pricing functions that offer the price for the two tokens, which make their users always price takers, while users of order book exchanges can be both makers or takers.

Uniswap as well as most AMMs have infinite liquidity¹, while order book exchanges don’t. The liquidity of Uniswap v1 & v2 is provided throughout the price range [0,∞]².

Uniswap as well as most AMMs have price slippage³ and it’s due to the pricing function, while there isn’t always price slippage on order book exchanges as long as an order is fulfilled within one tick.

In an order book, each price (whether in green or red) is a tick. Image source: https://ftx.com/trade/BTC-PERP

¹ though the price gets worse over time; AMM of constant sum such as mStable does not have infinite liquidity

² the range is in fact [-∞,∞], while a price in most cases won’t be negative

³ AMM of constant sum does not have price slippage

2. Tick

The whole innovation of Uniswap v3 starts from ticks.

For those unfamiliar with what is a tick:

Source: https://www.investopedia.com/terms/t/tick.asp

By slicing the price range [0,∞] into numerous granular ticks, trading on v3 is highly similar to trading on order book exchanges, with only three differences:

The price range of each tick is predefined by the system instead of being proposed by users.

Trades that happen within a tick still follows the pricing function of the AMM, while the equation has to be updated once the price crosses the tick.

Orders can be executed with any price within the price range, instead of being fulfilled at the same one price on order book exchanges.

With the tick design, Uniswap v3 possesses most of the merits of both AMM and an order book exchange! 💯💯💯

So, how is the price range of a tick decided?

This question is actually somewhat related to the tick explanation above: the minimum tick size for stocks trading above 1$ is one cent.

The underlying meaning of a tick size traditionally being one cent is that one cent (1% of 1$) is the basis point of price changes between ticks, ex: 1.02 — 1.01 = 0.1.

Uniswap v3 employs a similar idea: compared to the previous/next price, the price change should always be 0.01% = 1 basis point.

However, notice the difference is that in the traditional basis point, the price change is defined with subtraction, while here in Uniswap it’s division.

This is how price ranges of ticks are decided⁴:

Image source: https://uniswap.org/whitepaper-v3.pdf

With the above equation, the tick/price range can be recorded in the index form [i, i+1], instead of some crazy numbers such as 1.0001¹⁰⁰ = 1.0100496621.

As each price is the multiplication of 1.0001 of the previous price, the price change is always 1.0001 — 1 = 0.0001 = 0.01%.

For example, when i=1, p(1) = 1.0001; when i=2, p(2) = 1.00020001.

p(2) / p(1) = 1.00020001 / 1.0001 = 1.0001

See the connection between the traditional basis point 1 cent (=1% of 1$) and Uniswap v3’s basis point 0.01%?

Image source: https://tenor.com/view/coin-master-cool-gif-19748052

But sir, are prices really granular enough? There are many shitcoins with prices less than 0.000001$. Will such prices be covered as well?

Price range: max & min

To know if an extremely small price is covered or not, we have to figure out the max & min price range of v3 by looking into the spec: there is a int24 tick state variable in UniswapV3Pool.sol.

Image source: https://uniswap.org/whitepaper-v3.pdf

The reason for a signed integer int instead of an uint is that negative power represents prices less than 1 but greater than 0.

24 bits can cover the range between 1.0001 ^ (2²³ — 1) and 1.0001 ^ -(2)²³. Even Google cannot calculate such numbers, so allow me to offer smaller values to have a rough idea of the whole price range:

1.0001 ^ (2¹⁸) = 242,214,459,604.341

1.0001 ^ -(2¹⁷) = 0.000002031888943

I think it’s safe to say that with a int24 the range can cover > 99.99% of the prices of all assets in the universe 👌

⁴ For implementation concern, however, a square root is added to both sides of the equation.

How about finding out which tick does a price belong to?

Tick index from price

The answer to this question is rather easy, as we know that p(i) = 1.0001^i, simply takes a log with base 1.0001 on both sides of the equation⁴:

Image source: https://www.codecogs.com/latex/eqneditor.php

Let’s try this out, say we wanna find out the tick index of 1000000.

Image source: https://ncalculators.com/number-conversion/log-logarithm-calculator.htm

Now, 1.0001¹³⁸¹⁶² = 999,998.678087146. Voila!

⁵ This formula is also slightly modified to fit the real implementation usage.

3. Concentrated liquidity

Now that we know how ticks and price ranges are decided, let’s talk about how orders are executed in a tick, what is concentrated liquidity and how it enables v3 to compete with stablecoin-specialized DEXs (decentralized exchange), such as Curve, by improving the capital efficiency.

Concentrated liquidity means LPs (liquidity providers) can provide liquidity to any price range/tick at their wish, which causes the liquidity to be imbalanced in ticks.

As each tick has a different liquidity depth, the corresponding pricing function x * y = k also won’t be the same!

Each tick has its own liquidity depth. Image source: https://uniswap.org/blog/uniswap-v3/

Mmm… examples are always helpful for abstract descriptions 😂

Say the original pricing function is 100(x) * 1000(y) = 100000(k), with the price of X token 1000 / 100 = 10 and we’re now in the price range [9.08, 11.08].

If the liquidity of the price range [11.08, 13.08] is the same as [9.08, 11.08], we don’t have to modify the pricing function if the price goes from 10 to 11.08, which is the boundary between two ticks.

The price of X is 1052.63 / 95 = 11.08 when the equation is 1052.63 * 95 = 100000.

However, if the liquidity of the price range [11.08, 13.08] is two times that of the current range [9.08, 11.08], balances of x and y should be doubled, which makes the equation become 2105.26 * 220 = 400000, which is (1052.63 * 2) * (110 * 2) = (100000 * 2 * 2).

We can observe the following two points from the above example:

Trades always follow the pricing function x * y = k, while once the price crosses the current price range/tick, the liquidity/equation has to be updated.

√(x * y) = √k = L is how we represent the liquidity, as I say the liquidity of x * y = 400000 is two times the liquidity of x * y = 100000, as √(400000 / 100000) = 2.

What’s more, compared to liquidity on v1 & v2 is always spread across [0,∞], liquidity on v3 can be concentrated within certain price ranges and thus results in higher capital efficiency from traders’ swapping fees!

Let’s say if I provide liquidity in the range [1200, 2800], the capital efficiency will then be 4.24x higher than v2 with the range [0,∞] 😮😮😮 There’s a capital efficiency comparison calculator, make sure to try it out!

Image source: https://uniswap.org/blog/uniswap-v3/

It’s worth noticing that the concept of concentrated liquidity was proposed and already implemented by Kyper, prior to Uniswap, which is called Automated Price Reserve in their case.⁵

⁶ Thanks to Yenwen Feng for the information.

4. Range orders: reversible limit orders

As explained in the above section, LPs of v3 can provide liquidity to any price range/tick at their wish. Depending on the current price and the targeted price range, there are three scenarios:

current price < the targeted price range

current price > the targeted price range

current price belongs to the targeted price range

The first two scenarios are called range orders. They have unique characteristics and are essentially fee-earning reversible limit orders, which will be explained later.

The last case is the exact same liquidity providing mechanism as the previous versions: LPs provide liquidity in both tokens of the same value (= amount * price).

There’s also an identical product to the case: grid trading, a very powerful investment tool for a time of consolidation. Dunno what’s grid trading? Check out Binance’s explanation on this, as this topic won’t be covered!

In fact, LPs of Uniswap v1 & v2 are grid trading with a range of [0,∞] and the entry price as the baseline.

Range orders

To understand range orders, we’d have to first revisit how price is discovered on Uniswap with the equation x * y = k, for x & y stand for the amount of two tokens X and Y and k as a constant.

The price of X compared to Y is y / x, which means how many Y one can get for 1 unit of X, and vice versa the price of Y compared to X is x / y.

For the price of X to go up, y has to increase and x decrease.

With this pricing mechanism in mind, it’s example time!

Say an LP plans to place liquidity in the price range [15.625, 17.313], higher than the current price of X 10, when 100(x) * 1000(y) = 100000(k).

The price of X is 1250 / 80 = 15.625 when the equation is 80 * 1250 = 100000.

The price of X is 1315.789 / 76 = 17.313 when the equation is 76 * 1315.789 = 100000.

If now the price of X reaches 15.625, the only way for the price of X to go even higher is to further increase y and decrease x, which means exchanging a certain amount of X for Y.

Thus, to provide liquidity in the range [15.625, 17.313], an LP needs only to prepare 80 — 76 = 4 of X. If the price exceeds 17.313, all 4 X of the LP is swapped into 1315.789 — 1250 = 65.798 Y, and then the LP has nothing more to do with the pool, as his/her liquidity is drained.

What if the price stays in the range? It’s exactly what LPs would love to see, as they can earn swapping fees for all transactions in the range! Also, the balance of X will swing between [76, 80] and the balance of Y between [1250, 1315.789].

This might not be obvious, but the example above shows an interesting insight: if the liquidity of one token is provided, only when the token becomes more valuable will it be exchanged for the less valuable one.

…wut? 🤔

Remember that if 4 X is provided within [15.625, 17.313], only when the price of X goes up from 15.625 to 17.313 is 4 X gradually swapped into Y, the less valuable one!

What if the price of X drops back immediately after reaching 17.313? As X becomes less valuable, others are going to exchange Y for X.

The below image illustrates the scenario of DAI/USDC pair with a price range of [1.001, 1.002] well: the pool is always composed entirely of one token on both sides of the tick, while in the middle 1.001499⁶ is of both tokens.

Image source: https://uniswap.org/blog/uniswap-v3/

Similarly, to provide liquidity in a price range < current price, an LP has to prepare a certain amount of Y for others to exchange Y for X within the range.

To wrap up such an interesting feature, we know that:

Only one token is required for range orders.

Only when the current price is within the range of the range order can LP earn trading fees. This is the main reason why most people believe LPs of v3 have to monitor the price more actively to maximize their income, which also means that LPs of v3 have become arbitrageurs 🤯

I will be discussing more the impacts of v3 in 5. Impacts of v3.

⁷ 1.001499988 = √(1.0001 * 1.0002) is the geometric mean of 1.0001 and 1.0002. The implication is that the geometric mean of two prices is the average execution price within the range of the two prices.

Reversible limit orders

As the example in the last section demonstrates, if there is 4 X in range [15.625, 17.313], the 4 X will be completely converted into 65.798 Y when the price goes over 17.313.

We all know that a price can stay in a wide range such as [10, 11] for quite some time, while it’s unlikely so in a narrow range such as [15.625, 15.626].

Thus, if an LP provides liquidity in [15.625, 15.626], we can expect that once the price of X goes over 15.625 and immediately also 15.626, and does not drop back, all X are then forever converted into Y.

The concept of having a targeted price and the order will be executed after the price is crossed is exactly the concept of limit orders! The only difference is that if the range of a range order is not narrow enough, it’s highly possible that the conversion of tokens will be reverted once the price falls back to the range.

As price ranges follow the equation p(i) = 1.0001 ^ i, the range can be quite narrow and a range order can thus effectively serve as a limit order:

When i = 27490, 1.0001²⁷⁴⁹⁰ = 15.6248.⁸

When i = 27491, 1.0001²⁷⁴⁹¹ = 15.6264.⁸

A range of 0.0016 is not THAT narrow but can certainly satisfy most limit order use cases!

⁸ As mentioned previously in note #4, there is a square root in the equation of the price and index, thus the numbers here are for explantion only.

5. Impacts of v3

Higher capital efficiency, LPs become arbitrageurs… as v3 has made tons of radical changes, I’d like to summarize my personal takes of the impacts of v3:

Higher capital efficiency makes one of the most frequently considered indices in DeFi: TVL, total value locked, becomes less meaningful, as 1$ on Uniswap v3 might have the same effect as 100$ or even 2000$ on v2.

The ease of spot exchanging between spot exchanges used to be a huge advantage of spot markets over derivative markets. As LPs will take up the role of arbitrageurs and arbitraging is more likely to happen on v3 itself other than between DEXs, this gap is narrowed … to what extent? No idea though.

LP strategies and the aggregation of NFT of Uniswap v3 liquidity token are becoming the blue ocean for new DeFi startups: see Visor and Lixir. In fact, this might be the turning point for both DeFi and NFT: the two main reasons of blockchain going mainstream now come to the alignment of interest: solving the $$ problem 😏😏😏

In the right venue, which means a place where transaction fees are low enough, such as Optimism, we might see Algo trading firms coming in to share the market of designing LP strategies on Uniswap v3, as I believe Algo trading is way stronger than on-chain strategies or DAO voting to add liquidity that sort of thing.

After reading this article by Parsec.finance: The Dex to Rule Them All, I cannot help but wonder: maybe there is going to be centralized crypto exchanges adopting v3’s approach. The reason is that since orders of LPs in the same tick are executed pro-rata, the endless front-running speeding-competition issue in the Algo trading world, to some degree, is… solved? 🤔

Anyway, personal opinions can be biased and seriously wrong 🙈 I’m merely throwing out a sprat to catch a whale. Having a different voice? Leave your comment down below!

6. Conclusion

That was kinda tough, isn’t it? Glad you make it through here 🥂🥂🥂

There are actually many more details and also a huge section of Oracle yet to be covered. However, since this article is more about features and targeting normal DeFi users, I’ll leave those to the next one; hope there is one 😅

If you have any doubt or find any mistake, please feel free to reach out to me and I’d try to reply AFAP!

Stay tuned and in the meantime let’s wait and see how Uniswap v3 is again pioneering the innovation of DeFi 🌟

Uniswap v3 Features Explained in Depth was originally published in Taipei Ethereum Meetup on Medium, where people are continuing the conversation by highlighting and responding to this story.

👏 歡迎轉載分享鼓掌

Show starts after one more math question

Quite a number of people have asked me to share about this controversial question. When I did a short video on this problem, it caused quite a bit of a...discussion, to say the least. After noticing some common wrong concepts around this topic, I decided to make this problem the topic of my first Math Lesson video on YouTube. (I have posted original math songs before, if you're curious. Yes, I wrote them lol)

Feel free to comment below if you have questions or suggestions :)

Exciting projects are in the works so stay tuned!

'Til next time!
- Mr F Chung

#ControversialQuestion #OrderOfOperations #MrFChung

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ファイナル・アンサーの詳細希望の方は下記のフォームに登録しておいてください。１週間〜２週間程度でご案内できると思います。

https://forms.gle/i6kfiwtBhghWEzkQA

英語学習で課題として取り組むのに適しているのは、最初から、自分の今のレベルの１つ上のレベルの課題をやる事ですが、従来は「i+1」と言われてきましたが、今回わかりやすいように、「１負荷」という風に僕が言い換えました。

全てのことに当てはまりますが、何かをレベルアップさせたい時というのは負荷がないといけないんですが、日本人の生徒さんは英語学習となると、負荷を避ける傾向にあります。負荷は必須なんですね。でも「i+1」という愕然として概念よりも、だから「負荷」をかけなければいけない事を常に念頭に置いてもらいたいので、あえて+1ではなく、「1負荷」という言い方にしました。今、自分がやっている勉強は+1かどうか、よりも「1負荷」かどうかの方がイメージしやすいですよね。

で、その「１負荷」とは常に「会話量」、「説明量」を軸にしないといけません。文法の難しさ、単語の難しさではなく、「会話量」、「説明量」だけを軸にしてください。

そしてスピーキング・レベル・シートを参考に会話量、説明量を増やす試みを提案死しましたが、さらに昨日の動画では、「１負荷」というのは、「え〜できるかな〜」っていうくらいが良いとお話ししました。そして、最初から日常会話でよく出てくる会話に１０文で答える練習をしてもいいと思うという話をしました。

そして、僕の１０文のスクリプトをお見せして、１０文だけど、みなさんが知らない英文法もなければ、英単語もないし、英語表現もない事を確認してもらいました。

それを確認してもらった理由は、多く話すために、会話量、説明量を意識して勉強するのに必要な英文法の知識、英単語の知識、英語表現の知識は既に備わっているというのをみなさんに確認してもらいたかったからです。また１から英文法を勉強しなきゃ、英単語を勉強しなきゃ、英語表現をもっと勉強しなきゃ、という発想を捨てて欲しいからです。

で、後は、前回書き出した質問に5W1Hを意識して、あともう2つとても重要な要素をプラスしてもらえれば、完璧です。

5W1Hは昨日も言いましたけど、皆さん、知ってるけど使わないんですよ。ここではスクリプトを作成するための「要素」って呼びますが、そういう要素の知識って実は皆さん持っているんです。知識としてはたくさん持っています。でも、今まではそれらを単純に知識として知っていただけで、それを「スキル」として意識してこなかったから使ってなかっただけです。そういった要素をたくさん引っ張り出せば、全然１０行なんか難しくありません。

で、今回はその要素を２つ教えます。まず、１つ目。例えば、Do you like sushi?って聞かれたら、大抵の方はYes.で終わってしまいます。でもそこで、確かに5W1Hはちょっと使いづらいですよね。そういう時は、Eから始まる要素です。Examples＝例です。例をあげるんです。お寿司、全てのネタが等しく好きって人はいないはずなので、大好きなものを嫌いなものを言っておくことはできますよね。

それがないと、
A: Do you like sushi?
B: Yes.
A: Me too. I love squid. Do you like squid?
B: No.
A: Oh, OK. Well, I also like octopus very much. Every time I go to a sushi restaurant, I always order octopus first. Do you like octopus?
B: No.
A: Oh, OK. I also like hokkigai. Kappa sushi has the best hokkigai. Do you like hokkigai?
B: No.

最初からExamples＝例として大好きなもの、嫌いなものを言ってくれてたら、もう少し自然な会話になってたと思うんですが、もうこれ、「一人で盛り上がってごめんね」状態ですよね。

なので、Examplesを言いましょう。

もう１つの要素もEで始まります。なんでしょう。

はい、正解。Episodeです。Episodeがない話ほど、盛り上がらないものはないですよ。また会話で見てみましょう。

A: Do you cook?
B: No. I’m not good at cooking.
A: Oh, really? What have you tried cooking?
B: Brownie.
A: You made brownie?
B: Yes.
A: How was it?
B: Bad.

これにエピソードを入れるとどうなるか。ちょっと時間の関係で日本で説明しますね、僕のBrownie事件。

ブラウニーが食べたくて、スーパー行ったんですけど、ブラウニーがなくて、代わりにブラウニーミックスが売ってたので、それ買ったんですね。で、家帰って、粉と油を混ぜて容器に入れて、焼こうと思って箱の裏を見たら、180度で40分って書いてあるんですよ。そんなに待てないんですよ。だから、I did some quick math in my head. 180 x 40が合計熱量としたら、210 x 25で同じくらいの熱量になるんじゃないかって思ったんですよ。まあ、僕の勘ピューターが弾き出したわけですよ。で、結果は想像つきますよね。

で、今のがDo you cook?に対する僕のエピソードでした。

この話を僕はセミナーとかでよくするんです。けど、みなさんの反応はどんな感じだと思います。「先生、おちゃめですね」で終わる人が８割です。今、この動画見てる人ももしかしたら、それで終わりの人もいるかもしれません。

僕がわざわざ自分のPerfect Humanのイメージを壊して、Episodeの重要性について、Episodeを入れることによってみなさんの英語のレベルが高まるスキルを教えているのに、「イムラン先生お茶目」っていう知識で終わってるんですよ。必要なのは、それをスキルとして使えるようになることですよね。

なので、ここで一度まとめます。

みなさんの英語のレベルアップに重要な要素は３つです。会話量、説明量を増やすための重要な３要素は
１）5W1H、２）Examples、３）Episode です。
それを念頭に昨日の１０トピックに対する自分の答えを考えてみてください。

ここから、細かくステップを解説します
1まずQuestion 1に関して、この３要素を意識しながら、話すことを日本語で書き出してください
2 思いつかない人は、英語の先生かネイティブの友達にお金を出して、その人のスクリプトを書いてもらってください
3 出来上がったら、それをGoogle翻訳とかPapagoというアプリなどを使って、英語にしてください
4そのスクリプトを英語の先生かネイティブの友達にお金を出して添削してもらってください
5添削されたものが返ってきますが、それがその質問に対するあなたのファイナル・アンサーです
6それをひたすら音読して覚えてください
7 ある程度覚えたら、ネイティブにその話をぶつけてみてください。通じるかどうか、試してみてください。添削してもらってるから
8 それを１０質問分やってください

それを１０質問分やると、どうなるとおもいます？

昨日の動画を見たファイナル・アンサーの生徒さんから、今朝メールをもらいました。まだ質問は８までしか公開していないですが、昨日の動画で９、１０を公開しちゃったんですね。そしたら、居ても立ってもいられなくなり、自分でおそらくイムラン先生はこんな感じのスクリプトにするじゃないかなと思って書いてみちゃいました！って送ってきたんですよ。

どういうことかわかります？８個もこの流れでやっていると、こういう風に話すんだろうなって、想像がつくようになるんですよ。わかってくるんですよ。こういう例を入れたり、こういう風にエピソードを入れたりするんだろうなってというのは、もうわかっちゃいます。

だから、１０個もやれば、どんな質問でも大体、こんな風に答えるんだろうなっていうのが、わかってきます。これが知識としての学習じゃなくて、スキルとしての学習をした場合の結果です。しかも、１０個だから、１週間に１つで、２ヶ月半。今まで、どれくらい勉強してきたかわからないですけど、その何十分の１の期間で、今の何倍も話せるようになるんですよ。

中には、このフローを見て、「えー、お金出してスクリプト作ってもらうの？お金出して、添削してもらうの？」って人もいると思います。

そういう文も正しい英文を自分で書けるようになるためにやっぱり英文法勉強した方がいいんじゃないかなって思う人もいると思うんですよね。

正しい文が書けるようになるには、正しい文が何なのか、わからないといけないですよね。自分で文を作って、それが正しいかどうか、間違っていたら、どういう風に言うのが正しいのか、聞いちゃえば、５分で済むんですよ。その程度の添削なら５分でできるから。

それを何年も勉強して、できるようになる方を選ぶんですか？しかも、中には今まで何年もそれをやってきて、できるようにならなかった人もいると思うんですけど、だから、普通に考えて、これからもならないですよ。

５分で済むことに、もう何年も費やしたいですか？

もっと自分の時間、自分の人生、自分の生活を大切にしてください。

僕はみなさんに、英語を勉強してもらいたいんじゃないんですよ。英語を話してもらいたいんですよ。早く話しちゃってもらいたいんですよ。いつまでもモジモジしてないで、思い切って話してもらいたいんですよ。

だから、５分で添削してもらえば、５分後に自分のファイナル・アンサーが出来上がるんだから、それでよくないですか。だって、それ繰り返していたら、正しい文も書けるようになりますよ、英文法勉強するよりも早く。

今回、また長くなりましたが、これで僕が伝えたいことは全てです。この通りにやってもらえれば、ファイナル・アンサーをやっているみなさんのように、必ず結果が出ます。出ないわけがない。

で、ここで絶対にメールとかコメントが来るので、先に言っておきます。英語の先生にスクリプトを作ってもらうのとか、添削してもらうのとか、その他諸々、どうすればいいかわからないから、先生をどう探せばいいですか？という質問が絶対来るんですよ。

そういう人には、もうファイナル・アンサーをやってもらうしかないです。そもそもファイナル・アンサーはそういう人のために作ったんです。自分で添削する人を見つけられない人は、代わりにうちで全部やるので、というのがファイナル・アンサーです。

最後は僕の心の兄のこの言葉で締めくくれればと思います。

知覚動考

知っても、覚えても、動かなければ、結果はわからない。とりあえず動いて結果を見て考えようという言葉なんですが、私の心の兄は、読み方が実は違うんです。

ともかくうごこう

なんですね。今日、僕の話を聞いてもらって、次のステップに進んでいただけることをお祈りしております。

コペル英会話という英会話スクールをやってます。
http://coper.biz/

ブルーフレイムという、英語セミナー、英語ツアー、大人向け留学サービス、親子留学、スキル習得留学
などをやっている会社もやっています。
http://www.blueflame.jp

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コラボや仕事の依頼も直接メールでお願いします。
imran@imran.jp