A Fraction of What Lies Ahead (ha ha i'm hilarious)

Everyone hates fractions, they are so confusing, and what even does a lowest common denominator mean? Well sit yourself down and pay attention, because I am going to explain the shit out of fractions.

The first thing you need to keep in mind for fractions is that fractions are division rewritten. is . That's important when you need to reduce fractions, especially for mixed fractions.

(Sidenote: you scrubs in lower division math got it rough - nobody in higher division even cares about mixed fractions. I'm going to take a moment to point and laugh at you.)

What is a mixed fraction? That's something like , which means three and one half. This is an "improper fraction" rewritten as a "proper fraction."

These words are in quotes because the idea is ridiculous and improper fractions are way easier to work with than proper fractions and as a math major I disagree with the classification NOBODY CARES MOVING ON

Improper fractions are fractions where the top (the numerator) is a bigger number than the bottom (the denominator). WHOA WHAT THE FUCK HOLD UP. Denominator? Numerator?

Look guys you gotta know these things, we can't do a vocab lesson every time. (Sigh, but we're going to.) The problem with these names is that they're really fucking easy to confuse with each other, so it helps to have a way to remember what the hell these words actually mean.

If the fraction bar is the surface of the ocean, the numerator is the number of dudes in a boat who are about to get dominated by the greatest great white that ever lived, the Denominator. FRACTIONS ARE JAWS.

Okay that was stupid let's move on. Improper fractions: top bigger than bottom. Proper fractions are the opposite: the denominator is bigger than the numerator. is an improper fraction and is a proper fraction. Mixed fractions are also "proper," because they are the result of rewriting an improper fraction into a proper one.

Proper no longer means anything as a word. I hate this post.

Here is a thing which is true:

Why is this true? It is true because fractions are division, and to explain it I am going to have to bring up the good old standby of long division. DO YOU KNOW HOW TO LONG DIVIDE? BECAUSE SO HELP ME IF I HAVE TO EXPLAIN THAT TOO...

Fine. I can't even find a way to include a long division symbol so I'm just going to write it as a close-parenthesis I HOPE YOU'RE HAPPY.

You're dividing by the number on bottom, so is 2 )7 (DO YOU SEE THAT, ME BEING CREATIVE RIGHT THERE?)

2 goes into 7 three whole times - that means you can subtract 2 from the original number three times before the number being subtracted from is bigger than 2.

Okay, so what the hell do we do with that 1 that's left over? That's smaller than 2, so you can't just divide into it, but remember what division really is? ~*Fractions*~! You can write the remainder as , which means you can write the whole thing as ! You have now turned an improper fraction into a mixed proper fraction! DON'T YOU FEEL SPECIAL.

The thing about mixed fractions, though, is that they are a pain in the ass to work with inside a problem. This is why people don't use them past Algebra II (you scrubs). So how do you turn a mixed fraction back into an improper fraction so you can actually work with it? You pretty just do what we did backwards. It's not that hard.

is really , but you can't add fractions unless they have the same number in the denominator. So you need to get a 2 underneath that 3 somehow.

Hey guess what's really cool about the number 1? You can multiply it in anywhere and not change a damn thing about the problem's answer! Know what's even cooler? You can write 1 as anything over itself. And remember last post where I said that you can rewrite 3 as ? That's important now!

(multiply by 1 in the form of to get the same denominator)

(now you can add the numerators because the denominators are the same)

OH MAN WASN'T THAT AWESOME? We just worked backwards to get our original improper fraction!

Yeah we did a whole bunch of work to get back to exactly where we started. That's not important. What's important is that this applies to any mixed fraction you will ever meet, and also any case where you need to add a whole number to a fraction. This is how you write whole numbers into fractions with a specific denominator.

But I'm fucking sick of fractions, so we'll talk about lowest common denominators and shit next time.

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* Again, codecogs.com is great for providing the equations EXCEPT THE LONG DIVISION ONE THAT I HAD TO MAKE MYSELF I HATE EVERYTHING

Complaining About Textbooks = Best Use Of My Time

I know I said the next post would be about fractions, but fuck fractions. Today I'm going to talk about textbooks and fractions can suck it.

You heard me, fractions!

As a math major, I am of course currently enrolled in math courses. Two of them, in fact, and for the first time since sophomore year of high school, my math classes are actually collecting homework. This is a totally foreign idea to me, you have to understand, but it has the effect of me actually doing my HW in a timely fashion. (I am the best math student.)

While doing my homework for differential equations, I came across something in the textbook that was, at first, nothing more than a minor annoyance, but the more I think about it, the more it makes me ANGRY WITH RAGE.

There's this thing that textbooks do sometimes where they will introduce a new concept during the exercises. In my calculus book, it's some minor idea that never really comes up again beyond this section of the text. It's a novelty that you're not expected to remember for future problem-solving.

In my differential equations book, it was a core method for solving differential equations. HALF THE CHAPTER was problems based on this idea, which was introduced and explained IN PROBLEM 30.

WHY COULDN'T THIS HAVE BEEN PUT IN THE CHAPTER TEXT? Why would you take a VITAL IDEA and secret it away inside the practice problems instead of introducing and explaining it in the chapter text????? This is like if my calc book had introduced the idea of epsilon-delta proofs in the middle of a problem set. GUYS THAT COMES UP AGAIN LATER WHY DID YOU DECIDE TO MAKE IT EASY TO MISS


(Yes I am using a printed copy of my textbook I AM THE BEST MATH STUDENT.)

I do not think I am correctly portraying just how angry I am about this poor organizational decision. I am so angry about this, you guys. So angry. I am so angry that I am about ready to punch a kitten.

In fact, I think I will.



DO YOU SEE THAT BOYCE AND DIPRIMA? DO YOU SEE THE CONSEQUENCES OF YOUR DECISIONS? THIS IS WHAT YOU HAVE WROUGHT.

MAYBE NEXT TIME YOU WRITE A MATH TEXTBOOK YOU WILL MAKE BETTER DECISIONS HMMM?

The Four-Operations Conspiracy, or How I Learned To Stop Worrying and Love the Commutative Property

We all know that addition is commutative--

Hold up. What's that? You don't even know what "commutative" means? I'm not surprised - I didn't even learn that this was a thing until my Calculus III professor used the word to explain that dot products are commutative while cross products are not (wait please don't go I promise those don't show up in this post, honest!)

Okay, so what the fuck is the commutative property? It's actually just a name for something you already know is true, because you've been doing it since your very first math class ever without even realizing it.

Basically, the commutative property means that 3 + 5 = 5 + 3. Addition is commutative. Multiplication is also commutative. 2 × 6 = 6 × 2

But those are examples of it. To define it, just remember that if something's commutative then that means you can switch things around as much as you want without changing the answer.

WHY IS THIS IMPORTANT?

I'm getting there, goddammit.

Now that you know addition and multiplication are commutative, you can extrapolate that subtraction and division are not, because we all know that 3 - 5 ≠ 5 - 3 and 8 ÷ 2 ≠ 2 ÷ 8. Pretty fuckin' basic, you guys.

But I'm here to tell you something that will blow your fucking mind.

This is a lie.

Subtraction and division don't even EXIST. They are not even REAL. They are nothing but illusions given to students to make things seem simpler. Subtraction and division are nothing but a vast mathematical conspiracy!

That's a pretty bold claim there, Guindo! This is a thing you are thinking to yourself, obviously, because I do not think about myself in the third person - usually.

I know I just turned your world upside down but quit freakin' out for a second and I'll explain. You see, numbers can be positive or negative (but Guindo what about imaginary numbers SHUT UP I'LL GET THERE LATER this post isn't about that). When you subtract, what you're really doing is adding a negative number.

Oh man, are you confused yet? Let me make it worse.

Check out multiplication. Multiply two positive numbers and your answer is positive. Multiply two negative numbers and your answer is positive. Multiply a negative and a positive, and your answer is negative.

In handy visual format:

+ × + = +
- × - = +
+ × - = -

Think of a minus sign as a negative one being multiplied into the number.

3 - 5 = 3 + (-1)(5)

(Also you can signify multiplication with parentheses NOW YOU KNOW)

Now it's addition! Instead of subtracting five, you're adding negative five. Mindblowing shit, I know.

Here's the cool thing: now that you're adding, you can use the commutative property to switch things around.

3 - 5 = 3 + (-1)(5) = (-1)(5) + 3 = -5 + 3

GOD DAMN! THAT'S AWESOME!

Why is that, uh, important? At all? Fuck you, that's why. It's important because it means you can rearrange subtraction and that's vital when you start working with polynomial functions (HOLY SHIT WHAT ARE THOSE look I'm just setting up a foundation you guys quit freakin' out on me).

BUT WHAT ABOUT DIVISION? YOU SAID THAT WAS A LIE TOO!

Yes, it is a lie! It is a lie in the same way that subtraction is a lie: it's multiplication written differently. But to understand this, you need to understand fractions.

OH GOD NOT FRACTIONS. I HATE FRACTIONS. Yeah, yeah, so does everyone. We'll cover that in more detail later, but right now all you need to know is this: multiplying and dividing by 1 doesn't change your number, right? So 1 × 3 = 3 and 3 ÷ 1 = 3.

Well, all division can be written as a fraction. So let's rewrite 3 ÷ 1 = 3 as That also means that any time you see a 3, you can write it as (because they are equal and you can do this with things that are equal to each other).

When you divide, what you are really doing is multiplying by the inverse. Inverses are a complex thing that I don't feel like talking about now because this post is long enough as it is, but for now just keep in mind that an inverse of a number is that number as a fraction, flipped upside-down. For example, the inverse of is

Let me show you what I mean.

(fractions are multiplied straight across, top times top and bottom times bottom)

So there you have it, division is multiplication by the inverse. That means that you can rewrite division as multiplication by 1 over whatever, and just like with subtraction, you can re-organize shit now and move that little fraction wherever the fuck you want.

Now that you've had the illusions of subtraction and division shattered for you and your entire world has been torn asunder by the revelation, NEXT TIME WE'LL TALK ABOUT ~*FRACTIONS*~!!

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* Equations provided by codecogs.com, sorry I took the links out of the nice html you provided!

A Math Blog?!

Ugh, Guindo, you say - or maybe you don't say, but pretend somebody did - why would you even want to write about math! It is so tedious and boring and you never use that stuff in real life, really. Why should I care if you are writing a blog about math?

Well, imaginary naysayer, maybe if you would kindly STUFF IT for a minute, I will tell you why you should care.

Here is a list of things you can explain or represent with math:

• FUCKING EVERYTHING

Here is a list of things that are possible because mathematicians sat down and figured some shit out:

• FUCKING EVERYTHING

Without math you couldn't even be sitting there reading this post about it. Guess what computers and the internet are both made of? MATH, ASSHOLE. Yeah that's right. Forget the wires and metal and plastic and shit, IT'S PURE CONCENTRATED MATH.

"But Guindo I hate math :(!" WELL TOO FUCKING BAD BECAUSE EVERYTHING AROUND YOU IS IMPOSSIBLE WITHOUT IT. Now go re-evaluate your life or something while I explain why people hate math.

The only subject with a worse rap than math is probably history. Or maybe chemistry. Fuck history, even if everything around you is made of that, too. And also fuck chemistry even if it is math because seriously fuck chemistry.

Anyway. Math is a subject that you cannot understand unless you understand all of the stuff that came before. Don't know how to divide? Good luck struggling through the rest of your arithmetic lessons without that. Didn't quite grasp the definition of a derivative in Calc I? Whoops you go back to that in Calc III for stuff you can't do with derivation shortcuts, GUESS YOU'RE FUCKED.

It doesn't help that most math teachers are really terrible at explaining math to people who don't already know what they're talking about. It really doesn't help that a lot of elementary school teachers - you know, the ones teaching kids the fundamentals that are vital for everyday life? - also grew up hating math, so they did the bare minimum in college and then have to turn around and TEACH IT.

So the cycle looks something like this:

• Bad math teachers make kids hate math
• Kids grow up hating math
• Young adults go to college and become teachers who hate math
• Teachers who hate math are bad math teachers
• Bad math teachers make kids hate math
• ad nauseum

Well, math is fucking awesome, and it doesn't deserve that treatment. It deserves a fair chance, and I am here to pick up the slack for your inadequate math teachers. BECAUSE I AM SO GENEROUS.

WHAT YOU CAN EXPECT FROM THIS BLOG:

• Math.
• NOT sounding like a fucking textbook - seriously nobody actually does math like that; textbook writers have sold their souls to demons for the complete inability to do anything like a normal person
• Explanations of why, instead of expecting rote memorization of technique
• Low level math stuff like arithmetic as well as higher level stuff - don't worry, I'll mark it, so you won't be sitting there flinching at every use of an integral sign and wondering why nothing about your life makes sense anymore
• Belligerence and probably a lot of swearing, uhhh, if you didn't notice already

Some ground rules:

• Don't be a dick
• I get to be a dick, though. DEAL WITH IT.
• If you're not clear on an explanation, ask. I'm belligerant but informative, and I'll edit posts for clarity and to answer questions.
• Feel free to suggest topics for future posts in the comments. No guarantee I'll write them but ideas are welcome.
• I probably won't respond to comments except to answer math questions so don't expect it.
• This will be on the test.

P.S. if you're British you can just pretend I said "maths" all over this post. It's okay, I'm sure you're imaginative enough to handle that.