#math proof

[xyymath]

Things Math Professors Say

"The proof is trivial." (Oh, cool. Guess I’m just an idiot then.)

"Left as an exercise." (Translation: You’ll never solve this in a million years.)

"It’s obvious, really." (Sure, if you’re a demigod.)

"By inspection." (Stares harder at problem… still nothing.)

"For small values of epsilon." (How small? Subatomic? Microscopic? Vibes?)

"WLOG (Without Loss of Generality)." (Oh, we’re just assuming it doesn’t matter now? Alright.)

"Details omitted." (Because apparently, you don’t need to understand it.)

"By the usual argument." (Which you somehow don’t know because you weren’t born in 1702.)

"Assume the rest holds." (That’s some impressive optimism right there.)

"The usual abuse of notation." (Why does this feel like an emotional wound?)

"Almost surely correct." (But also possibly wrong? Cool, thanks for the clarity.)

"A non-rigorous approach." (I thought math was supposed to be precise?!)

"Assume it’s obvious." (Buddy, NOTHING about this is obvious.)

"The reader may verify." (No, the reader may CRY.)

"To the interested reader." (Guess I’m not interested enough, huh?)

"Well-behaved functions only." (We’re function-shaming now?)

"Obvious to the trained eye." (Guess I’ll never make it out of amateur league.)

"A trivial case analysis." (Trivial to WHO??)

"Integrate by parts, twice." (Bold of you to assume I got it the first time.)

"As you can clearly see." (Oh, I clearly see my FAILURE, alright.)

"It works in practice too." (Unlike me, who barely works at all.)

"Assume a spherical cow." (Are we doing math or abstract sculpture?)

"A standard result." (Not in my standards, pal.)

"We skip the tedious algebra." (No, no, please—I wanted to suffer MORE.)

"Assume non-zero solutions exist." (Okay, now we’re just assuming life works out.)

"The usual topology." (Bro, I don’t even know the unusual topology.)

"Finitely many cases left." (Just kidding, there’s 72.)

"By virtue of symmetry." (Virtue? I have none left.)

"Don’t worry about the constant." (The constant is probably my GPA dropping.)

"Assume continuity." (I’m assuming my brain is breaking.)

"Smooth functions only." (Guess I’ll leave, I’m clearly not smooth enough.)

"The simplest non-trivial case." (Simplest? NON-TRIVIAL? Pick a side!)

"Epsilon goes to zero." (Epsilon isn’t the only one losing it.)

"And the rest follows." (Where? Straight to my breakdown?)

[simplydnp]

who's hand is in this picture?

A Mathematically Rigorous Proof That I Spent Too Long Writing

welcome to university math: dnp hand edition

(no, don't leave, you'll be fine i promise)

to begin, we need a statement to prove. we have two options:

- the hand is dan's hand

- the hand is phil's hand

now, for most proofs in university math, you are told a true statement, and you must show why it is true using logic rules, definitions, and theorems. but, we do not know which of these statements are true, so we have to find out.

to prove that a statement is true, we must show that it is always true for the situation presented. to show a statement is false, we must present a single instance where the statement is false (also known as a counter example).

a quick not scary math example:

definition: a prime number is only divisible by 1 and itself.

statement: all prime numbers are odd

(this is false, because 2 is a prime number and it is even. you don't even need to check if there's any others, all you need is one single case where it isn't true to disprove it)

so now that we have a little background on proofs and how to prove and disprove them, we go back to our two statements.

the thing with this situation is, one of them must be true (unless you're gung-ho on someone else holding dan's face while phil takes a picture on his phone of dan in his glasses, in which case, i applaud your commitment, but in actuality this proof will cover that option too)

the full statement we have is: dan is touching his face or phil is touching dan's face

now, because this is Real Life and we have a picture where a hand is touching dan's face, we know already that one of these options is true (as mentioned above) but! using symbolic logic you could also come to this conclusion.

this type of statement is an 'or' statement, and if you're curious, you can look into 'truth-tables' and see why, but at least one of the options must be true.

back to the proof at hand (bah-duhm-tss)

okay. now, proofs also must be 'general' in order to mean anything, really. these are statements of truth of the universe, not just for individuals. so, we will prove this generally.

we have 2 people involved, so individual 1 (dan, the owner of the face and potential face toucher) will be labelled as 'D' , and individual 2 (phil, the possible face toucher who does not own the face) will be labelled as 'P'. thus, this can be true for any such D and any such P.

so with our 'or' statement, in order to prove it, we pick one of the options and say that it is not true, and we have to show then that the other is true.

step 1: let's assume this is not P's hand. (assumption)

step 2: thus, it must be D's hand. (what we take from our assumption)

step 3: now, if it is D's hand, we look at what a hand on one's own face is capable of appearing like. (a definition or true fact about step 2)

the position in the given photo shows the hand with a thumb on the cheek, and a finger on the forehead. so, we find an example of a person with their fingers in the same position (or close to) and see if this supports our claim.

consider:

now, with this image, you can clearly see how the subject's right hand has the thumb on the temple and index finger on the top of their head, however, it is a close enough position for our case.

from the view of the camera, the closest finger to the camera is the edge of the pinkie. in fact, it will always be the closest finger to the camera in this position, assuming the subject has all fingers and no additional appendages.

step 4: we now compare this to our photo (we verify if this holds to our claim or contradicts it)

in our photo, the closest appendage to the camera is the edge of the thumb.

step 5: thus, it cannot be the case that D is touching their own face. (what the evidence says)

step 6: as we assumed it was not P's hand and have shown it cannot be D's hand, and as this is an 'or' statement both of these claims cannot be false, we can therefore conclude it must be P's hand. (our conclusion: re-stating the statement and assumptions and conclusion)

step 7: we verify that P is true (optional step but in beginner proofs you generally show why your case works)

to do this, i will show a picture of a person touching another's face, and compare it to our image.

consider:

now, this image is not exactly the same, similar to above. however, P's left thumb is on the cheek, with their index on D's temple. the closest appendage to the camera (if it were in a similar perspective as our original) would be the edge of the thumb.

comparing it to our original:

our comparison holds.

thus, we can conclude that the true claim in this statement is that P must be touching D's face, which, in particular means that:

phil is touching dan's face in the image

thank you for partaking in phannie mathematics. we now know. i am not sorry.

bonus:

phil has a hitchhikers thumb and dan doesn't so why was this necessary at all 🤡

[felixcloud6288]

Did you know if a number is divisible by 3, you can add all the digits of the number and the sum is divisible by 3? For example, 327 is divisible by 3 and 3+2+7 = 12, which is divisible by 3.

For the Proof on why this is, click Read More

Prerequisite Knowledge

Numbers

Yes, I need to explain how numbers work for this proof.

Let's go back to elementary school. You're being taught basic addition of multi-digit numbers. You're given the number 54,321. 54,321 = 50,000 + 4,000 + 300 + 20 + 1

Let's expand that a little.

54,321 = 5*10,000 + 4*1,000 + 3*100 + 2*10 + 1

Now I''m going to take an aside and mention exponents, just in case.

Normal notation for exponents are a base number with a smaller number floating on the top right corner of your base value. Since Tumblr can't support that notation, I'll use a^b as the notation.

When you have an exponent, you have to multiply the base number by itself however many times the exponent number is. For example, 3^4 = 3*3*3*3 = 81.

Also, if the exponent value is 0 or smaller, you start dividing by the base number instead. So 3^0 = 3/3 = 1.

Now that I've explained that, let's go back to earlier.

54,321 = 5*10,000 + 4*1,000 + 3*100 + 2*10 + 1

Each digit in 54,321 is beling multiplied by an exponential power of 10.

54,321 = 5*10^4 + 4*10^3 + 3*10^2 + 2*10^1 + 1*10^0

This expanded form of an arbitrary number is going to be necessary for the proof.

Modulo and Equivalence

Let's go back to elementary school again. Remember how in division, you'd give a remainder as part of your answer? 12 / 5 = 2 remainder 2.

The modulo operator returns the remainder when you divide two numbers. In programming, % is used for modulo so I'll use that for notation. So 12 % 5 = 2.

An expansion of modulo is equivalence. Two numbers are equivalent under a modulo value when both numbers have the same remainder when divided by the modulo value. Under modulo value 5, 7 and 12 are equivalent because 7 % 5 = 2 and 12 % 5 = 2.

The normal notation for equivalence is the equal sign with 3 lines instead of 2. I'll use == for the notation here.

Since 7 and 12 are equivalent under mod 5, the notation is 7 == 12 (mod 5)

If a number is evenly divisible by a modulo power, it's equivalent to 0. 5 / 5 = 1, therefore 5 % 5 = 0, therefore 5 == 0 (mod 5)

There are some special rules about numbers that are equivalent. We'll let a, b, c, and m be arbitrary numbers.

If a == b (mod m), then a+c == b+c (mod m) If a == b (mod m), then a*c == b*c (mod m) If a == b (mod m), then a^c == b^c (mod m)

The Proof

Now let's move onto the proof.

Let's say we have some arbitrary whole number X which is divisible by 3. That means there is some whole number c such that X = 3*c. So if X is 327, c would be 109 because 109*3 = 327.

Now let's rewrite X into the expanded form from the Numbers section. We'll let d(0) refer to the least significant digit (The one at the far right). d(1) will be the next least significant digit and so on until we reach d(n), the most significant digit.

X = d(n)*10^n + d(n-1)*10^(n-1) + ... + d(1)*10^1 + d(0)*10^0

Since X is divisible by 3, that means the long form of X is equal to 3c.

d(n)*10^n + d(n-1)*10^(n-1) + … + d(1)*10^1 + d(0)*10^0 = 3*c

Now let's start doing some modulo.

10 == 1 (mod 3)

By the earlier rules mentioned in the Modulo section, we can multiply and exponentiate each term (The parts between the +) and they're still equivalent. So now we'll replace each term with an equivalent term under mod 3.

d(n)*10^n + d(n-1)*10^(n-1) + ... + d(1)*10^1 + d(0)*10^0 == d(n)*1^n + d(n-1)*1^(n-1) + … + d(1)*1^1 + d(0)*1^0 (mod 3)

1 exponentiated to any power is 1 so

d(n)*10^n + d(n-1)*10^(n-1) + ... + d(1)*10^1 + d(0)*10^0 == d(n) + d(n-1) + … + d(1) + d(0) (mod 3)

Meanwhile 3*c == 0 (mod 3). Since 3*c = X, then anything equivalent to X is equivalent to 0.

d(n) + d(n-1) + … + d(1) + d(0) == 0 (mod 3)

At this point we can say the sum of all the digits of X can be divided by 3 because the sum would have no remainder.

So in conclusion, if the sum of all the digits of a number can be divided by 3, then the number can be divided by 3.

[catscomputerscreativewriting]

It's so funny to me that people think of Math/Mathematicians as being hyper-logical and rational. Like, have you seen some of the wild things hiding in the Math?

Did you know there are non-computable numbers?? (https://en.wikipedia.org/wiki/Chaitin%27s_constant)

Did you know that there are things that are true, but we can't prove them??? (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)

Did you know that we can prove that something exists, and yet never actually figure out what that thing is?? (https://mathworld.wolfram.com/NonconstructiveProof.html)

Math is crazy. Math is wild. Math hardly makes sense, and when you think you understand the weirdest parts of it, everyone who hears you explain it to thinks you're a gibbering lunatic.

"In mathematics you don’t understand things. You just get used to them." - von Neumann

(please share more unhinged math with me, i want to see more scary math)

[tea-for-patchouli]

For your enjoyment, a lovely mathematical paper which can fit in a single image.

[mentalknot]

Kids these days are quite spoiled… they get to add, subtract, multiply, and divide any numbers they want without specifying their field or proving that the operations they’re using are well defined… do they just expect that the teacher will do all the work ahead of time??

Utterly preposterous…

[the-way-astray]

yeah so fitz is actually still sixteen in the latest book

[anghraine]

Nothing like attempting basic division to sharpen my hatred of "everyone is mentally disabled in their own way :) don't let labels define you :) :)" anti psych posts

[pv1isalsoimportant]

(Semi-regularly updated) list of resources for (not only) young mathematicians interested in logic and all things related:

Igor Oliveira's survey article on the main results from complexity theory and bounded arithmetic is a good starting point if you're interested in these topics.

Eitetsu Ken's list for resources on proof complexity, computational complexity, logic, graph theory, finite model theory, combinatorial game theory and type theory.

The Complexity Zoo for information on complexity classes.

The Proof Complexity Zoo for information on proof systems and relationships between them.

Computational Complexity blog for opinions and interesting blog posts about computational complexity and bunch of other stuff.

Student logic seminar's home page for worksheets on proof complexity, bounded arithmetic and forcing with random variables (great introduction for beginners).

Jan Krajíček's page is full of old teaching materials and resources for students (click past teaching) concernig logic, model theory and bounded arithmetic. I also recommend checking out his books. They are basically the equivalent of a bible for this stuff, although they are a bit difficult to read.

I also recommend the page of Sam Buss, there are downloadable versions of most of his articles and books and archive of old courses including resources on logic, set theory and some misc computer science. I especially recommend his chapters in Hnadbook of Proof Theory.

Amir Akbar Tabatabai's page for materials on topos theory and categories including lecture notes and recordings of lectures.

Andrej Bauer's article "Five stages of accepting constructive mathematics" for a funny and well-written introduction into constructive mathematics.

Lean Game Server for learning the proof assistant Lean by playing fun games.

[xypheris]

Exam question: Prove the following.

Me, ten minutes into writing the proof:

Well, I know this isn’t going anywhere, but I’m gonna make it look like I know what I’m doing.

[max1461]

"Visual proofs", more commonly known as "examples",

[xyymath]

Reduced Row Echelon: We Need to Talk

I don’t know who needs to hear this, but there is a HUGE difference between "ROW ECHELON FORM" and "REDUCED ROW ECHELON FORM."

Row Echelon Form? That’s when all the non-zero rows are stacked up top like overachievers, zeros chilling at the bottom, and every leading entry is to the right of the one above it. Simple enough.

Reduced Row Echelon Form? Oh, buddy. It’s Row Echelon Form’s glow-up. Every leading entry is now a 1 (living their best life as pivots), and everything above and below them is a 0. It’s like Row Echelon Form decided to tidy up its room and go full-on minimalist.

Why the confusion, you ask? BECAUSE MATH PROFESSORS LOVE TO SAY, “Now we’ll reduce the matrix to Row Echelon Form,” so naturally, I thought "Reduced Row Echelon Form" was just the bougie version of the same thing. Like, come on, you’re throwing reduction in the phrasing—it’s confusing!

So here I am, thinking I’m killing it in class, and suddenly I find out there’s another level of reduction? Like, hello, I’m not a washing machine—I didn’t sign up for the extra rinse cycle.

UGGGHHH.

[felixcloud6288]

I have been itching to teach someone some math for a while and I desperately need to scratch it.

To start, look at this totally amazing gif I once made for a discrete math class.

This is the proof that the square root of 2 is irrational (infinitely long, non repeating decimal value which cannot be represented as a fraction of two numbers). This proof is actually one of the most important proofs out there because it opened the door to new fields in continuous mathematics.

People also died trying to prove this because there was a mathematic religious cult (the Pythagoreans) that believed irrational numbers do not exist.

Read more for discussion and proof explanation.

Welcome aboard. I'll try to make this as understandable as possible. First we'll do some onboarding so you know how the proof methodology works, then we'll do the actual proof.

Prerequisite Knowledge

Implication

Here's a basic implication statement in mathematic notation:

p->q

In human words that means "If statement p is true, statement q will also be true."

Here's a quick exercise. Think of a standard-issue cat. The cat has no unusual features about it, it is as normal and average as can be.

Does your cat have four legs? How about whiskers? A mouth?

I bet I'm right and that's because those are things that are naturally true if you are thinking about a cat.

So back to the implication definition. If p = "I am thinking of a standard-issue cat" and q = "I am thinking of something with four legs, whiskers, and a mouth." then

"I am thinking of a standard-issue cat" -> "I am thinking of something with four legs, whiskers, and a mouth."

Or in human,

"If I am thinking of a standard-issue cat, then I am thinking of something with four legs, whiskers, and a mouth."

While not necessary for the proof, I want to make something clear, p->q is NOT the same as q->p. Just because standard-issue cats have four legs, whiskers, and a mouth doesn't mean everything with those characteristics is a cat.

Proof by Contradiction

So now that you know what implication is about, here's a question:

What happens when p being true leads to a statement that can only be false?

The answer: You have a contradictory statement and p cannot be true.

Now recall that a standard-issue cat has four legs. What if I told you I am thinking of something with 57 legs. Well a cat can't simultaneously have only 4 legs and 57 legs. Therefore, I can't be thinking of a standard-issue cat because if I was, then two contradictory statements would be true.

So proof by contradiction works as follows:

Assume the opposite of what you want to prove

Show what would be true if that assumption were true

Repeat until you have two contradicting statements

Close the proof by saying the assumed statement is false

Definition of even numbers

Okay. I'll try to explain this slowly, since Tumblr doesn't understand what odd and even numbers are.

Let's say X is some arbitrary even number. That means numbers like 0, 2, 4, 6, 8, 10, etc, and -2, -4, -6, -8, -10, etc. X can be any even number. Just fill in the blank.

If X is even, there's some other whole number (whole number means you don't need a decimal. 1.5 or any other number that needs a . in it is NOT a whole number.) that when multiplied by 2 equals X.

0 is even because 2 * 0 = 0. 2 is even because 2 * 1 = 2. 17 is NOT even because 2 * 8.5 = 17 and 8.5 is NOT a whole number. Negative numbers can also be even. -10 is even because 2 * -5 = -10.

Similarly, if you take any whole number and multiply it by 2, the result will be even.

Greatest common divisor

The greatest common divisor (gcd for short) is the largest number that can evenly divide two numbers. The gcd of 9 and 21 is 3 for example. If two numbers have nothing in common, their gcd is 1.

I'll use gcd(x,y) as shorthand for the gcd of two numbers.

A fraction is in lowest terms when the gcd of the numerator(top value) and denominator (bottom value) is 1.

2/3 is in lowest terms because gcd(2,3) = 1. 4/6 is not in lowest terms because gcd(4,6) = 2

Rational vs Irrational numbers

A rational number can be represented as a fraction of two whole numbers in lowest terms. .25 is rational because it is equal to 1/4 1.3333333..... is rational because it is equal to 4/3 10 is rational because it is equal to 10/1

Irrational numbers cannot be represented as a fraction of two whole numbers. Pi is a classic example. Irrational numbers are non-repeating, meaning there's never a point where the value is repeating the same sequence forever. Don't worry about wrapping your head around this. Human brains aren't made to comprehend infinite. Just understand that you can't write it as a fraction of two whole numbers.

Misc

I'm just going to fill in some additional things here in case you're rusty on high school math.

The square root of a number is whatever value times itself equals the original number. The square root of 16 is 4 because 4 * 4 = 16. Since Tumblr doesn't support math notation (as far as I can tell), I'll use sqrt(x) to mean the square root of a value.

The opposite of the square root is the exponential power of 2. Normally, the exponent is represented as a smaller-size number floating to the right of the number (called a superscript) but since Tumblr doesn't support math notation, I'll use x^2 to mean a value to the power of 2.

Exponents mean you are multiplying a number by itself as many times as the power. So 4^3 = 4 * 4 * 4 = 64.

If you square a square root, they cancel each other out. (sqrt(25))^2 = 25 because sqrt(25) = 5, and then 5 * 5 = 25.

If you multiply and divide something by the same value, they cancel each other out. 3 * 2 / 2 = 3 because 3 * 2 = 6 and then 6 / 2 = 3.

When you have an equation, you can do the same thing to both sides and the equation is still equal. If a/2 = b, then I can multiply both sides by 2 and the equation is now a = 2*b.

The actual proof

So we want to prove sqrt(2) is irrational.

To start, we'll use Proof by Contradiction and say sqrt(2) is rational. This IMPLIES sqrt(2) can be represented as a fraction of two whole numbers. We don't care what those numbers are so we'll just use a and b as placeholders.

sqrt(2) = a/b

And since a/b is the fraction representation of sqrt(2) this implies

gcd(a,b) = 1

Now we'll move onto some equation finagling. square root and square power are opposites so if we square both sides, we can remove the square root part.

sqrt(2) = a/b (sqrt(2))^2 = (a/b)^2 2 = (a/b)^2

Exponential power means we're multiplying something by itself however many times the power is so (a/b)^2 = a / b * a / b. We can rearrange these because they're in the same priority on PEMDAS. So a/b*a/b = a*a/b/b = a^2 / b^2

2 = (a/b)^2 2 = a^2 / b^2

Now we can multiply both sides by b^2 to get rid of the division

2 = a^2 / b^2 2b^2 = a^2

a^2 and b^2 are whole numbers. a^2 is equal to 2 times a whole number. This means a^2 is even. a^2 = a * a and a is either even or odd. When you multiply two odd numbers, you get another odd number. When you multiply two even numbers, you get an even number. So a must be even.

If a is even, then there is some other whole number (We don't care what it is so we'll use c as a placeholder) such that a = 2 * c.

And if a = 2*c, then a^2 = (2*c)^2 = 2*c*2*c = 4c^2

2b^2 = a^2 = 4c^2 2b^2 = 4c^2

We can divide both sides by 2.

2b^2 = 4c^2 b^2 = 2c^2

And now we can do the same song and dance for b^2 that we did with a^2 and conclude b must be even.

And now we get to the contradiction part of Proof by contradiction. We earlier said gcd(a,b) = 1 because a/b was a fraction in lowest terms. But we also showed that a and b are even which means gcd(a,b) is at least 2. So we have two contradictory statements gcd(a,b)=1 and gcd(a,b) > 1.

So if sqrt(2) was rational, it would lead to a scenario where a fraction both is and is not in lowest terms. Therefore, we conclude sqrt(2) is irrational.

Now that you have this forbidden knowledge beware the Pythagoreans, lest you too meet your end.

Note: Any genuine questions asking for clarification will be answered in the comments. Any "Um, ackually" comments will be ignored unless you actually are pointing out a genuine mistake I made, which I will correct and acknowledge.

[necromimetics]

I’m an enjoyer of applied math. Hypothetically. The problem is half the time I do applied math they’re like “so first we’re going to make 25 assumptions” and my pure math brain immediately violently recoils

[kcl1627]

Okay one thing that we as math people should start doing, is to start using symbols from languages which are noy english or greek, into our proofs, because why can't i let ख = <something>, why does it always have to be x. Nope we shall start using more symbols,or rather any symbol that we want to, that already doesn't have a predefined meaning, for example ♤ or ☆.

[mentalknot]

Sharing of Proof Between Friends

After spending over nearly eight hours of each day in a mathematics department for two years straight, I’m shocked that many high schoolers believe math is a solitary pursuit.

In reality, this community seems to be one of most welcoming and collaborative academic communities I’ve found. Let me share some moments:

“Hey! How’s it going?”

“Alright, been stuck on this interesting question my friend emailed me the other day… he said the first part of the proof is pretty easy, but I’ve been at it for 10 hours…”

“Never trust a mathematician who throws around the word ‘easy,’ c’mon let’s try it together at the board.”

Thirty minutes later, they had completed the proof, and sat back with wide smiles to admire their work. In truth, there was rarely a conversation that didn’t eventually turn to math in that department…

“Yeah, and I heard that the guy cheated with his best friend’s sister… wild right?”

“Yeah… not to interject, but I have this representation theory question… would you all be willing to take a look?”

The conversation took an immediate turn with collective enthusiasm. I have been lucky to have my own “collaborative math” moments since returning to my undergraduate studies, and do my best to share this part of “math culture” with younger students curious about the major.

“So that’s the proof that motivates our paper! It’s quite short, but there’s something about it I love.”

“Wait… but you’ve only done half, and this is a biconditional statement, let’s try it together.”

There was a reason my mentor never encouraged me to look at the other side of the proof… it was far more “ugly,” but tons of fun to piece together with a fiend. We looked back at our work after forty minutes with satisfaction before returning to our neglected problem sets…

And finally, I tried to assist a student with a calculus question using the “process of questioning” the research world had taught me:

“I need to find a closed-form equation for this geometric series… but I can’t seem to get the alteration sign?”

“Try writing out the first six terms, do you see anything that you could simplify? Look at the denominator specifically…”

“Well, they’re all multiples of three…”

“Try pulling that three out, any more similarities?”

“The numbers multiplying the threes are powers of two! But I still need that alternating sign…”

“Remind me, what happens when you raise a negative number to an odd/even power? Try it with (-1)^n”

“If it’s odd, the number stays negative, and positive if even… so if I add this to the denominator, the sign is alternating depending on the index n!”

“YES! This little (-1)^n trick comes up everywhere, it’s a nice ‘tool’ to hold on to if you decide to take more math.”

The exchange was wonderful… and motivated me to review the calculus I’d excitedly ran past when I was younger. I wish this type of discourse was taught more expansively.