The equals sign

In a previous post, we talked about the difference between numbers and numerals.  There are often many numerals that express the same number.  For example, the numerals “4” (Hindu-Arabic), “IV” (Roman), “٤” (Arabic), and “四” (Chinese) all express the same number, the number four.  But let’s suppose we just use the Hindu-Arabic numeral system, and not any of the others.  Then is there anything we can write down other than “4” that expresses the number four?

Yes there is!  All of the following express the number four:

4numeral

four1

four2

four3

four4

four5

All of these sequences of symbols express the number four, and they all are in the Hindu-Arabic numeral system.  So we don’t need to go outside the Hindu-Arabic numeral system to find other ways of expressing the number four.  However, only the first of these, “4”, is a numeral.  “1+3” is not a numeral, it is rather two numerals “1” and “3” with a plus sign “+” in between them.  What do we call sequences of symbols like these that express numbers but are not necessarily numerals?  We call them expressions.  All of the sequences of symbols pictured, even “4”, are expressions, but only “4” is a numeral.

Two expressions may either express the same number or express different numbers.  For example, “2+2” and “1+3” express the same number, four, but “1+1” and “2+3” express different numbers, namely two and five respectively.  We use the equals sign to state that two expressions express the same number.  For example, “2+2=1+3” states that “2+2” expresses the same number as “1+3”.  This is true, because the expressions “2+2” and “1+3” do indeed express the same number, the number four.  We can take any two expressions from the above list and put an equals sign in between them in order to state the truth that they both express the same number.


Let’s look at bit more at what we are doing here.  What is going on in this sequence of symbols?

equation.PNG

Is this sequence of symbols a numeral?  No, it has many numerals in it, instances of “1”, “2”, and “3”.  Is it an expression?  No, it has two expressions in it, “2+2” and “1+3”.  It is a new kind of thing, an equation.  What does this equation mean?  It doesn’t express a number like numerals and expressions do.  Rather it states a truth, the truth being that the expressions “2+2” and “1+3” express the same number.

We call “2+2=1+3” an equation because by writing it we equate “2+2” and “1+3” — we consider them to be equal.  “Equate” and “equation” are not just words that are used in mathematics, they are used in ordinary English as well.  An insecure person may “equate criticism with insult”, and if we are talking about this trait of theirs, we might refer to it as “their equation of criticism with insult”.  However, this ordinary English usage is not to be confused with the precise mathematical usage about two expressions expressing the same number.  I just gave the ordinary English usage so you can see that the words “equate” and “equation” are not exclusive to mathematics, and that their mathematical usage has some bearing on their general usage in English.

What if we put an equals sign between two expressions that don’t express the same number?

equation2.PNG

This equation is false.  The expressions “1+2” and “2+3” don’t express the same number.  Nonetheless, it is still an equation.  Equations can be true or false.  They don’t have to be true in order to be equations.  The insecure person mentioned above equated criticism with insult, but and this equation was not correct, but it was an equation nonetheless.

What if we put an equals sign between two instances of the same expression?

equation3.PNG

This is an equation too.  Is it true or false?  Well, the expression “1+2” on the left of the equals sign expresses the number three, and the expression “1+2” on the right of the equals sign also expresses the number three.  Since the expressions on either side of the equals sign express the same number, the equation is true.  In general, we can always make a true equation by writing the same expression on the left and the right of an equals sign.


Please notice that what I am saying about the equals sign is different than what you might have been taught in school.  In school, you might have been taught that “1+1=2” means that when you add one and one, you get two.  In this interpretation, the equals sign is placed between a procedure, “1+1”, and a result, “2”.  Even if your teachers didn’t tell you that an equals sign is put between a procedure and a result, you still might have gleaned this interpretation from how it is used in exercises like “35×24=__”  where you are given a procedure to do (multiply thirty-five and twenty-four) and you are asked to write down the result in the blank space.  But here I am saying that the equals sign is not about results of procedures at all.  It is just about expressions expressing the same thing.  Consider the following equation:

equation4.PNG

This equation wouldn’t make any sense in school’s interpretation.  How could the result of the “procedure” “2” be “1+1”?  What would that even mean?  But in my interpretation (which is the correct one), this makes perfect sense.  The expressions “2” and “1+1” express the same number, the number two.

In fact, whenever you have a true equation, (say “1+1=2”), swapping the expressions gives you another true equation (for this case “2=1+1”).  The expression on the left side of the equals sign and the expression on the right side of the equals sign are interchangeable.  It doesn’t matter which expression you put on which side.  This is an important property of equality called symmetry.  In school’s interpretation, equality is not symmetric because the right side is distinguished as the “result of the procedure”, so it loses this crucial property.

(Not all equations in English are symmetric.  For example, if the insecure person above equates criticism with insult, that does not mean that they also equate insult with criticism.  I think that equating insult with criticism would be the mark of a very secure and very naïve person.  Here is a clear difference between equations in English and in math, and it shows that we can’t always take our intuitions from English and assume that things work the same way in math.  It also serves as an example of a general trend that things tend to be more simple (always symmetric) in mathematics than in English (not always symmetric).  I am reminded of the mathematician John von Neumann’s quote “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”)

Now that you know this, you can use it to troll your teachers.  If they give you a problem like “2+2=?”, you can answer like this:

equation

You would be completely correct, and if your teacher complains it is due to their own miseducation in their schooling and you can refer them to this post so that they can learn what the equals sign really means.  Let’s say your teacher gives you a problem you really don’t feel like doing, such as “355×224=__”.   You would be entirely correct if you just wrote:

equation5.PNG

Now you don’t have to do the problem at all, and you’re still correct!  If your teacher complains, ask if it is not enough that your answer be correct.  Is math not about being correct?  What more can they ask for? [sarcasm: math is about creativity, not simply being correct]

(A mathematically sophisticated teacher can get around this by writing something like “express 2+2 in the simplest form”.  Then you would have to write “4”, because it is the simplest way to express the number four.)


The equals sign is often misused outside of math, and this really irks me when I see it.  Before I got used to it, the meme phrase “mind=blown” really irked me.  If your mind is blown, that doesn’t mean it’s equal to blown!  Your mind and blown are still two different things!  Also, it’s really common in chemistry, if you’re talking about, say, molecular mass, to write something like “CH4=16″ to say that the molecular mass of CH4 (methane) is 16.  This is wrong!  Methane is not equal to the number 16!

I feel bad being mad about this because it is as if I am a nationalist of mathematics trying to uphold the sanctity of its symbols in the greater world.  Of course the equals sign is public property and people are free to make use of it however they like.  I just wish that real mathematical understanding was a bigger part of our culture.


Disclaimer:  In this post I have just explained one use of the equals sign.  There are other uses too, both within math and in other fields, like computer science.  Sometimes slightly different symbols, like “≈” and “≡” are used to tell apart these different uses.  Equality is also not the only concept of sameness.  There are other ways for two things to be the same.  I will talk about this stuff in later posts, I just wanted to mention it so that people don’t think that the notion discussed above is the be-all and end-all of the equals sign, equality, or sameness.

A common train of thought that prevents people from doing what they want to do

Here is a common train of thought:  “I want to do x, so I will put myself in a situation where someone else is forcing me to do something vaguely similar to x, as well as all this other stuff too.”  A good response to this train of thought is:  “Why not just cut to the chase and do x right now?  Then you won’t have the discomfort of being forced to do things, and you also will actually get to do x instead of something else that’s vaguely similar.”  Of course you often will not be able to do x without the help of others who are more knowledgeable, but you hopefully will be able to find such people to help you.

Examples:

  • “I want to be a great inventor like the ones I read about, so I will go to school for engineering.”  Being in school for engineering is a situation where people are forcing you to learn something that is not exactly the kind of invention you were reading about.  The curriculum is not designed to make you a great inventor, it is designed to make you marketable to companies that hire engineers.  If you want to be a great inventor, then why not just start inventing right now, and find more knowledgeable people to help you?  You could even go to a university building and start talking to professors, if you think they would be helpful.
  • “I want to help people in [some particular way], so I will work for a company that says it helps people in that way.”  Most likely your work at the company will not be what you set out to do.  Why not just start helping people in this way right now?  Probably you can’t do it alone, so you could try to get others to do it with you.  That’s better than joining others in doing something that doesn’t actually help people.
  • “I want to get closer to God / live a more meaningful life, so I will join this cult.”  The cult will not force you to get closer to the version of God that you believe in.  And the “meaningful life” that you live within it will only seem meaningful because of cult brainwashing, it is not something that you would consider meaningful now, before joining.  Spirituality and meaning in life can come from all sorts of places, they are not the exclusive province of any particular cult or belief system.  You can find them by talking to people, doing research, making friends, working on projects, and helping people (among other ways).
  • “I am interested in [some particular kind of kinky sex], so I will join the BDSM subculture.”  The BDSM subculture is a very bad place, read this and follow the links to see how.  If you join, you will be expected to conform to all sorts of abusive cultural practices completely unrelated to your kinks.  But they will try to indoctrinate you to think that they are related.  Also, BDSM is something very specific, it’s not the same as kink in general.  It’s definitely not what you’re into, unless you’re a sociopath.  If you’ve been reading BDSM propaganda, you might think they have a monopoly on kinky sex, and that joining them is the only way you will find it.  But really, there are many kinky people in ordinary life.  If “kink” is just sex that is not “normal”, then all sex is kink, because normal things are rare and absolutely normal things are nonexistent.  Finding people who “share your kinks” then translates to just finding people who are sexually compatible with you.  This is not anything that requires joining a cult like BDSM, it is something that humans have been doing for tens of thousands of years before BDSM existed.
  • “I want to learn how candles work the way they do, so I will read a general chemistry textbook.”  The state of chemistry and chemical education today is such that it is not at all geared towards answering basic questions we have about the world.  Instead it is geared towards making pharmaceuticals, refining petroleum, and other things like that which benefit rich people.  A general chemistry textbook probably won’t even mention how candles work. It will rather introduce you to the grand system of modern chemistry, which could be used to help you understand the world, but generally is used instead for things that benefit rich people.  It is a noble task to repurpose modern chemistry so that it actually helps people understand the world, but don’t suppose that it will be quick or easy.  If you want to learn how candles work without going through this long process, I suggest this adaption of Faraday’s lectures.

You don’t need to go to a bad place to do something you want to do.  If you want to do it, then just do it and find people to help!  You don’t need to be forced to do something that’s actually different.

And if you do need to go to the bad place for some reason, like money or parents, then don’t delude yourself into thinking that you’re going to it for some other reason (like wanting to do x when the bad place doesn’t even force you to do x, but rather something else) instead.

Numbers and Numerals: Some more examples

Here are some more examples that illustrate the difference between numbers and numerals.  This post is a continuation of the previous post about numbers and numerals.

An example

Here is a big numeral for a small number:

2nu

Here is a small numeral for a big number:

bignumber.png


Another example

Here is an infinitely long numeral for a finite number, pi:

pi.png

This is actually not the whole numeral, because it is too big to reproduce here.  I put “…” at the end to indicate that it continues on past the portion I have written.  Click here to see more of it.  Even the linked page doesn’t have the entire numeral, because the entire numeral is infinitely long.  It would occupy more space than the whole internet has!

On the other hand, here is a finite numeral for an infinite number:

infinity

 

Numbers and Numerals

This is a picture of an elephant:

African_Bush_Elephant.jpg

This is a picture of the word “elephant”:

elephantword

An actual elephant is a different thing than the word “elephant”.  Here are some differences.

  • An elephant is an animal, but the word “elephant” is not an animal, it is a word.
  • The word “elephant” has 8 letters, but an elephant doesn’t have letters at all.  (Look at the picture of the elephant:  there are no letters!)
  • An elephant has a trunk, but the word “elephant” has no trunk because it is just a sequence of 8 letters.
  • The word “elephant” is in the English language, but an actual elephant like the one in the picture has no particular relation to the English language.

An elephant is an animal that would exist whether humans were there to call it an “elephant” or not.  The word “elephant” is a sequence of 8 letters that some humans use to refer to this animal.

If you didn’t know English and you saw the two above pictures, you wouldn’t think they had anything to do with each other.  One is a picture of an animal, and the other is a picture of some weird symbols in a row.  Why would someone unacquainted with English suppose that these had anything to do with each other?   It is only the tradition of the English language and writing system that ties together the animal and the word.  There is no necessary connection.  In fact, it would be conceivable for there to be another language in which this same sequence of letters instead referred to something else, say, the sun.


The difference between numbers and numerals is similar.

Here is a picture of the numeral four:

4numeral.png

I would like to have a picture of the number four for comparison, but the number four is an abstract concept that is hard to make a picture of.  Just think that if you had never seen the symbol “4”, you might still have the concept of four.  You might notice that things sometimes come in fours:  there are four sides to a square, most humans have four limbs, and there are four seasons.  You would be able to think about things that are in fours, about fourness, without having the numeral four.  The number four is this concept of fourness, and it is very different from the numeral four.

Here are some differences between the numeral four and the number four:

  • The numeral four is angular  (fanciful fonts aside), but the number four is not angular — how could it be angular when it is a concept rather than a shape?
  • The number four is even.  This means that four things, say, four apples, can be divided into two groups evenly:  just put two in one group and two in the other group.  This couldn’t be done for five apples, on the other hand:  one group would always have to have more apples in it than the other.  So the number four is even and the number five isn’t.  We are not allowing apples to be cut or made into applesauce.  But the numeral four is not even.  What would it even mean for a symbol to be even?  Maybe that we could divide it evenly into two groups?  Looking at the symbol, I’m not sure how this would be done.  And even if we could, we would have to cut the symbol or make it into symbolsauce, and this is not allowed, just like it wasn’t allowed for five apples.

Just like there is no necessary connection between an elephant (the animal) and the word “elephant”, there is no necessary connection between the number four and the numeral four.  What do this symbol and this concept really have to do with each other?  If you saw the numeral four for the first time, you wouldn’t think of the number four:  there is nothing in the symbol that obviously evokes fourness.  You had to be told at some point that the numeral four represents the number four.  And if you had never seen the numeral four, there is very little chance you would think of it just from the number four.


Let’s talk more about the number four.  We said that this is the concept of fourness.  To understand this better, we need to talk about a philosophical idea called universals.

Russell Dale, a teacher of mine, told me a great “formula”, so to speak, that explains universals:

A universal is a one that embraces many.

Let’s talk about what this means through some examples.  An example of a universal is blueness.  There are many blue things, just look around you wherever you are.  You can probably find lots of things that are blue.  All of these things have blueness.  But blueness is a single thing, a one.  Thus blueness is a one (a single thing) that embraces many (the many blue things).  Therefore blueness is a universal.

Another example is doghood, the quality of being a dog.  Doghood is a single idea, but there are many dogs.   Any dog has doghood just as much as any other dog, so although doghood is a one (a single idea), it embraces many dogs.  So doghood is a universal.

Universals are ubiquitous:  you can probably think of many more examples.

We need one more term before we get back to talking about the number four.  That term is “instance”.  An instance of a universal is just one of the many things that it embraces.  Any blue thing is an instance of blueness and any dog is an instance of doghood.

Now back to the number four.  The number four is a universal.  There are many places where it crops up:  we can have four apples, four limbs, there are four seasons, etc.  All of these situations have fourness.  Thus fourness, or the number four, is a single concept that embraces these many situations — it is a one that embraces many, a universal.  Each of these situations is an instance of the number four.

When we say that the number four is even, what we mean is that any instance of the number four can be divided into two groups evenly.  We went over how four apples could be divided into two groups evenly, but the same could be done for four pears, four tractors, four limbs, the four seasons, etc.

Now it is clear that the number four and the numeral four are different.  The number four is this universal that embraces all sets of four things, and the numeral four is this symbol “4”.


We have so far been talking about the numeral four.  But there are really many numerals four.  Here are some of them.

The Hindu-Arabic numeral four:

4numeral

The Roman numeral four:

4roman

The Arabic numeral four:

4arabic

The Chinese numeral four:

4chinese

The Greek numeral four: (This is the system that Euclid used.  The overbar serves to distinguish Greek numerals from Greek letters.)

4greek.png

Each of these symbols is a numeral, and each of them is used to represent the number four.

The point of showing you these different numerals for four is so you can see that the number four is the same no matter how we choose to express it — no matter which of these numerals we use.  It is an abstract concept that exists independently of how we choose to write it down.  The symbol “4” has no more to do with this concept than the symbol “IV”, the symbol “٤”, or the symbol “四”.


Now let’s talk about the difference between numerals and digits.  Consider the following numeral:

42numeral.png

This numeral is interesting because it has two symbols in it, “4” and “2”.  These symbols are called the digits of the numeral.  Each digit would be a numeral itself if it appeared alone, but when both of these digits appear next to each other they combine to make a single numeral.

Order is important in a numeral with multiple digits.  If we put these digits in the opposite order, we get a different numeral:

24numeral

Why do we get a different numeral depending on the order?  Because the meaning of a digit changes based on its position in the numeral.  If the 4 is last, as in “24”, it just means four, but if it is first, as in “42”, it means forty.  It is a peculiar feature of the Hindu-Arabic numeral system that the meaning of a digit depends on where it is in the number.  This is not the case in all numeral systems.  For example, in the Greek numeral system that Euclid uses, the numeral 42 can be written as either “μβ” or”βμ“:  order doesn’t matter.  This is because “β” represents two and “μ”  represents not four, but forty.  Unlike Hindu-Arabic numerals, Greek numerals use two different symbols for a digit that means four and a digit that means forty.  In Greek numerals,”μ” means forty and”δ” means four.  Hindu-Arabic numerals, on the other hand, just use “4” for both.  Whether the digit “4” means four or forty depends on where it occurs in the numeral.  Since the value of a digit depends on its place in the numeral, Hindu-Arabic numerals are called a place-value system.

In Hindu-Arabic numerals, the digit “4” can mean four, forty, four-hundred, four-thousand, etc. all depending on where it occurs in a numeral.  If it weren’t for our numeral system, we wouldn’t see such a strong connection between the numbers four, forty, four-hundred, etc.  We would just see them as a list of quantities.  Of course there is some connection between these numbers, because if we take four and we multiply it by ten we get forty, and if we multiply that by ten we get four-hundred, etc.  But what is so special about ten?  Why not have a numeral system where eleven serves this same function?  In fact, the Babylonians used a system where sixty serves this function, and we still have remnants of this today.  For example, there are 60 seconds in a minute and 60 minutes in an hour.  Computer scientists often use a system where 2 plays this function, called binary. The number that serves this special function in a numeral system is called the base.

I can use this to explain why numerology seems like bullshit to me.  A common technique in numerology is to take a number and add up the digits.  For example, 356→3+5+6=14.  But in 356, the digit 3 doesn’t mean three, it means three-hundred.  And 5 doesn’t mean five, it means fifty.  The only reason we see such a strong connection between three and three-hundred, and between five and fifty, is that the base of our numeral system is 10.  There is no reason it had to be this way.  So I don’t trust numerology because a lot of its arguments would come out completely differently in a different numeral system.

Edit (2/1/17):  My friend Junyi Sun suggested a good illustration of this last claim.  In Chinese numerals, 356 is “三百五十六”.  These characters, in sequence, mean three-hundred-five-ten-six.  So a numerologist using Chinese, rather than Hindu-Arabic numerals might think to add up 3+100+5+10+6 instead of 3+5+6.


Suppose someone says that their favorite number is 444, because they find the three fours in a row to be aesthetic.  We could say to them:  “You actually like the numeral 444, not the number, because it is the way it is written that you like, not the actual quantity.  We doubt that you would get the same aesthetic feeling from a pile of 444 apples.  Also, we’re not sure that you would still like the Roman numeral for it, which is CDXLIV.  No, we think it is really the Hindu-Arabic numeral that you like, and not the number.”

So if you have a favorite number, is it really the numeral or the number that you like?  Do you think that you would still have the same favorite if we used a different numeral system, like Roman numerals?

For a few more examples of the difference between numbers and numerals, see the next post.

School is bad at recommending books

[Scope: This is not just about reading books, it can apply to studying in general.]

Imagine someone you know  — let’s call them X — recommends you a book.  X is adamant that you read this book, and they keep insisting that you read it.  Finally you give in and read the book.  After finishing the book, you call up X to talk about it.  “I finished the book you recommended!”, you say to them excitedly.  X, however, is skeptical, and insists on asking you some questions about the book to make sure that you read it.  After satisfying them, they simply say “OK, great.  You read it.”  And they hang up.

You would feel strange in this scenario.  Why was X so adamant that you read the book if they don’t even want to talk about it with you afterwards?  This experience might make you hesitant to take this person’s book recommendations in the future.

However, this is exactly what school does.  School is X, who is adamant that you read and learn certain things, but doesn’t want to talk to you about them afterwards.  It only wants to verify that you read/learned them, and then it moves on.  School is bad at recommending books (and things to learn).

This is why I advise that when given the choice between a book that school recommends and a book that a friend recommends, always pick the book that the friend recommends.  You wouldn’t take X’s book recommendations, so you shouldn’t take school’s either.   A friend knows something about you and cares about you.  They will pick a book that is relevant to where you are in your education and in your life, and they will discuss it with you so that it becomes part of your shared experience with them.

Studying something requires a big investment of time and energy.  It’s not wise to make this investment for the sake of people like X, who don’t give a shit about you or your experience studying and whose real loyalty belongs to the employers they are selling you to.  (The purpose of school is to manufacture employees.)  Value your time and energy, and spend it on people who deserve it.

The point of education should be to build a shared knowledge-base with your friends, which will help you understand and change the world.  This world is so confusing that we are lost without this.  Without education, we don’t know what to do or think, so we just do and think whatever everyone else is doing because it’s easier.  This is why we say that education is liberatory.  (By “education” I obviously don’t mean what goes on in schools.)  Let your friends educate you, and don’t spend so much time being distracted with the next irrelevant book X demands that you read.

 

Note:  When I talk about a friend, this is not limited to other students.  Adults can be friends.  In fact, adult friends are often great at recommending books because they’ve simply read more on account of being alive longer.

How to Memorize the Amino Acids Without Mnemonics II: Glycine, Alanine, Serine, Cysteine, Threonine, Proline

[This is the second post in a series designed to help you memorize the amino acids without mnemonics.  It covers glycine, alanine, serine, cysteine, threonine, and proline, with a recurring theme of stereochemistry.

The first part is here, and it is the introduction.  You should read it if you are wondering why there is so much detail here above and beyond simply stating what the amino acids are, and why I think this excess of detail will help you memorize the amino acids.  I’ll just say here that it is to help you fit them into a larger system in your mind, and you don’t have to memorize all this detail in order to memorize the amino acids.

The third part of this series will cover phenylalanine, tyrosine, valine, leucine, isoleucine, methionine, and tryptophan.  The fourth part will cover histidine, lysine, arginine, aspartate, asparigine, glutamate, and glutamine.  After I publish the later parts, I will update the contents of these brackets accordingly.]

Which amino acids are “the amino acids”?

An amino acid is any molecule that has both a carboxylic acid group and an amine group, so there are infinitely many possible amino acids.  Here is an example:

aa-1b

When an amino acid is dissolved in water, as it often is in biology, most of the amine groups become protonated and most of the carboxylic acid groups become deprotonated. It then looks like this most of the time:

aa-3

Though it would be technically correct to call the protonated amine group an “ammonium” group and the deprotonated carboxylic acid group a “carboxylate” group, as shown in this picture, this is seldom done.  Unless they are specifically talking about protonation states, chemists will talk loosely and refer to the “amine group” or the “carboxylic acid group” of an amino acid, even when it is dissolved in water.

So far, it seems like there must be a lot of amino acids.  However, when we talk about memorizing the “amino acids”, we are just talking about the proteinogenic amino acids, which are amino acids that are incorporated into proteins during translation, the process whereby ribosomes read RNA and construct proteins accordingly.  All proteinogenic amino acids are α-amino acids, which means that their amine group is located on the α-carbon, which is the carbon bonded to the carboxylic acid group.  (The β-carbon is the carbon bonded to the α-carbon, and the γ-carbon comes after that, etc.)

aa-4

Here are some proteinogenic amino acids connected together in a protein.

aa-5

When amino acids are in a protein like this, they are called “amino acid residues”.  When they are not in a protein, they are called “free amino acids”.

All proteinogenic amino acids also have an α-hydrogen, a hydrogen bonded to the α-carbon.

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So far we have talked about 3 of the 4 substituents on the α-carbon: the carboxylic acid group, the amino group, and the hydrogen.  The fourth substituent is what differs in the different amino acids.  In principle, there are infinitely many possibilities for this fourth subsitutuent, but only 20 of these are proteinogenic.  So we are going to talk about these 20 in what follows.

Serine

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We have already shown the proteinogenic amino acid serine in the pictures above.  Its side chain is hydroxymethyl, or -CH2OH.  In 1865 E. Cramer first isolated serine from a silk protein called sericin.  Cramer named both serine and sericin, and he named them after sericus, the Latin word for “silk”.

The 3-letter abbreviation of serine is “Ser”, and the 1-letter abbreivation is “S”.  The purpose of these abbreviations is to make it easier to list long sequences of amino acid residues in proteins.

Serine is the only amino acid to feature a primary alcohol (an -OH group on a primary carbon, which is a carbon bonded to 1 carbon atom and 2 hydrogen atoms).  We will get to other amino acids, which feature a secondary alcohol (threonine) and an aryl alcohol (tyrosine).  These three are the only hydroxy amino acids.  It is remarkable that there is exactly one of each of these different kinds of alcohols, so that a small number of amino acids can cover a wide range of functions.

Serine exists in two forms: l-serine and d-serine:

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These forms are called enantiomers, from Greek ενάντιος/enantios meaning “opposite” and μέρος/meros meaning “part”.  How are they opposite?  One of them is the mirror image of the other, just as your left hand is the mirror image of your right hand.  The enantiomers of alanine are distinguished by the “l-” and “d-” prefixes.  Only l-serine occurs biologically.  In fact, all proteinogenic, and most biological, amino acids are l-amino acids.  This is an amazing fact!  How did things evolve so that life uses only l-amino acids?  There is much speculation on this.

To understand what the designations and d mean will require a digression.

Digression to explain what and d mean

Consider the following sugars:

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These are drawn in Fischer projection, which means that the vertical lines are viewed as curling backwards into the screen, and the horizontal lines are viewed as curling forwards out of the screen.  In order to show this, we draw bonds that are coming out of the screen towards you as wedges and we dash bonds that are going into the screen away from you.

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The four substitents of the central carbon atom point, approximately, to the vertices of a tetrahedron centered at it:

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The “d-” and “l-” prefixes distinguish the two enantiomers of glyceraldehyde.  I don’t really understand this atm, but they refer to how the compound interacts with plane-polarized light:  d-compounds rotate it clockwise (which is called “dextrorotation”, hence “d“) and l-compounds rotate it counterclockwise (which is called “levorotation”, hence “l“).  

The problem with the d/l nomenclature is that at the time that it was invented, there was no way to figure out which of the two possible structures corresponded to d and which corresponded to l.

Due to this inadequacy, the German chemist Emil Fischer started to use the d and l designations in a different way.  He was dealing with sugars and the molecules that could be synthesized from them, and in this area the inadequacy mentioned really mattered.  First he guessed (!) that the structure of d– and l-glyceraldehyde were as shown, as opposed to the other way around.  There was a 50% percent chance that he would be right.  (In 1951 it was determined by X-rays that he was right!)  Once he had made this guess, Fischer began to call molecules that were synthetically related to d-sugars, such as d-glyceraldehyde, d themselves, and similarly for l, regardless of how they interacted with plane-polarized light.

In 1905, the Russian-American chemist Martin André Rosanoff realized the ambiguity of these two opposing usages of d and l, and proposed to separate them.  He suggested that d and l be restricted to their original usage, relating to plane-polarized light, and that the Greek letters δ and λ be used instead for Fischer’s new usage, to refer to molecules that were synthetically related to d– and l-sugars.  He also resolved another ambiguity:  sometimes the same molecule could be synthesized from both an l-sugar and a d-sugar, so that it wasn’t clear whether to call it δ or λ.  He resolved this ambiguity by restricting the possible syntheses that were allowed.

What we do today is a further variation of Rosanoff’s contribution.  First of all, instead of using the Greek letters δ and λ, we use the small caps d and l.  Secondly, instead of referring to a synthetic relationship between the molecule in question and d– or l-glyceraldehyde, d and l refer to a resemblance in the structure, regardless of how the molecule was synthesized.  This is called the Fischer-Rosanoff convention, after Fischer and Rosanoff.

Here is how this works in the case of serine.  We draw serine in Fischer projection, so that it looks like glyceraldehyde.  The enantiomer that “looks like” d-glyceraldehyde is called d, and the one that “looks like” l-glyceraldehyde is called l.

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For more information on this (I think, fascinating) subject, read the articles I read to write this section!

  • D. W. Slocum, D. Sugarman, and S. P. Tucker 1971:  “The Two Faces of d and l Nomenclature”, Journal of Chemical Education, Volume 48, Number 9, pp. 579-600.
    • This is a survey article, which cited the following one.  The Journal of Chemical Education is great!  It is one of the few places that chemists actually generalize and say why they care about the things they care about.
  • M. A. Rosanoff 1906:  “On Fischer’s Classification of Stereo-Isomers”, Journal of the American Chemical Society, Volume 28, pp. 114-121.
    • This is Rosanoff’s original article, where he proposed the nomenclature reform.
  • C. S. Hudson 1941: “Emil Fischer’s Discovery of the Configuration of Glucose”, Journal of Chemical Education, Volume 18, Number 8, pp.353-357.
    • Not directly relevant, but an exposition of the absolutely beautiful work Fischer was doing, and why it so heavily involved stereochemistry.

Note:  Just as life primarily uses l-, rather than d-amino acids, it also primarily uses d-, rather than l-sugars.

Alanine

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If we replace the hydroxyl group in serine by a hydrogen atom, we obtain alanine.  Its side chain is just a methyl group (methyl is -CH3).

To see why l-Alanine is called l, we show it next to l-glyceraldehyde:

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It doesn’t line up as perfectly as serine, because there is no hydroxyl anymore, but we still line it up the same way, because that’s the Fischer-Rosanoff convention.

It was first synthesized, not isolated.  The German chemist Adolph Strecker synthesized it from acetaldehyde in 1850, which in his time (and in German) called simply “Aldehyd”.  Here was his synthesis, along with his names for the compounds involved and their modern names:

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Strecker named alanine “Alanin” because it was derived from “Aldehyd”.

The synthesis of alanine from acetaldehyde in fact generalizes to the synthesis of any amino acid from the corresponding aldehyde, and this synthesis is named (aptly) the Strecker synthesis.  For example, here is how Fischer, in 1902, synthesized serine from the corresponding aldehyde:

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Alanine is the most “normal” amino acid.  Thus it often functions as a “control” in experiments on proteins, in the sense that biochemists will mutate a residue of a protein to alanine to see what happens when the specific function of that residue is absent.

The 3-letter and 1-letter abbreviations are just what you would think:  “Ala” and “A”.

Glycine

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Glycine is the simplest amino acid.  Its side chain is merely a hydrogen atom.

Glycine is achiral, which means that it is superimposable over its mirror image.  To see this, we can draw the following Fischer projection of it:

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If we reflect this about a vertical mirror, we get the same thing!  Thus l-glycine and d-glycine are actually the same, so we just say “glycine”, without any prefix.  Due to its achirality, glycine does not rotate plane-polarized light either clockwise or counter-clockwise, so it is called optically inactive.

On account of its minimal side chain, glycine can rotate more freely than other amino acids, so a protein has enhanced flexibility at glycine residues.  This is why it is alanine, and not glycine, that functions as a “control”, as I mentioned in the previous section.

Glycine was isolated from gelatin in 1820.  It was named after the Greek word γλυκύς/glykýs, meaning “sweet-tasting”, because of its sweet taste.  This is the same root as “glucose”, “glycogen”, and many other words having to do with sweetness.

The 3-letter and 1-letter abbreviations, again, are just what you would think: “Gly” and “G”.

Cysteine

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Cysteine is like serine except instead of oxygen it uses the congener sulfur (a congener of an element is an element in the same column of the periodic table).  Its side chain is a primary thiol, CH2SH.  (A thiol is like an alcohol but with sulfur instead of oxygen.  In general, the “thio” prefix denotes that an oxygen is replaced with a sulfur.  It comes from the Greek word for sulfur, θεῖον/theîon.)

It was first described in 1810 by Wollaston, and given its name by Berzelius.  Its name comes from “cyst”, which is an old term for the bladder, since cysteine was first isolated from urine.

Cysteine also exists in a dimerized form, as cystine.  Cystine is really two amino acids, which have a disulfide side chain -CH2S-SCH2– connecting them.  Here is how it would appear as a residue in a protein.

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It is not clear what the difference in pronunciation is between “cysteine” and “cystine”.  I have heard chemists pronounce “cysteine” simply as /sɪs.tin/ (sis-teen), but I have also heard them make a big point of the distinction by saying /sɪs.tə.ʔin/ (sis-tuh-een).

Cystine adds another level of structure to proteins, because it can cross-link two different sections of the protein together.  This is how hair keeps its shape, as straight or curly or whatever. Perms work by chemically breaking the disulfide bonds and reforming them in new places corresponding to a different hairstyle.

Here is a general schematic of disulfide bonds breaking and reforming in new places (Cysteine residues/free amino acids are indicated by “Cys”):

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When disulfide bonds are not forcibly broken as in a perm, they generally do not interchange.  But when free cystine is exposed to acid, the disulfide bonds do tend to interchange.  Surprisingly, this interchange causes transitions between l– and d-cystine!  Thus the longer free l-cystine is exposed to acid, the more it will approach a 1:1 mixture of  l– and d-cystine, which is called a “racemic mixture”.  This interconversion between l– and d-cystine is called racemization, and cystine is the only proteinogenic amino acid that racemizes under acidic conditions.  Cystine racemizes because in the process of disulfide interchange the α-hydrogen atom sometimes leaves the molecule briefly and is replace by another hydrogen atom on the other side, like so:

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The question marks indicate that this is taking place at sometime in the middle of disulfide interchange, so I’m not exactly sure what the sulfur is bound to.  For most amino acids this wouldn’t happen because the α-hydrogen wouldn’t ever leave, but something about the process of disulfide interchange makes this more feasible.  One theory is that at some point in the disulfide interchange the sulfur is very electron-deficient, which stabilizes the negative charge on the α-carbon in the transition state.

3-letter and 1-letter abbreviations: “Cys” and “C”.  This might seem too obvious to need to state.  However, later on the abbreviations aren’t as simple.  For example, the 1-letter abbreviation of glutamate can’t be “G” because “G” is already taken by glycine, so it is “Q” instead, because of the pun “Q-tamate”.

Note: If the next congener of oxygen after sulfur, namely selenium, is used, we get selenocysteine, a rare amino acid.  It is debated whether selenocysteine is proteinogenic or not, because, although it is coded by RNA, there are some differences between the way it is coded and the way the standard 20 amino acids are coded.  For example, the nucelotides that code for it also form the “stop codon”, the sequence that tells ribosomes to stop adding new residues to a protein, and release it. The abbreviations for it are “Sec” and “U” (the latter was apparently chosen because it was the only letter left).

Threonine

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Threonine, as was briefly mentioned above, is a secondary alcohol.  This means that it has a hydroxy group on a carbon that is bonded to two other carbons and a hydrogen.  The side chain of threonine is serine with a methyl added to the β-carbon, so it really is the minimal way to create a secondary alcohol amino acid.

You may realize that the β carbon has 4 different substituents, so that there are two non-equivalent possible configurations of l-threonine.

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A carbon that can have two non-equivalent configurations, like this, is called a “chiral carbon”, from Greek χείρ/kheír, meaning “hand”, as in right- and left-handed.  Thus, threonine is the first amino acid we have discussed to have two chiral carbon (the only other such proteinogenic amino acid is isoleucine).  The two configurations of l-threonine are not enantiomers, because the enantiomer of l-threonine would be d, rather than l.  We call these two configurations of  l-threonine “diastereomers”, rather than “enantiomers”, for this reason.

Only one of threonine’s two diastereomers appears in nature, and its name, “threonine”, as we will see, tells us which one!

Remember how the “l” designation referred to a comparison with “l-glyceraldehyde”?  Well, an old form of stereochemical nomenclature extends this comparison, so that diastereomers are named after larger sugars whose structures they are comparable to.

Consider the 4-carbon aldoses, that is, the sugars of the form CHO(CHOH)2CH2OH.  There are four such sugars: d-erythrose, l-erythose, d-threose, and l-threose.  Here they are shown with d– and l-glyceraldehyde, from which they get their d and l designations by the Fischer-Rosanoff convention:

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The ” d” and “l” refer to whether the 3-carbon’s configuration is like d– or l-glyceraldehyde, and the rest of the name describes the 2-carbon’s configuration.  In erythrose, both of these carbons have the same configuration (in a Fischer projection).  In threose, they have opposite configurations.  The name “erythrose” comes from the Greek word ἐρυθρός/erythrós for “red”, because it is red in the presence of alkali metals.  In a common trick of chemical nomenclature, the diastereomer threose is named by switching the letters of “erythrose”.

Finally, let’s get back to threonine.  We can draw the proteinogenic diastereomer in a “Fischer projection” to look kind of like  d-threose.  This is why it is called “threonine”, rather than “erythronine”.  (Here is the paper where they figure out that it is threo- instead of erythro-.)

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Confusingly, we have drawn an l-amino acid to look like a d-sugar!  This is possible because the α-carbon of l-threonine now corresponds to the 2-carbon of the sugar, which is not the carbon of the sugar that makes it a d-sugar.  Here we draw  l-threonine and d-threose along with their corresponding glyceraldehydes:

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Notice that it is the 3-carbon of d-threose that makes it d, but the α-carbon of l-threonine that makes it  l.  These carbons don’t correspond to each other in our correspondence, so there is no contradiction.

IUPAC noticed this confusion, so they recommended subscripts on  “l” and “d” symbols to disambiguate situations like this.  With this system, threonine is ls, but dg.  The subscripted “s” stands for serine, and it indicates that threonine is  l in the usual system of amino acid nomenclature.  This is because, as we noted above, serine is the amino acid that most naturally can be drawn to look like glyceraldehyde.  The subscripted g stands for glyceraldehyde, a sugar, and it indicates that the sugar threonine most closely corresponds to is d.

“Erythronine” never became an accepted trivial name, because it is very rare biologically, so the name is generally used instead is “allothreonine”, from Greek ἄλλος/állos, meaning “other”.  In general, the prefix “allo-” picks out whatever the other diastereomer than the standard one is.

Threonine is the first amino acid we have discussed that has C-beta branching, which is when the β-carbon is bonded to more than two atoms that aren’t hydrogen.  C-beta branching makes it harder for residues to form α-helices.

3-letter and 1-letter abbreviations: “Thr” and “T”.

Proline

Here is proline:

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It is often said that proline is not technically an amino acid, but rather an imino acid.  This is because the side chain connects to not only the α-carbon, but to the nitrogen atom as well, so that the nitrogen is bonded to two carbons.  However, according to IUPAC, the usage of “imino acid” for this situation is obsolescent (declining).  The general term “imine” (not specifically about carboxylic acids) refers to a C=N double bond, and the usage of the term for nitrogen bonded to two different carbons is obsolete (not just obsolescent).  So I guess the obsolete usage is tolerated when applied to carboxylic acids, because people like to call proline an imino acid!

The side chain of proline consists of three methylenes (methylene is -CH2-), the last of which joins to the nitrogen.  This makes a 5-membered ring: Cα, N, and the 3 methylenes.  If this ring stood alone, as (CH2)4NH, it would be called pyrrolidine.

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“Proline” is a contraction of “pyrrolidine”.  “Pyrrolidine” itself comes from “pyrrole”, which is the aromatic compound (CH)4NH.

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The name “pyrrole”  comes from the Greek πυρρός/pyrrhós, meaning “red”, because of the “red color developing when pyrrole vapor acts on a pine splinter moistened with hydrochloric acid” (Elsevier’s Dictionary of Chemoetymology), and the “idine” suffix in “pyrrolidine” denotes saturation (adding the most hydrogen atoms possible while mantaining the structure otherwise).  We previously encountered another Greek word meaning “red”: ἐρυθρός/erythrós, which “erythrose” comes from.  The difference between these two “reds” is that πυρρός/pyrrhós is a “fiery red”: it comes from πυρ/pyr, meaning “fire”, which the root of English “pyro-“.

The ring in proline severely restricts the rotation of the Cα-N bond.  This is an effect that can be considered opposite to that of glycine, whose trivial side chain allows for much more rotation.  Proline thus has a unique structural function.  It is used for making quick turns, I think.  Many statements about amino acid residues must be qualified by “except proline”.  For example, there are many proteases which cut proteins at sites where there are specific residues, except if proline is there!

Why does proline have a 5-membered ring, as opposed to some other size?  A general rule-of-thumb is that 5- and 6-membered rings are the most stable, so when given the choice, rings tend to have 5 or 6 members.  The homolog of proline with a 6-membered ring is called pipecolic acid, and it does appear in biology, but not in proteins, so it is not a proteinogenic amino acid.  One possible explanation for nature selecting proline over pipecolic acid is that proline is more rigid since its ring is smaller, so it better executes its unique structural function.

3-letter and 1-letter abbreviations: “Pro” and “P”.

How to Memorize the Amino Acids Without Mnemonics I: Introduction

[This is the first post in a multi-part series.  I’m not sure how many parts it will be yet, but it will tentatively be four parts.  This part is the introduction. The second part covers glycine, alanine, serine, cysteine, threonine, and proline, with a recurring theme of stereochemistry.  The third part will cover phenylalanine, tyrosine, valine, leucine, isoleucine, methionine, and tryptophan.  The fourth part will cover histidine, lysine, arginine, aspartate, asparigine, glutamate, and glutamine.  After I publish the later parts, I will update the contents of these brackets accordingly.]

Memorizing Without Mnemonics

Q:  How can you memorize the amino acids without mnemonics?

A:  By encountering lots of interesting information about them, in the form of a narrative.

Analogy:  When you read a novel, you memorize the characters and their characteristics.  This is because you encounter a lot of interesting information about them, in the form of a narrative.

Note:  You don’t have to memorize this interesting information to memorize the amino acids, you just have to encounter it.  In general, the information you encounter will always be more detailed than the information you remember.

In this post, I will present lots of interesting information about the amino acids, in the form of a narrative, in order to help you memorize them.  I will choose information that I think is interesting, which can be idiosyncratic at times.  (I happen to think nomenclature and etymology is very interesting, but synthesis not so interesting.  (I don’t find the kind of synthesis that chemists do very interesting.  I do, however, find biosynthesis and prebiotic synthesis very interesting.))

Why this could work:  Encountering information about the amino acids forces you to think about them.  When you think about them, you keep them in your mind, i.e. you remember them, for the duration of time that you are thinking about them.  If this information is interesting to you, then you will probably think about it again later, after you read it, and this will give your mind even more practice remembering the amino acids.  If the information here is not interesting to you, then it won’t have this benefit, but it may still help you find other information that is.  Why must the information be in the form of a narrative?  Because we think in narratives.  Narratives connect information together so that if we forget one part, we can retrieve it from the other parts.  The information about each amino acid will refer not just to that amino acid, but to others as well, in order to connect them in your mind.

Why mnemonics wouldn’t work as well:  A mnemonic is fake, made-up information about something.  Just like real information, a mnemonic about the amino acids would make you think about them, and thus it would help you remember them.  However, this information wouldn’t be as interesting as real information, and it would apply exclusively to the amino acids. The real information contained here applies far beyond the amino acids.  It will help you memorize other things in biology and chemistry that you may want to memorize in the future.  And it will connect these later things to the amino acids, so that they too will help you remember the amino acids.  In general, things get easier to memorize as you learn more in a certain field, provided you memorize them based on real information rather than mnemonics.

When learning, we try to build a large, well-connected network of ideas in our mind.  An idea that is just based on a mnemonic is a miser that doesn’t participate in this network.  Not giving any support to other ideas, the other ideas reciprocate by not supporting it.

Why school doesn’t do this:  In school, students are basically handed a list of the amino acids and their properties and asked to memorize it.  This is because the logic of school dictates that every morsel of information you encounter should be memorized well enough to answer questions on it.  To do the approach here, it is necessary that you encounter a lot of information that you don’t memorize and won’t be able to answer questions on.

Another note:  Just because you are not going to remember the information here doesn’t mean you shouldn’t think through it fully.  In fact, you won’t really “encounter” it unless you think through it.  The difference between memorizing and encountering is obscured by school, which would have us believe that we haven’t actually encountered information unless we can pass a test on it.

Why Memorize the Amino Acids?

One might claim that it’s more important to simply know stuff about the amino acids, and that it doesn’t really matter too much if you have their structures and properties thoroughly memorized.  I agree that knowing stuff about them is preferable to the sterile type of memorization done in school.  However, if you correctly memorize them, assisted by your knowledge of them, you will have the essential building blocks of life on Earth at your disposal, whenever you might want to think about them!  Furthermore, the amino acids are an essential part of the vocabulary of biology and chemistry, and you will be able to read in these fields a lot more fluently if you have them memorized.

In the Next Part of this series…

I will actually start giving interesting information about the amino acids, in the form of a narrative.  I will cover the first 6 amino acids: glycine, alanine, serine, cysteine, threonine, and proline.

Continue to part II…

 

How to Make College More Tolerable

[Scope: Although this is about college, it really applies to school in general, and I’m sure it applies to work as well.]

So you’re in college, a place where people force you to do things.  How do you do the things you care about when you are being forced to do other things?  The most obvious answer to this is to drop out.  This is also the most effective answer.  But suppose you don’t want to drop out for some reason, or you can’t.  You don’t have to suppress yourself just because you’re in college.  You can still do the things you care about, it’s just a lot harder.

The two approaches to college

In most cases, you have some control over which things you are forced to do:  you can pick your classes.  You don’t have much control, because you are not going to find a class that forces you to do exactly what you care about, and you wouldn’t even know this ahead of time anyway.  There are two approaches in this situation.

Approach 1: Pick classes that align as closely as possible with what you care about.  Figure out how they fulfill the requirements as an afterthought.  You can even ask sympathetic professors and administrators to bend the requirements for you. Then do what you care about, and try to fit it into the requirements of these classes.

Approach 2: Fulfill the requirements with classes that are as non-demanding as possible.  Then do the absolute minimum for them and spend the bulk of your time on what you care about.

These approaches differ in how they distribute the bullshit.  In approach 1, everything you do has a dilute concentration of bullshit.  In approach 2, you take the bullshit in one concentrated dose.  They also differ in how you go about your assignments.  In approach 1, you try to ignore the dilute bullshit and do your assignments thoughtfully, as if they were something you were choosing to do.  In approach 2, you try to get them out of the way as fast as possible and you don’t make any effort to learn from them.

Although they seem opposite, both of these approaches are subversive.

Approach 1 flouts what is expected of students in school.  Regardless of what particular people in school say, the system of school does not expect students to do thoughtful projects that they care about; it is perfectly happy to reward students who simply follow directions and do what they are told.  In fact, the system cares so much about students following directions that it often penalizes students following approach 1, who deviate from the directions in order to do something more important to them.

Approach 2 also flouts expectations.  Even though success in school is determined by requirements, grades, and exams, the students are still expected to uphold the pretense that school is about education.  Students following approach 2 disregard this pretense by working directly towards satisfying their requirements, rather than working towards learning the material and hoping the requirements will accurately reflect their prowess.

In short, approach 1 flouts the actual expectation and approach 2 flouts the pretended expectation.  School would have students in the No Man’s Land between the approaches, where they chase grades and requirements while convincing themselves that what they are doing is actually learning.  This is a toxic land to be in, but it is perfect training for work, where you again chase requirements and ratings while convincing yourself that you are “contributing to society” (see this video).  This is exactly where you don’t want to be if you are trying to make school tolerable, and do the things you care about in spite of being there.  Hence…

Whatever you do, DO NOT MIX UP THESE APPROACHES.

Decide once and for all whether you are using approach 1 or approach 2.  If you mix them up, you will start to become confused between what you care about and what you are being forced to do.  You will start caring about bullshit and getting things you care about done as fast as possible.  Don’t mix them up.  If you are going to do something, either do it well and don’t waste your time, or do it fast and waste as little of your time as possible.

A good way to not mix up these two approaches is to just choose one and use it for everything.  However, this may be impractical if certain classes or assignments are more suited to one approach than the other.  If you do the other extreme and choose between the approaches on an assignment-by-assignment basis, it may be difficult to keep them apart.  So you need to find a balance.  Just make sure that the approach you are using at a given time is clearly defined.  There should never be any ambiguity.

The difficulty of approach 1 is to remind yourself that you do indeed care about what you are doing, and to do it authentically, without taking shortcuts just to fulfill requirements.  In other words, the difficulty of approach 1 is to avoid falling into approach 2.

The difficulty of approach 2 is to remind yourself that you are just doing it for the grade, and to not get duped into thinking that it has any meaning.  In other words, the difficulty of approach 2 is to avoid falling into approach 1.

Note: Another commonality between these two approaches is that they both prioritize things that are actually important over grades.

Always Put Yourself First

If done correctly, both of the above approaches let you do what you really care about.  If you start prioritizing school requirements over what is really important to you, you know something is wrong.  School has gotten to you, and it has started to affect your priorities.

If there’s something important to you that you feel inspired to do, but you have schoolwork to do, do the important thing instead, or at least first!  Your schoolwork will always be there, but your inspiration won’t.  You are more important than school, and you don’t need to feel guilty for putting yourself first.

Also, if studying for a test is really stressing you out, instead of procrastinating for hours, simply choose that you’re not going to study for it until the hour before.  Odds are you wouldn’t have done any better on the test if you had been studying a lot, and you will save yourself a lot of mental anguish.

Eliminate Guilt

School-related guilt is pervasive in the minds of people in school.  The guilt that we’re not working hard enough for school, we are procrastinating too much, we are really behind in classes, we’re missing homework, we’re missing class, etc.  You can’t relax, because when you do, you feel guilty that you are not doing work.  You can’t do work either without getting repeatedly interrupted by distracting thoughts about work for other classes you’re not doing right now, and wow you’re so behind in those classes what are you doing not working on them.  You certainly can’t do things you care about, because it feels SO irresponsible to put that much energy into a non-school thing when you have so many school things that need to get done.

The way to counter school-related guilt is to decide ahead of time what work you are going to do and which classes you are going to attend, and set reasonable expectations for yourself.  On Sunday, before the week starts, write down all the homework you have to do, including late homework.  Make a plan for which homework you are going to do, and which day.  You don’t have to catch up completely in classes you are behind in.  You don’t have to do everything you are assigned.  You don’t even have to attend every lecture.  Pick just a few school-related things you are going to do each day, and give yourself room to work on things that are not related to school.

This works because you won’t feel guilty for not doing a homework that you never intended to do, or missing a class that you planned to miss.  You will probably end up doing what you were doing anyway, but with more control and less guilt.

Even at times of heaviest workload, the schoolwork allotted for a given day should be beneath your capacity.  You need some time every day to do things that are not related to school, and if you don’t schedule such time it will happen by accident in the form of procrastination, and not be as rewarding.

Take a “Sabbath” from School

School has an insidious tendency to bleed out of the time allotted for it and permeate your entire life.  Even when you are not doing schoolwork, you are probably still feeling like you should be doing it.  The way to counter this is to precisely define school’s allotment, and not permit it to go outside this allotment.  In other words, decide which times you are allowed to do homework and which times you are not.

One such policy is to take a “Sabbath” from homework.  Decide that, say, on Fridays, you are not allowed to do homework, or anything directly related to schoolwork.  You might not have done homework on a Friday anyway, but now that you are not allowed to, it’s not your fault anymore.  Even if you wanted to do homework, you couldn’t.  In other words, you get to do the same thing you might have done anyway, except without guilt.

This doesn’t have to be weekly.  You could decide that you are not allowed to do homework before midnight, or perhaps anytime during the weekend.  You could even decide that there is only one day per week that you are PERMITTED to do homework.  Just figure out a rule, and make it precise.

Why School is Still Intolerable For Me

First of all, I haven’t mastered the above techniques.  I mix up approaches 1 and 2, I miss classes by accident and feel guilty, etc.  Second of all, the above techniques can only make school more tolerable, they can’t actually make it tolerable.  School is inherently intolerable, just as is any authoritarian system.  I only have so much time in my life, and school steals it.  Not just the time that I’m in class, not just the time that I’m doing homework, but all my time.  This is one the one hand because of the guilt feelings I already mentioned, which prevent me from doing non-schoolwork when I have schoolwork to do (which is always).  On the other hand, this is because any big projects I might try to do in my free time get interrupted by school, so I refrain from even starting them.  Hopefully there’s a way to get around this latter problem, because even though I may be ending school soon, the problem will recur in other forms later in my life.  More on these points in later posts.

My Bullshit Rationalizations for Going to College and Why They’re Bullshit

So I’m in college now, and I recently realized that I don’t want to be in college.  I have suffered greatly from college over the 3 years I’ve been here, but I somehow never questioned the idea that I should be here in the first place.  Here are the rationalizations that kept me from doing so.

Rationalization 1: To be part of a mathematical community

This rationalization was the reason I told myself I was going to college, when I initially went there.  Throughout high school, I did all sorts of beautiful mathematics by myself, and I was very frustrated that I had no one to share it with.  I figured that going to college was the only way I could find such people.

Why it’s bullshit:  As I recently realized, I don’t need to be in a college to find such people, I simply need to be near a college.  I can go to the college and meet undergraduates, graduate students, and professors, and even audit classes if I want.  Then I could get all the good parts of going to college without having to deal with grades, tests, or requirements.

I just looked at a facebook conversation with Maymay from 2013, when I was still in high school, and I told them this rationalization.  They tried to dissuade me of it, using the same refutation I recently figured out!  Why did I forget this refutation so completely?  Probably because it was convenient for rationalizing college.  When I came up with it again myself recently, I thought back to the conversation with Maymay, thinking “I wish Maymay had mentioned that I could meet mathy people by just being near a college rather than in one; then I wouldn’t have gone to college!”  But looking back at the conversation, they did say that!  Lol

Rationalization 2: To get the knowledge of capitalism, and then use it against capitalism

This rationalization was created in my third year of college (this year).  It’s really dumb, so be warned.  The colonists in America used to run boarding schools for Native Americans, to indoctrinate them into taking on European ideals and Christianity (https://en.wikipedia.org/wiki/American_Indian_boarding_schools).  I read (I forgot where) that some Native Americans would actually choose to go to these schools, in order to learn about the Europeans and then later use that knowledge against them.  I started saying to my friends that I liked to think that I am doing something similar by being in school, since the science that I’m studying has generally be used for evil (e.g. military), but has the potential to be used for good.

Why it’s bullshit:  So many reasons.  Firstly, scientific knowledge is totally accessible without being in a school.  Knowledge is no longer very tied to a place now that the internet is a thing.  And, as stated above, I could always talk to professors or audit classes to get oral knowledge.  Secondly, my situation is not at all analogous to that of the Native Americans.  I didn’t come from a community that is resisting capitalism, that decided it was prudent to send someone to the schools of the capitalists to figure out various specific things that everyone knew they needed to know.  I never was part of such a community in any big way, so I went to college not knowing what was needed from the capitalist knowledge base.  Moreover, I made this rationalization retroactively, after I had already been in college for 3 years!  It’s not like I, ahead of time, decided the best thing for resisting capitalism was to go to college and learn various specific things.

So Why Did I Actually Go to College?

Enough with the bullshit rationalizations.  Here are the real reasons, I think.

  1. It was easier.  It’s always easier to do the thing that everyone else is doing, no matter how bad it is, because people already know how to do it.  There are large support networks for college students, and lots of advice for how to “be a college student”.
  2. I wanted to get away from my parents.  At that time, I still considered myself a child, and I didn’t think I was ready to find housing and deal with all that adult stuff myself.  Colleges have a very paternalistic arrangement, where they feed you, clean your dorm, and put way more restrictions on what you can do than a normal landlord would.  For example, you can’t make holes in the wall, and big furniture you buy must be approved.
  3. I had had a fantasy of how great college would be when I was a child, and I still hoped that it would be like that, in spite of all the stuff I read about how college sucked.  When I was very young, like somewhere from 5 to 10, my parents would tell me about what college was like, and it made me really really want to go there.  It sounded amazing to learn interesting things with your friends, living together in a place devoted to learning.  Even though I learned something about what college was really like, this fantasy retained.
  4. Not going to college would have been difficult to negotiate with my parents.  Or rather, whatever I might have done instead of college would have been difficult to negotiate.  College was a monolith they could agree to fund once and for all.  Whatever else I might have done would be funded on a case-by-case basis.  This was not a sufficient reason on its own, as I could have negotiated shit with them, but I’m sure that, coupled with the other reasons, it made it harder to not go to college.

However, these reasons are just conjectural.  I’m not really sure why I decided to go to college.  Or why, after every semester going terribly, I convinced myself that the next semester would be better.  At least I’ll be done soon, and then I can enter the real world.

Memorization

A motivating example

How memorable is this sequence of 31 numbers?

4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 9, 6, 10, 10, 12, 11, 11, 10, 12, 12, 11, 6.

For most people, it’s not very memorable.  Maybe they could memorize it, but it would take a while, and they wouldn’t remember it for any length of time.

You may find it more memorable if you found a pattern in it.  For example, you might notice that one part of the sequence is just another part with some number added to it.

Look at this part of the sequence, starting at the 2nd number: 3, 3, 5, 4, 4, 3, 5, 5, 4 .  If we add 7, we obtain this part, starting at the 21st number: 10, 10, 12, 11, 11, 10, 12, 12, 11.

So having this pattern, we have reduced the amount of information we have to remember.  Instead of memorizing every single number in the later part of the sequence, we just remember that it is the earlier part with 7 added to each number.  Now instead of having to remember the 9 numbers of the later part, we only need to remember the instruction “add 7 to the numbers starting at number 2”.  This instruction is considerably easier to remember than the 9 numbers of the later part, so by finding this pattern we have made the sequence easier to remember.

What if told you that this sequence is just the number of characters in each word of “zero one two three four … twenty twenty-one twenty-two … thirty”?  Now, we have reduced the amount we have to remember even further.  Instead of having to remember 31 numbers, we only need to remember 1 simple instruction, “find the lengths of the English words for the numbers from 0 to 30”, and 1 clarification, “write 21, 22, etc. as ‘twenty-one’, ‘twenty-two’, etc. and count the hyphen when computing the length”.  This is much easier to remember!

This also explains the pattern we found.  The earlier part of the sequence is the word lengths in “one two … nine”, and the later part is the word lengths in “twenty-one twenty-two … twenty-nine”.  Now it is clear why we can add 7 to each number in the earlier part to get the later part: “twenty-” is 7 characters long.

So a completely non-memorable sequence of seemingly random numbers can be made more memorable if a pattern is found in it, and it can be made completely memorable if we find a simple rule that generates the whole thing.  Furthermore, this rule will explain why that pattern we saw was observed.

Criticism of School

This example lets us make a good analogy for what is done frequently in school.  In school, we are often asked to memorize the sequence, and not told the simple rule that generates it.  This makes it take a lot longer to memorize the sequence.  It also makes memorizing the sequence a complete waste of time, since we will forget it right away after the test.

For example, in my high school I found students memorizing the following facts about lines separately:

The definition of slope: If (x_1, y_1) and (x_2, y_2) are distinct points on a line in the plane, then m = \frac{y_2-y_1}{x_2-x_1} is the slope of the line.

Slope-intercept form: If m is the slope of a line in the plane and b is the y-intercept (i.e. the line intersects the y-axis at y-value b), then y = mx +b is an equation for the line.

Point-slope form: If (x_0, y_0) is a point on a line in the plane, then y=y_0+m(x-x_0) is an equation for the line.

The students did not realize that all of these facts can all be derived from each other, so that all you really need to remember is the definition of slope and a few pointers to do the derivations.  (Maybe I’ll do these derivations in another post.  Please comment saying so if you’re interested in this.)

This kind of confusion is not just because of bad teachers not explaining how facts are related, it comes from school itself.  School does banking education, which is where students are thought of as banks into which knowledge is deposited.  Teaching means depositing certain knowledge into these human banks, not imparting understanding.  Banking education follows from the hierarchical and authoritarian nature of schools, in which the teacher has more power than the students, and talks much more than the students.  The idea of “banking education” for describing school comes from the great educator Paulo Freire.  For more information, see his book Pedagogy of the Oppressed, especially the 2nd chapter. (Here is a copy that’s on the internet atm: http://faculty.webster.edu/corbetre/philosophy/education/freire/freire-2.html)

In banking education, it’s not important that students understand the relations between facts, it’s only important that students memorize the facts.  If the teacher spends any time talking about the relations between them, they do it quickly and preface it by saying that the students don’t “need to know” it since it won’t be on the test.

How to Memorize Well

Memorization isn’t all bad.  Because of brainwashing by school, I believed that it was for a long time, but I after I memorized the amino acids recently, I realized that memorization can be amazing if it’s done the right way.  Some ideas are really beautiful or useful and it is a great feeling to have them in your mind at the ready whenever you might call upon them.

To memorize things well, we must meet three criteria:  it shouldn’t take too long to memorize them, we should be able to remember them for a long time, and we should be able to easily integrate them with other things we might memorize.  This is why it is good to find patterns or rules such as the ones found in the example above, because such patterns or rules help us meet the first two criteria.

The third criterion is best satisfied by patterns or rules that actually explain why something is the case.  For example, sometimes the notes on the lines of the treble clef are taught with the acronym “every good boy does fine”:

visualizing-intervals-and-mnemonics-21721636

This is a pattern found in the notes on the lines of the treble clef.  It turns out that, taken in ascending order, the letters of these notes are E, G, B, D, and F, the same letters as the letters at the beginning of each word of “every good boy does fine”.  Sure, I guess this is a pattern, and arguably easier than just remembering the letters E, G, B, D, and F not tied to anything.

But there is a much easier pattern that better meets the third criteria.  The treble clef is also called the “G clef”, because the line that it curls around expresses the note G.  Music notation has notes on the spaces as well as the lines (see figure below), so the lowest line is two notes below the G line.  Going down two notes, we get G F E, so the lowest line is E.  Going up, we have to remember that there are only 7 notes, so there is no H, and instead the note about G is A.  So going up from G, we have GF, in which the notes on the lines are indicated in bold.

lesson-3-notes-on-treble-staff

This pattern might take longer to explain, but it imparts more understanding, and it explains why the first pattern holds.  Also, it helps us achieve the third criterion better:  we can integrate it with other things we also want to memorize if we’re playing music, like the fact that there are only 7 notes that loop around, and the fact that the treble clef is also called the G clef.

In general, it is better to find out the actual reasons why something is the way it is, and use those reasons to memorize it, instead of using some stupid mnemonic that has nothing to do with anything.

 

An Interesting Problem

When you first saw the sequence above, you probably didn’t think that such a simple rule exists that generates it.  But it turned out that such a rule did exists, because that’s actually how I created the sequence.  What about a sequence of randomly generated numbers?  Could we find a rule that generates it and use this rule to remember it better?

13, 2, 12, 1, 6, 7, 15, 3, 9, 10, 6, 2, 2, 3, 7, 6, 15, 4, 15, 6, 3, 10, 3, 1, 7, 15, 1, 11, 7, 12, 3

I generated this sequence with https://www.random.org/integers/.  It is also 31 numbers long, and each number is between 1`and 15.

What are the odds that we could find a simple rule that, by coincidence, happens to generate a random sequence?

To figure this out, let’s look at the number of possible sequences.  Each number in the sequence has 15 possibilities, and there are 31 numbers, so the number of possible sequences is 15^31\approx 3\times 10^36 (if you don’t see how I got this, comment and I’ll explain).  So if we think of a simple rule, the odds that that rule will happen to generate this random sequence are 1 in 3\times 10^36.   This is extremely unlikely.

So we couldn’t reliably find a rule that generates a random sequence just from trying various rules.

What are the odds that a simple rule exists, even if we may not be able to find it?  Are all sequences generated by some simple rule, however hard to find?

I will talk more about this issue in later posts.